Paper Title: Line tension from dual-geometry sessile droplet measurements: Combining contact angle size dependence data for axisymmetric and cylindrical droplets to determine the line tension.




Line tension from dual-geometry sessile droplet measurements]Line tension from dual-geometry sessile droplet measurements: Combining contact angle size dependence data for axisymmetric and cylindrical droplets to determine the line tension


d.tatyanenko@spbu.ru



Department of Statistical Physics, Faculty of Physics, Saint Petersburg State University
  7–9 Universitetskaya nab., St. Petersburg, 199034, Russia



§ ABSTRACT 
We perform a thermodynamic analysis of various contributions to the size dependence of the contact angle cosine for axisymmetric and cylindrical sessile droplets. This shows that a widely used method to determine the line tension from the slope of the contact angle cosine dependence on the three-phase contact line curvature (for axisymmetric droplets) provides a certain combination of the line tension, adsorptions at the three interfaces, and the macroscopic contact angle. To extract the leading-order contact-line-related contribution and determine the line tension, we propose a simple technique using contact angle size dependences for axisymmetric and cylindrical droplets at the same conditions.




[
    Konstantin D. Apitsin
    August 5, 2024
=========================






§ INTRODUCTION


The contact angle of a large sessile droplet can be found from the well-known classical Young equation <cit.>

    cos =  -

with σ the surface tension of a corresponding interface,  the macroscopic contact angle; α, β, and γ mark the liquid, gaseous and solid phases, respectively, and double Greek superscripts mark the corresponding interfaces. The surface tensions here are thermodynamic (defined as the surface excesses of the grand potential per unit area of the interface). For small droplets, the thermodynamic line tension κ (defined as the line excess of the grand potential per unit length of the three-phase contact line <cit.>) affects the contact angle <cit.>:

    cos =  -  - κ/.

Here θ is the contact angle, r is the radius of the three-phase contact line, subscript “a” refers to axisymmetric droplet (see Fig. <ref>(a)); the line tension κ is implied to be constant. Eq. (<ref>) is usually called the modified Young equation.

Subtracting the modified Young equation (<ref>) from the classical  one (<ref>) and dividing the both parts of the resulting equation by , one usually arrives at <cit.>

    cos - cos = κ/.

This equation is widely used for contact-angle-based measurements of the line tension in laboratory experiments (see, e.g., <cit.>) and computer modeling (e.g., molecular dynamics simulations <cit.>). According to Eq. (<ref>), the slope of the graph cos vs 1/ must be equal to - κ/. The surface tension  is typically measured or calculated separately.

However, some authors have noted that Eq. (<ref>) is generally not correct <cit.>, since it assumes the corresponding surface tensions in Eqs. (<ref>) and  (<ref>) to be equal, whereas they correspond to different equilibrium sessile droplet sizes, which implies different values of the chemical potential(s). There are arguments that adsorption at the solid–liquid interface and its effect on the surface tension  affect cos in the same manner and with a similar magnitude as it is measured in experiments <cit.>, so the line tension itself is not necessary to explain the dependence of cos on 1/. Consideration with use of an interface displacement  model shows <cit.> that the contributions of the line tension and the adsorption at the solid–gas interface (and its effect on the surface tension ) to the  dependence of cos on 1/ can be comparable in the presence of a precursor film. Some authors <cit.> suppose only  to depend on the droplet size and Eq. (<ref>) to be correct, but the slope of the graph cos vs 1/ to give a quantity not equal to - κ/.  Sometimes the resulting “line tension” is called the apparent line tension <cit.>. General thermodynamic analysis shows <cit.> that the value of this slope has contributions both from the line tension and the adsorptions at all three interfaces (see also <cit.> for similar arguments).
             

How to “extract” the line-tension contribution and determine the value of the line tension from the measurements or calculations of cosθ vs 1/r? Thermodynamic analysis shows <cit.> that, for cylindrical droplets [long liquid “channels”, or “filaments”, or “ridges” attached to the substrate surface, see Fig. <ref>(b)], there is no κ/r term in the equation corresponding to (<ref>). It means that dependence of the contact angle  on the size of a cylindrical droplet (, its half-width  on the substrate) is governed by dependences of the surface tensions and the droplet size on the chemical potential(s). This suggests the idea to compare or combine data of size dependence of cosθ for axisymmetric and cylindrical droplets at the same conditions to exclude the surface-tension-related contributions and determine the line tension.

In section <ref>, we present a thermodynamic analysis of various contributions to droplet size dependence of the contact angle for both axisymmetric and cylindrical droplets in a system with a single-component fluid and an ideal smooth homogeneous solid substrate in absence of external fields. Based on it, in section <ref> we propose a simple technique that allows to determine the line tension from a combination of contact angle size dependences for axisymmetric and cylindrical droplets in a manner similar to use of Eq. (<ref>).

Cylindrical droplets are often used in modeling, e.g., in molecular dynamics (MD) simulations <cit.> due to convenient use of periodic boundary conditions along their longitudinal axes and due to expected independence of the contact angle of (or a weak dependence on) the droplet width. However, they are hardly available for experimental measurements due to Plateau–Rayleigh-type instability <cit.>. Thus, the technique we propose is mainly oriented to modeling, especially with methods not allowing for direct calculation of the line excess of the grand potential (e.g., MD simulations). The dimensions of such simulated axisymmetric and cylindrical droplets, i.e., R, r, and θ, can be “measured”, and these data can be used to determine the line tension.





§ THERMODYNAMICS AND CONTACT ANGLES





 §.§ Equilibrium conditions for sessile droplets

For thermodynamic description of a small sessile droplet on a partially wettable substrate, we consider a materially open system of a fixed volume including the droplet itself, the substrate, and the surrounding gas (vapor). The liquid and gaseous phases are supposed to be a single-component fluid at a given values of chemical potential μ and temperature T (that supposed to be constant).  The grand thermodynamic potential of the system decomposes into bulk, surface, and line parts:

    Ω =    - p^α - p^β + ω^γ V^γ
        +   +  +  + κ L

with V the volume, A the area of the interface, L the length of the three-phase contact line, p the pressure in the bulk phase, ω the bulk density of the grand thermodynamic potential (in fluid phases equals -p). Other quantities as well as notations of phases and interfaces has been explained above.

Let us consider two types of droplet geometry: (1) axisymmetric droplet [shaped as a spherical segment in our “macroscopic” thermodynamic description, see Fig. <ref>(a)] and (2) a fixed-length piece of a cylindrical droplet [shaped as a segment of a circular cylinder in our “macroscopic” thermodynamic description, see Fig. <ref>(b)]. In both geometries, the cross-section of the droplet in our “macroscopic” thermodynamic description is a circular segment, therefore,

    r = R sinθ.

Thus, the shape and size of the droplet can be described by two independent variables, e.g., (R, r), (R, θ) or (r, θ).

In both geometries, choosing the  dividing surface to be the surface of tension, one can obtain equilibrium conditions for sessile droplets by expressing all the volumes, areas and the length of the three-phase contact line in (<ref>), and solving one of the following sets of two equations


    ∂Ω(R,r,T,μ)/∂ R = 0     and     ∂Ω(R,r,T,μ)/∂ r = 0,
       ∂Ω(R,θ,T,μ)/∂ R = 0    and     ∂Ω(R,θ,T,μ)/∂θ = 0, 
       ∂Ω(r,θ,T,μ)/∂ r = 0    and     ∂Ω(r,θ,T,μ)/∂θ = 0,


and then taking into account the geometric relation (<ref>). Since T and μ are constant, the pressures and the surface tensions in (<ref>) will be constant at such variations (for , this also requires / R = 0 which is true namely for the surface of tension chosen as the  dividing surface).

The droplet-size-dependent surface tensions  and  for spherical and cylindrical interfaces can generally differ, though their difference is expected to be very small at the same value of μ (see subsection <ref>).


Let us chose R and θ as independent variables describing the sessile droplet. Introducing non-variable quantities for the total volume occupied by the fluid ≡ V^α + V^β and the total area of the substrate ≡ + and using relation (<ref>), one can express the volumes, areas, and the contact line length for an axisymmetric droplet as

    = π^3/3 (2 + cos)(1 - cos)^2,
       = 2πsin, 
     = 2π^2 (1 - cos),
       = π^2 sin^2 ,
     =  - ,
       =  - .


Decomposition (<ref>) of the grand potential then takes form

    (,) = - ( p^α - p^β) π^3/3 (2 + cos)(1 - cos)^2 
        + 2 π^2 (1 - cos) + ( - ) π^2 sin^2 
        + 2πsin - p^β - .

Application of the equilibrium conditions (<ref>) gives

    [t][8.5cm]
    - π^2 ( p^α - p^β) (2 + cos) (1 - cos)^2 
     + 4 π (1 - cos) + 2 π( - ) sin^2 
     + 2 πsin + 2 π∂/∂sin^2  = 0,
    
          - π^3  ( p^α - p^β) sin^3  + 2 π^2 sin + 2 πcos
        + 2 π^2 (  - ) sincos + 2 π^2 ∂/∂sincos = 0.

To exclude the pressure difference p^α - p^β, we multiply the first equation by (2 π)^-1sin^2   (1 - cos)^-2 and then subtract the second equation multiplied by (2 + cos)(2 π^2 sin)^-1 <cit.>. After some trigonometric simplifications and substitution of the result into one of the above equilibrium conditions, we arrive at the Laplace equation and the generalized Young equation as equilibrium conditions for an axisymmetric sessile droplet <cit.>

    p^α - p^β = 2/, 
    cos =  -  - / - /


For a cylindrical droplet,

    = ^2/4 (2  - sin 2),
     = ,
       = sin,
     =  - ,
       =  - ,

and the length  of the three-phase contact line does not depend on  and  [see Fig. <ref>(b)] and supposed to be constant. Contact angle  here is in radians. Decomposition (<ref>) of the grand potential then  takes form

    (,)    = - ( p^α - p^β) ^2/4  (2  - sin 2) + 
        + ( - ) sin +  - p^β - .

Application of the equilibrium conditions (<ref>) gives

    [t][8.1cm]
    - ( p^α - p^β)  ( - sincos) + 
     + ( - ) sin + ∂/∂sin = 0,
    
       [t][8.1cm]
    - ( p^α - p^β) ^2 sin^2  + 
      +  ( - ) cos + ∂/∂cos = 0.

Here we have employed trigonometric equalities  sin 2 = 2 sincos and 1 - cos 2 = 2 sin^2. To obtain a simpler equation for p^α - p^β, we multiply the first equation by cos and subtract the second equation multiplied by sin. After some simplifications and substitution of the result into one of the above equilibrium conditions, we arrive at the Laplace equation and the generalized Young equation as equilibrium conditions for an cylindrical sessile droplet, slightly different from Eqs. (<ref>) and (<ref>):

    p^α - p^β = /, 
    cos =  -  - /.


The absence of the coefficient 2 in the Laplace equation  (<ref>) for the cylindrical droplet can be explained by the fact that its  interface is curved only along one direction, whereas the spherical droplet has equal curvatures along two directions. The line-tension terms in the generalized Young equations (<ref>) and (<ref>) originate from variations of the line term κ L in (<ref>). For an axisymmetric droplet, = 2π and ()/ = 2π + 2π / giving the two terms / + / in Eq. (<ref>). For a cylindrical droplet,  does not depend on  and ()/ =  / giving the only term / in Eq. (<ref>).

For both geometries, we suppose the line tension to depend not only on T and μ, but also on the droplet size. This dependence is usually overlooked and, in this case, Eq. (<ref>) transforms into the modified Young equation (<ref>), whereas Eq. (<ref>) transforms into formally classical Young equation (<ref>) (with surface tensions, however, depending on the chemical potential, i.e. on the equilibrium droplet size, that makes the contact angle  different from , see subsection <ref>).

We suppose the contact line radius (half-width) r to be the only appropriate size variable the line tension can be considered to depend on. It is not clear for the derivation we used in this paper, and some authors suppose the line tension to depend also on the contact angle θ <cit.>, corresponding “notional” derivatives also appear in the work <cit.>. However, a variational derivation of Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), which considers all three variables R, r, and θ to be independent in some sense, gives an argument in favor of r as the mathematically appropriate size variable for κ in both geometries <cit.>.

Simple dropping the size-dependent term κ/ r in the generalized Young equations (<ref>) and (<ref>) make them inexact (even if κ is supposed to depend on the droplet size). It can be seen in a model system if the line tension is calculated directly <cit.>. Also, the generalized Young equation is the simplest way to calculate κ/ r if θ and r are found and the surface and line tensions (the latter for axisymmetric droplet only) are calculated directly <cit.>.




 §.§ Size dependence of contact angle and relations between pressures, surface tensions, droplet size and chemical potential


Comparing the generalized Young equation (<ref>) or (<ref>) with the classical Young equation (<ref>), one should take into account that these two equations are written for droplets corresponding to different values of the chemical potential μ. The classical Young equation corresponds to a macroscopically large (strictly speaking, infinite) drop, i.e. liquid–gas bulk phase coexistence at the binodal. Let us mark the values of the quantities corresponding to the binodal with subscript “∞”. The surface tensions at the binodal then should be denoted as , , and , and the classical Young equation (<ref>) should be rewritten as

    cos =  - .


Subtracting the generalized Young equations (<ref>) and
(<ref>) from  Eq. (<ref>) we obtain, for an axisymmetric droplet,

    cos - cos = δ + / + / r
    and a similar equation for a cylindrical droplet:cos - cos = δ + /.

Here δ≡ - with ≡ - and ≡ -. To make these equations more similar to Eq. (<ref>), let us introduce δ≡ -. Then, Eqs. (<ref>) and (<ref>) can be rewritten as

    ( cos - cos ) = δ + δcos + / + /, 
        ( cos - cos) = δ + δcos + /.


Now we see, that, in the exact counterpart of Eq. (<ref>) for the difference cos - cosθ, we have 4 terms for an axisymmetric droplet (instead of 1) and 3 terms for a cylindrical droplet (instead of 0). How do they depend on the droplet size [the radius (half-width) of the droplet on the substrate]?

For the  liquid–gas interface, let us write down the isothermal Gibbs adsorption equations:

    = - μ,

where  is the adsorption (the surface excess of matter per unit surface area) at the  interface for corresponding geometry. Since the surface of tension is chosen as the dividing surface in both geometries, these equations do not contain (/ R)  R terms.

For each of the solid–fluid interfaces, similar equations can be written. Generally, they are more complicated (see, e.g., Sec. 6.3 in <cit.> and Sec. 3.4.3 in <cit.>), but in some simple cases (e.g., for non-deformable substrate and such a choice of the solid–fluid dividing surfaces that the amounts of matter of the immobile component of solid do not change in the solid phase at any change of μ) they take <cit.> the same simple form as (<ref>), so we can write down

    ≡(  - ) = (  - ) μ.

Integrating Eqs. (<ref>) and (<ref>) from the value  at the binodal to the current value of μ, we obtain, introducing δμ≡μ -,

    δ = - ∫_^μ(μ') μ' ≃ - δμ,  
    δ = ∫_^μ(  - ) μ' ≃(  - ) δμ.

Since () = () =, the asymptotic expression in (<ref>) is the same for spherical and cylindrical surfaces at the same value of μ, but the exact integral is generally not if (μ) ≠(μ). Thus,  and  generally differ, but at least in the second order in δμ:

    -  = δ - δ = ((δμ)^2).


The last asymptotic expressions in Eqs. (<ref>) and (<ref>) are valid for small values of δμ. How is this quantity related to the droplet size? This can be estimated using the Laplace equation (<ref>) or (<ref>) for the axisymmetric or cylindrical droplet, respectively. Applying the isothermal Gibbs–Duhem equations p = n μ to liquid and gaseous phases, respectively, we obtain p^α - p^β≃δμ with  the number density of molecules in α liquid phase. Here we have neglected the compressibility of the liquid (≃) and the number density of molecules  in the gas (≪). With use of the Laplace equations (<ref>) and (<ref>), and geometric relation (<ref>), we obtain

    2/ = 2sin/≃δμ, 
    / = sin/≃δμ.

We now see that δμ∝ 1/R = (1/r), i.e., small values of δμ correspond to large droplets, and δμ→ 0 expectedly corresponds to R →∞ and r →∞.

Thus, for the correction terms in Eqs. (<ref>) and (<ref>) we have

    δ≃2sin/(  - )  = (δμ) = (1/r), 
    δ≃ - 2sin/ = (δμ) = (1/r), 
    /≃/ = (δμ) = (1/r), where = at μ = (→∞) is the line tension of the macroscopic (infinite) droplet, andδ≃sin/(  - ) = (δμ) = (1/r), 
    δ≃ - sin/ = (δμ) = (1/r).


As for the terms κ/ r, it can be shown <cit.> that they are ( (δμ)^2 ) = ( 1/r^2 ). For sufficiently large droplets, sinθ can be approximated with sin, with a relative error (1/r), that makes Eqs. (<ref>), (<ref>), (<ref>), and (<ref>) valid with sin replaced by sin. The same applies to cos in the correction terms δcosθ in Eqs. (<ref>) and (<ref>).

Taking this into account, we can now write down the linear in 1/r approximations for Eq. (<ref>) and (<ref>):

    cos - cos≃/, 
    cos - cos≃/,




where

    ≡ + 2sin/(  -  - cos) 
    and   ≡sin/(  -  - cos)

are the quantities sometimes called the apparent line tensions <cit.> of axisymmetric and cylindrical droplets, respectively (see below).

Eqs. (<ref>) and (<ref>) show that the dependence of cos on 1/, even in the first order in 1/, is determined not only by the line tension, but also by the adsorptions at the three interfaces. In fact, Eq. (<ref>) should be treated as Eq. (<ref>) often is. This means that the slope of the graph cos vs 1/ at 1/→ 0 does not give the line tension itself but rather a combination of the line tension, the surface tension of the droplet free surface, adsorptions at the three interfaces, and the macroscopic contact angle. Some authors use the term “apparent line tension” meaning the value of the line tension calculated from the dependence of cosθ on 1/r (denoted further as cosθ (1/r)) at 1/r → 0 as if Eq. (<ref>) were the correct linear approximation. Then Eq. (<ref>) gives the value of the apparent line tension of axisymmetric droplets. More interesting, Eqs. (<ref>) and (<ref>) show that, for cylindrical droplets, cos also depends on 1/ in the first order in 1/. This has been found in simulations <cit.> and attributed to size dependence of the surface tension  expressed using the Tolman length <cit.> ≈/.  As it is seen from Eq. (<ref>), dependences of other surface tensions on the droplet size via the chemical potential could also made their contributions to the apparent line tension of cylindrical droplets.




§ LINE TENSION FROM DUAL-GEOMETRY CONTACT ANGLE DEPENDENCES ON DROPLET SIZE


One can easily notice that the apparent line tension (<ref>) of cylindrical droplets, proportional to the slope of the cos(1/)  graph at 1/→ 0, is exactly half of the second, adsorption-related, term in the apparent line tension (<ref>) of axisymmetric droplets, proportional to the slope of the cos(1/) graph at 1/→ 0. Thus, combining data for dependences cos(1/) and cos(1/), one can determine the line tension of a macroscopic droplet as

    =  - 2.


Using this relation, we have calculated the line tension  of a macroscopic droplet from the apparent line tensions  and  of sessile droplets simulated by Kanduč et al. <cit.> using data from Fig. 2(c) in <cit.>. As one can see in Fig. <ref>, the values of  calculated with Eq. (<ref>) are in good agreement with ones obtained in <cit.> by taking into account Tolman length <cit.> and “line tension stiffness” <cit.> corrections. It is clearly seen, that the values of  and  are comparable in these simulations that makes  visibly different from . In contrast, simulations of Isaiev et al. <cit.> show very weak dependence of cos(1/) comparing to cos(1/) (see Fig. 4 therein). This corresponds to ≪ and, thus, ≈.

             

Eqs. (<ref>)–(<ref>) suggest another idea. With dual-geometry sessile droplet measurements, it is not necessary to use the classical Young equation at all. Subtracting the generalized Young equation for axisymmetric droplets (<ref>)  from the one for cylindrical droplets (<ref>) we obtain

    cos - cos = / + / - /.

Taking into account Eqs. (<ref>), (<ref>) with subsequent comments and κ/ r = ( 1/^2 ), the first-order approximation to the combined equation (<ref>) will be

    cos - cos≃/.

Here dependences of  on μ and  and of  on μ are no longer taken into account since they give corrections ( 1/^2 ).

All the quantities in Eqs. (<ref>) and (<ref>) must be taken at the same values of the chemical potential and temperature. The chemical potential is often unknown in simulations or measurements. But, according to Eqs. (<ref>) and (<ref>) and taking into account Eq. (<ref>), the same value of δμ corresponds to the relation ≃/2 or, in the same order in δμ, to ≃/2. The former relation is expectedly more accurate, since it relies only on the Laplace equations (<ref>) and (<ref>), whereas the latter also uses approximation sin≃sin.

Thus, to determine the line tension from the measured (in the experiment or modeling) dependences of the cosθ on 1/r for both axisymmetric and cylindrical droplets at the same conditions, it is sufficient to calculate the difference

    cos|_ = /2 - cos|_  or  cos|_ = /2 - cos|_

and take a linear fit of it vs 1/ at sufficiently large values of . The slope of this graph will give /:

    cos|_ = /2 - cos|_≃/, 
    cos|_ = /2 - cos|_≃/.


An additional advantage of this technique is that the graph of the difference cos - cos vs 1/ must start from a known point (0 at 1/ = 0), thus the slope of the linear graph remains the only fitting parameter. When plotting cos(1/) and cos(1/) independently, each of them must be approximated by a straight line starting from an unknown point cos(0), which must be the same for axisymmetric and cylindrical droplets (and this, in principle, should be somehow taken into account to reduce the number of fitting parameters from 4 to 3). Unlike experimental measurements, droplet simulations require more computational resources as the droplet size increases, so approaching droplet sizes whose contact angles is macroscopic can be resource demanding. However, macroscopic contact angle , can be found, e.g., using the classical Young equation (<ref>) and calculating  and  in three simulations of  planar interfaces <cit.>.


Experiments with nanosized droplets suggest another possible advantage of using dual-geometry techniques to determine the line tension. For nanosized droplets, the graph cos vs 1/ can be fairly linear but this linear fit may work well only for small droplets and give a false value of cos at extrapolation down to 1/→ 0 (see Fig. 5 in <cit.>) and the slope that does not correspond to asymptotic Eq. (<ref>). In many simulations, the droplets are also nanosized, so the same problem can be with simulated droplets, both axisymmetric and cylindrical, giving false values of  and . If Eq. (<ref>) is then used, this would ultimately give a wrong value of . However, not only the slope would be wrong in such situation, but, most probably, the extrapolated values of cos would be different for cos vs 1/ and cos vs 1/ indicating an inconsistency. With use of the technique we proposed, such a situation would appear as a set of points of the quantity (<ref>) vs 1/ that do not fit a straight line starting from (0,0). Both such indicators would simply mean that the droplets are not large enough for the asymptotic Eqs. (<ref>), (<ref>), and (<ref>) to work and, thus, cannot be used to determine the contact angle  and the line tension  of macroscopic droplets.




§ OVERVIEW OF RESULTS AND DISCUSSION


Thermodynamic analysis we have performed shows that droplet size corrections to the contact angle cosine for axisymmetric sessile droplets attributed often to the line tension only, are determined generally by the line tension , its dependence on the droplet size (directly on the droplet radius  on the surface and via the chemical potential) and also by dependences of all three surface tensions on the droplet size (via the chemical potential). The same applies to corrections to the contact angle cosine for cylindrical sessile droplets, except for the line tension term / that is absent in the generalized Young equation (<ref>) for cylindrical droplets.

In the first order in 1/ (<ref>), this gives the slope of cos vs 1/ to be proportional not to the line tension  of macroscopic droplets [as it is widely believed according to Eq. (<ref>)] but rather to the so called “apparent line tension” , a sum of this line tension  and a combination of adsorptions at the three interfaces, the number density of molecules in the liquid phase and the macroscopic contact angle (<ref>).

For cylindrical droplets, the contact angle does also depend on the droplet half-width  on the surface already in the first order in 1/ (<ref>), giving the apparent line tension . It is determined by the same combination of the adsorptions, the number density of molecules in the liquid phase and the macroscopic contact angle (<ref>) but lacks the numerical coefficient 2 compared to the axisymmetric case (<ref>).

This gives a possibility to determine the line tension  from  and  found in dual-geometry measurements (e.g., for droplets simulated at the same conditions, i.e. at the same temperature with the same interaction potentials, etc.) as (<ref>). It was illustrated with calculations of  from  and  of MD-simulated droplets <cit.> (see Fig. <ref>).

We have also proposed a technique for  dual-geometry measurements based on calculation of the slope of the difference cos - cos at the same value of the chemical potential vs 1/. It can be implemented as a calculation of this difference (<ref>) at = /2 or = /2. The slope will equal / with  the surface tension of the planar liquid–gas interface. This technique has some advantages discussed in section <ref>.

The Laplace and the generalized Young equations in both geometries, as well as the Gibbs adsorption equation (<ref>) for the liquid–gas interface, were obtained and considered for the surface of tension as the liquid–gas dividing surface. At an arbitrary choice of this dividing surface, all these equations are more complicated <cit.>. Even though most researchers use these equations in their simple forms, the real choice of the dividing surface is often different or even not specified explicitly. In simulations, it can be done, e.g., by a boundary detection algorithm, and different algorithms can give <cit.> visibly different dependences of cosθ vs 1/r. This can be considered as an additional source of errors that need to be estimated.

As for the solid–liquid dividing surfaces, we have used the analogue of the Gibbs adsorption equation (<ref>) for the difference ≡ - written under some conditions that are not necessarily met at an arbitrary choice of these dividing surfaces. However, if the same choice is made in both geometries, these quantities will still cancel, both with use of Eq. (<ref>) and in the proposed technique where Eq. (<ref>) will remain valid. In contrast, the line tensions  and  (including their macroscopic value ) do generally depend on the choice of the solid–fluid dividing surfaces. For any specific choice, the dual-geometry approach should work, giving the value corresponding to chosen dividing surfaces.

Finally, let us discuss if the proposed technique can be easily adapted to determine the size-dependent line tension  and/or  (not only their limit  at r →∞). Eq. (<ref>) does not contain solid–fluid surface tensions, but it contains the surface tensions . They can be calculated separately for free spherical/cylindrical droplets of the same radius  of the liquid–gas surface as for the sessile droplet of the same type of geometry (thus, at the same value of chemical potential). However, since (μ,) and (μ,) are different functions, the expression / - / on the right hand side of Eq. (<ref>) is still ( (δμ)^2 ) = ( 1/^2 ) and, therefore, cannot be neglected even in the second order in 1/. Thus, Eq. (<ref>) can hardly be used to obtain more detailed information on the line tension from dual-geometry measurements of cosθ(1/r).



The authors thank  Volkov and  Polovinkin for valuable discussions and comments on the manuscript. The research has been carried out with financial support of the Russian Science Foundation, grant no. 22-13-00151, https://rscf.ru/en/project/22-13-00151/https://rscf.ru/en/project/22-13-00151/






 Paper Title: Critical surface adsorption of confined binary liquids with locally conserved mass and composition.





Department of Physics, University of Milan, via Celoria 16, Milano 20133, Italy


Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany
Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany




§ ABSTRACT 
Close to a solid surface, the properties of a fluid deviate significantly from their bulk values.
In this context, we study the surface adsorption profiles of a symmetric binary liquid confined to a slit pore by means of molecular dynamics simulations; the latter naturally entails that mass and concentration are locally conserved.
Near a bulk consolute point, where the liquid exhibits a demixing transition with the local concentration as the order parameter, we determine the order parameter profiles and characterise the relevant critical scaling behaviour, in the regime of strong surface attraction, for a range of pore widths and temperatures.
The obtained order parameter profiles decay monotonically near the surfaces, also in the presence of a pronounced layering in the number density.
Overall, our results agree qualitatively with recent theoretical predictions from a mesoscopic field-theoretical approach for the canonical ensemble.





Critical surface adsorption of confined binary liquids with locally conserved mass and composition
    Felix Höfling
    August 5, 2024
==================================================================================================





§ INTRODUCTION









The investigation of fluids under nano-confinement has drawn substantial research attention in the past decades.
In this context, water doubtlessly belongs to the most prominent liquid substance whereas it is arguably also one of the most difficult one to understand.
To this end, molecular simulations have provided key insights into the properties of water, complementing experimental findings and theoretical analyses (see, e.g., Ref.  and references therein). This includes the calculation of phase diagrams, e.g., of liquid–vapour and fluid–solid equilibria <cit.>

and a putative liquid–liquid transition deep in the super-cooled state <cit.>, 
but also structural properties <cit.> and transport coefficients such as the dynamic shear viscosity <cit.>.

Confining a liquid in channels of only a few nanometers in diameter manifestly breaks the translational symmetry
and can significantly alter its physical properties relative to its behavior in bulk. For example, certain crystal structures can be suppressed and phase transition temperatures can be shifted <cit.>.
Confinement intricately modifies the relaxation dynamics of glass-forming <cit.> and polymeric <cit.> liquids, as well as water again <cit.>.
The presence of solid surfaces modifies the transversal density profiles showing, e.g., layering,
which can lead to surface-induced phase transitions such as
pre-freezing and pre-melting <cit.> and the rich scenario of wetting and drying transitions <cit.>; predictions that long-ranged dispersion forces lead to specific phase diagrams <cit.> were criticised recently <cit.>.
For complex liquids, consisting of anisotropic molecules, anchoring effects to the surface add another fascinating twist to the wetting problem <cit.>,
whereas orientational ordering can be observed for rod-like, stiff polymers <cit.>.

Going beyond single-component fluids, liquid mixtures exhibit phase separation, which in conjunction with confinement can lead to surface enrichment and domain growth of one component <cit.>.
The demixing transition of the bulk phase diagram is associated with a continuous phase transition and, thus, with the emergence of a macroscopic correlation length and relaxation time <cit.>.
Furthermore, the transversal pore dimensions delimit the growth of critical fluctuations so that the critical singularities near the consolute point appear rounded <cit.>.
It also gives rise to critical Casimir forces <cit.>
and to a splitting of the bulk universality class into few surface universality classes <cit.> (see below).
In particular, for the local excess concentration, playing the role of the order parameter (OP), one expects that its spatial dependence near a solid substrate exhibits critical scaling.
This surface-critical behaviour was corroborated within Monte Carlo simulations for
Ising-like lattice fluids <cit.> and for
single-component Lennard-Jones (LJ) liquids <cit.>, with the local density as the OP field.

Monte Carlo simulations on phase coexistence are typically carried out either in the grand canonical or in the Gibbs ensemble. Also the majority of theoretical studies on critical phenomena in confinement have been performed in the grand canonical ensemble.
Experimentally, however, the mass of each component of a mixture is locally conserved, the corresponding total masses are fixed, and thus the canonical ensemble appears as the proper description of confined liquids in thermal equilibrium.
Whereas the statistical ensembles are equivalent with respect to the mean values of thermodynamic observables, they are not with respect to their fluctuations <cit.>.
Further on the level of thermodynamic macrostates, the ensemble equivalence may not hold in the presence of long-ranged interactions (see, e.g., refs.  and  and references therein).
This issue becomes particularly relevant for confined fluids close to their critical points and, as a result, the choice of the ensemble can yield very different predictions for the behaviour of the fluid.
For example, the critical Casimir force emerging for a fluid confined in between two parallel walls with symmetric boundary conditions is repulsive for the canonical ensemble whereas it is attractive for the grand canonical ensemble <cit.>.



In the present study, we employ state-of-the-art molecular dynamics (MD) simulations to investigate the surface adsorption of a
confined binary liquid film at its bulk demixing (liquid–liquid) critical point.
The binary mixture considered is symmetric with respect to its components <cit.>, and both walls of the slit pore preferentially attract, with equal strength, the same component of the mixture.
Within the MD simulations, as well as in experiments, the partial masses of the two components are naturally conserved
so that the data obtained for equilibrium properties correspond to the canonical ensemble.
Specifically, we calculate spatial profiles of the excess adsorption and analyse their critical scaling.

Our findings are compared with existing mean-field predictions in the limit of infinite surface attraction strengths.

We note that computational studies of near-critical fluids are notoriously challenging due to critical slowing down <cit.>, a macroscopically large correlation length (entailing finite-size corrections <cit.>), and strong OP fluctuations (expressed by a high compressibility or generalised susceptibility); we have addressed these issues by performing a large number of independent simulations encompassing long time spans and huge particle numbers.




§ CRITICAL SURFACE ADSORPTION


Upon approaching the critical point of a continuous phase transition, the local OP near a boundary deviates in normal direction from its bulk value on the scale of the diverging bulk correlation length <cit.>.
The kind of thermodynamic singularities occurring in this surface layer depends on the boundary conditions for the OP such that each bulk universality class splits up into various surface universality classes.
Generically, fluids belong to the so-called normal surface universality class <cit.>, which is characterized by the presence of a symmetry breaking surface field inducing order at the surface even if the bulk is in the disordered phase.
Close to the critical point the bulk correlation length ξ_b diverges in the thermodynamic limit, thus becoming the most predominant length scale in the system.
In the following, we specialise to binary fluids near their bulk consolute point, given by the critical temperature T_c and the
critical composition (which plays a similar role as the critical density in case of a liquid–vapour transition).
In addition, we restrict the discussion to the mixed phase, i.e., to fluid temperatures T > T_c,
and we will often incorporate the temperature dependence via the bulk correlation length ξ_b = ξ_b(T).
For this choice of temperatures, wetting transitions are not expected to occur <cit.>.

The demixing OP of a binary liquid of species A and B in bulk is given by ϕ_b=x_A-x_B in terms of the bulk concentrations x_A and x_B of both species, defined as their corresponding mole fractions.
Upon confinement to a slit pore, these quantities depend on the perpendicular distance z from one of the confining walls,
so that the local OP reads Φ(z)=x_A(z)-x_B(z), where x_α(z) = ρ_α(z) / ρ(z) for α=A,B;
the partial number density ρ_α(z) counts the particles of species α in a narrow slab at position z, and
ρ(z) = ρ_A(z) + ρ_B(z) is the overall local number density.


For fluids in contact with a single wall, ξ_b competes with the range of the surface interaction of the fluid molecules;
in the case of a short ranged interaction, as studied here, this range is typically on the order of the molecular diameter σ.
A universal shape of the OP profile Φ(z), as function of the distance z to the surface, is anticipated except very close to the surface, i.e., universality holds for z ≫σ.
Depending on the distance z compared to the correlation length ξ_b, one distinguishes a standard regime with an exponential decay, Φ(z) ∼exp(-z/ξ_b) for z ≫ξ_b, and a critical regime exhibiting power-law behaviour, Φ(z) ∼ z^-β/ν for z ≪ξ_b.
The exponents β and ν denote the standard critical exponents of the three-dimensional Ising universality class; we use the reliable estimates <cit.> ν = 0.630 and β = 0.325.
In particular, they govern the critical singularities in the bulk behaviour of the correlation length and the OP, respectively <cit.>:
ξ_b(T) ≃ξ_0^+ ϵ^-ν in terms of ϵ = (T-T_c)/T_c ↓ 0, measuring the proximity to the bulk critical point, and ϕ_b(T) ≃ϕ_0 (-ϵ)^β for ϵ↑ 0,
where ξ_0^+ and ϕ_0 are non-universal amplitudes.

The OP profile exhibits critical scaling which, for a semi-infinite fluid, is condensed in a dimensionless scaling function P_1 with the rescaled distance ẑ := z/ξ_b as the only variable <cit.>:

    Φ(z;T>T_c) = φ_0 (ξ_b/ξ_0^+)^-β/ν P_1(z/ξ_b)  ,    z ≫σ .

The scaling function is universal, but depends on the boundary condition for the OP at the surface.
The scaling function obeys the asymptotics sketched above, namely
P_1(ẑ→∞) ≃exp(-ẑ) and
P_1(ẑ→ 0) ≃ c_0 ẑ^-β/ν with the universal surface amplitude c_0.
The excess mass of fluid adsorbed to the surface is quantified by the integrated OP profile,

    Γ(T) = ∫_0^∞ [Φ(z;T) - ϕ_b(T)]   z  ,

for which, upon inserting <ref>, one infers a critical power-law divergence <cit.>:

    Γ(T → T_c) ∼ξ_b^1-β/ν∼ |ϵ|^β - ν ,

provided that β < ν.
(In mean-field theories, where β = ν = 1/2, a logarithmic divergence is predicted).
These scaling predictions for the OP profile have been tested successfully in simulations of Ising lattices <cit.>.
Scaling has also been corroborated in simulations of one-component LJ fluids,
but the verification of the critical divergences has remained challenging <cit.>.

For fluids confined to a slit pore, surface effects due to both walls are present and, as a result, the distance D between the two walls enters the problem as another relevant length.
In particular, critical fluctuations are suppressed in the direction of the surface normal for ξ_b ≫ D, whereas the slit confinement is expected to be ineffective for ξ_b ≪ D, approximating the situation of a semi-infinite fluid delimited by a single surface.
The OP profile Φ(z) is now a function of the position z, the temperature T, and the pore width D;
it also depends on the strength of the surface interaction, and we may again use the bulk correlation length ξ_b(T) instead of T.
The dependence on the pore width D is accounted for by decorating the scaling function in <ref> with a second scaling variable, D̂ = D/ξ_b, which is further used to substitute ξ_b by D, i.e., we compare other quantities such as the position z to the pore width D.
In the strong adsorption limit (large surface field), the finite-size scaling form of the OP profile reads <cit.>:

    Φ(z;T,D) = φ_0 (D/ξ_0^+)^-β/ν P_2(z/D, D/ξ_b)  ,
         z ≫σ .

Thus, plotting
φ_0^-1(D/ξ_0^+)^β/νΦ(z;T,D) vs. z̅ = z/D for a fixed value of D̂ = D/ξ_b should yield a data collapse.
The new, two-parameter scaling function P_2 reproduces the behaviour near a single surface via
P_1(ẑ) = lim_D̂→∞D̂^-β/ν P_2(ẑ/D̂, D̂)
and it encodes the critical profile Φ(z; T_c, D) as the limit P_2(z̅, D̂→ 0).
The function P_2 also depends on the boundary conditions at the two surfaces; here, we focus on a slit pore with equal surfaces on both sides, namely symmetric, attractive boundary conditions.

We note that, due to the confinement, the critical divergence at the bulk critical values of temperature and composition is shifted towards a lower temperature and a more A-rich composition (assuming A-preferring, symmetric surfaces) <cit.>.
The phenomenon is related to capillary condensation and depends on the pore width D; it is particularly relevant for thin films, D ≪ξ_b.
More precisely, it may be expressed as a competition of the pore width D and the bulk correlation length ξ_b and can be incorporated in the scaling form (<ref>) by adding an additional scaling variable for the composition.
However, we will not explore these aspects here and restrict to T > T_c in the following, where T_c always refers to the critical temperature of the bulk fluid, which also enters <ref> through the scaling of ξ_b.


For a conserved OP field, the critical scaling of the OP profiles has been studied in depth recently within a mesoscopic theory based on the Ginzburg–Landau (GL) free energy functional, prescribing either a surface field <cit.> or the contact values at the two surfaces <cit.>. 
In these works, analytic calculations within a systematic perturbative approach yield OP scaling functions, which are corroborated by numerical solutions of the field equations;
in addition, these findings are supported qualitatively with data from Monte Carlo simulations for Ising lattices.
The obtained OP profiles are mean-field like, as is typical for GL theories, and suggest
that, for symmetric boundary conditions [see eq. (33) of Ref. ],

    Φ(z;T, D) ≃μξ_b^2 - Φ_m(T, D)  cosh(z - D/2/ξ_b)

in the inner part of the pore, viz., for ξ_b ≪ z ≪ D - ξ_b.

Here, we consider binary fluids at critical composition so that the chemical potential μ associated with the OP field is zero and the first term vanishes.
[A similar expression was obtained for Dirichlet boundary conditions at both surfaces [see eq. (24) of Ref. ]. However, the situations studied analytically in Refs.  and  differ from the present setup, which corresponds to the boundary conditions Φ(0) = Φ(D) = Φ_0 ≈ 1 and zero total OP, ϕ_tot = 0.]









With this, the prefactor Φ_m = Φ(z = D/2) equals the minimum value of the OP profile, which is attained in the central plane of the slit pore by symmetry.
Furthermore, the local conservation of the OP field implies a constraint on the total OP ϕ_tot, which is the spatial integral of Φ(z) and which vanishes in the mixed phase at critical composition:

    ϕ_tot := ∫_0^D Φ(z)   z = 0  .

Therefore and exploiting the symmetry Φ(z) = Φ(D-z) of the set-up, Φ_m < 0 is related to the excess adsorption Γ on each surface via

    Γ(T,D) := ∫_0^D/2 [Φ(z) - Φ_m]   z = -Φ_m D/2 .


For a finite pore width, D < ∞, the profile Φ(z; T, D) is expected to depend on the temperature T analytically near T_c, which implies that the scaling function P_2(z̅, D̂) is analytic in D̂^1/ν with D̂ = D / ξ_b [<ref>].
Hence, Φ_m(T, D) may be expanded for T ↓ T_c or, equivalently, for ξ_b(T) →∞:

    Φ_m(T, D) ≃φ_0 (D/ξ_0^+)^-β/ν[ P_2(1/2, 0) + C (D/ξ_b)^1/ν]  ,

with some constant C > 0.
In particular, Φ_m(T,D), and thus the excess adsorption Γ(T,D), do not diverge near the bulk critical point, which is a consequence of the OP conservation and at variance to the behaviour in the grand canonical ensemble <cit.> [<ref>].
Rather, for fixed D, the excess adsorption decreases linearly from its critical value at T_c as the temperature is increased (ξ_b(T) ≫ D):

    Γ(T ↓ T_c, D) + Γ(T_c, D) ∼ξ_b^-1/ν∼ T - T_c  .





§ SIMULATION MODEL AND METHOD


For the MD simulations, we use a symmetric binary liquid with well-known bulk properties <cit.>.
Specifically, we consider an equimolar mixture of two components A and B which interact symmetrically with each other via the Lennard-Jones (LJ) pair potential,

    u_LJ(r; ε,σ)=4ε [(σ/r)^12-(σ/r)^6]  ,

with parameters ε and σ.
The potential is shifted and smoothly truncated at a cut-off distance r_c=2.5σ, which renders very good numerical stability with respect to energy conservation <cit.>:

    U_μν(r)
      = [u_LJ(r; ε_μν, σ)-u_LJ(r_c; ε_μν,σ)]  f((r-r_c)/h)

for μ, ν∈{A, B}, h=0.005σ, and f(ζ≤ 0)=ζ^4/(1+ζ^4) and f(ζ >0)=0.
The interaction range σ is the same for all pairs μν, whereas the interaction strengths are chosen as ε_AA=ε_BB=2ε_AB=:ε.
For the units of length and energy, we use σ and ε, respectively, which implies the reduced temperature T^*:=k_B T/ε.
Of each species, we simulate N_μ particles such that species μ has concentration x_μ=N_μ/N and such that the total number density is fixed at ρ_b = N/V = 0.8σ^3,
in terms of N=N_A + N_B and the pore volume V.
We have previously determined the demixing phase diagram of this fluid
[We note that a force-shifted potential was employed in the studies in Refs. , which yields different values for non-universal properties, i.e., the critical temperature and the critical amplitudes.]
and the relevant critical behaviour in bulk <cit.>; in particular, the critical point was found to be
in the liquid phase at temperature T_c^* = 1.629±0.001 and concentrations x_A,c=x_B,c=1/2;
the non-universal amplitude of the correlation length ξ_b(T) was obtained as ξ_0^+ = (0.47 ± 0.02)σ,
and we have estimated for bulk OP amplitude φ_0 = 0.77 ± 0.02.

The liquid is confined in between two planar walls separated by a distance D.
The area of each wall is L× L and is set by the lateral dimensions of the simulation box, with periodic boundary conditions applied along the two directions parallel to the walls;
the volume of the slit pore is thus V=L^2 D.
The walls are located at nominal positions z=0 and z=D, with the z-axis of the Cartesian coordinates chosen normal to the walls.
The surface–fluid interaction is the same for both walls and is given by the LJ wall potential (see, e.g., Refs. ):

    U_sf^(μ)(z̃)=2πε_w/3[
        2/15(σ/z̃)^9
        - w_μ(σ/z̃)^3 ],

in terms of the distances z̃=z+σ/2 and z̃=D+σ/2 - z to each of the walls.
The offset σ/2 is introduced in the surface potential to better match the volume that is effectively available to the fluid particles with the nominal pore volume L^2 D.
The form of the potential results from an integral over the LJ pair potential, modelling the wall as an infinite half-space which is filled homogeneously with LJ particles.
The surface potential U_sf^(μ)(z̃) is further shifted and smoothly truncated at the cutoff distance r_c,sf=2.5σ as described above for the pair interaction.
The energy ε_w and the dimensionless parameter w_μ control the
overall attraction strength and the relative surface preference for one of the two fluid components, respectively;
U_sf^(μ)(z̃) has its minimum at z̃_min = (2/5w_μ)^1/6σ, for w_μ > 0,
with depth (2πε_w/9)√(10w_μ^3).
We use the same value ε_w for both walls of the pore, and we put w_A=1 and w_B=0, which amounts to symmetric boundary conditions with a preference for A particles.

All results presented here are obtained at or above the bulk critical temperature of the binary liquid, T ≥ T_c (after a quench from the mixed phase at T > T_c), and at the critical concentrations x_A=x_B=1/2,
i.e., for zero total OP, ϕ_tot = x_A-x_B=0.
Extensive MD simulations were carried out in the canonical ensemble with the Nosé–Hoover thermostat
(NHT) chain <cit.>, using an integration timestep of δ t=0.001t_0;
the unit of time is t_0=√(mσ^2/ε) in terms of the particle mass m, which is the same for A and B particles.
To mitigate the computational issues associated with the simulation of critical fluids, such as critical slowing down <cit.> and finite-size corrections <cit.>, we have used large system sizes for up to 
300000 particles over a long time span of e4t_0;
in addition, ensemble averages were taken over up to 20 independent initial configurations for each parameter set.
Specifically, we have used large simulation boxes of lateral length up to L=100σ and we have varied the pore width from D=10σ to 30σ.
The simulations were performed with the software HAL's MD package <cit.>, which exploits the high degree of parallelism of recent accelerator hardware <cit.>, e.g., graphics processors (GPUs), and which is an efficient and precise tool for the study of inhomogeneous fluids <cit.>.
For example, a simulation run for D=20σ over a period of 10^4 t_0  (e7 steps),
takes about 1h on a single GPU of type Nvidia A40 
and about 5.8h on the much older hardware generation, Nvidia Tesla K20Xm. 
















The simulation data were stored in the structured, compressed, and portable H5MD file format <cit.>.

Laterally averaged number density profiles ρ_μ(z) = N_μ(z) / (L^2 Δ z) were calculated in a post-processing step by counting the number N_μ(z) of particles of each species μ that are located in the interval [z - Δ z/2, z + Δ z/2) along the z-axis for a bin width of Δ z = 0.5σ.
For the total density profile ρ(z), shown in the inset of <ref>(b), we used a different technique to be described elsewhere, which is based on Fourier transforms, avoids the spatial binning and reduces the amount of data to be stored. The method includes a slight smoothing of the obtained profile ρ(z), here with a Gaussian filter of width σ/8.



§ RESULTS AND DISCUSSION






Within the MD simulations, we have obtained the surface adsorption profiles Φ(z) of binary liquids
confined to a slit pore which preferentially attracts component A at both of its walls with equal strength [<ref>(a)];
unless stated differently, we consider a strong surface attraction, ε_w=8ε [see <ref>].
The present study is restricted further to binary liquids at critical composition (x_A = x_B = 1/2) and in the mixed phase (T ≥ T_c), so that the temperature T controls the distance to the bulk critical point of the demixing transition.
Φ(z) serves also as the corresponding local OP, it is defined such that positive values indicate an excess of A particles.
Since the set-up exhibits a mirror symmetry, Φ(z; T, D) = Φ(D-z; T, D), with respect to the central plane (z=D/2), we will  often discuss merely the behaviour near one of the surfaces.

Two exemplary OP profiles Φ(z) for two temperatures, in the mixed phase and at criticality, are shown in <ref>(a).
At the higher temperature, T^*=5 ≈ 3 T_c^*, the profile is comparably flat in a large part of the pore but increases sharply upon approaching one of the surfaces (e.g., z → 0), the latter reflecting the expected surface enrichment of A particles.
Upon decreasing the temperature to T_c, the contact value Φ(z → 0) increases to its largest possible value, Φ(z → 0) = 1, which we attribute to the choice of a strong surface interaction (ε_w=8ε);
this value of Φ(z) corresponds to a pure layer of A particles next to the surfaces.
Concomitantly, the OP profile broadens as T ↓ T_c, reflecting the emergence of critical fluctutations.
However, the width of the surface enrichment layer remains finite (of the order of 3-4σ) despite the bulk correlation length ξ_b(T) diverging at T_c, which is a consequence of the confinement to a slit and of the aforementioned symmetry of Φ(z) at z=D/2 = 10σ.



Furthermore, the conservation of the OP implies that the integrated OP ϕ_tot does not change upon varying the temperatures; here, we use ϕ_tot=0.
It follows from <ref> that the OP value in the middle of the slit pore must be negative, Φ_m = Φ(z=D/2) < 0, to compensate the surface enrichment.
This is a specific feature of the canonical ensemble and unlike in the grand canonical ensemble, in which the local OP remains positive, Φ(z) ≥ 0, for symmetric surfaces both preferentially attracting component A.
For conserved OP, the value Φ_m thus serves as a measure of the total excess adsorption Γ(T,D) = -Φ_m(T,D)   D /2 [<ref>].
The latter is predicted to vary linearly with T near the critical temperature [<ref>],
which is corroborated by our simulation data (<ref>).

In contrast to the monotone decay of Φ(z) for 0 ≤ z ≤ D/2 (<ref>), the number density of the critical fluid (T=T_c) exhibits a pronounced layering near the surfaces (inset of <ref>):
the density profile ρ(z) shows rapidly decaying oscillations around the mean density ρ_b=0.8σ^-3, which have a large amplitude next to the surface at z=0 and which are barely visible in the middle of the pore, for z ≳ 4σ; the period of the oscillations is of the order of the molecular length σ.
Such oscillations hinder the scaling analysis of critical surface profiles, and their absence in the present OP profiles (i.e., in the concentration and thus the excess adsorption) underscores that, in this respect, binary liquids are favourable systems for the study of critical behaviour.
We note that the critical behaviour of the structural and dynamic properties of binary liquids is governed by two fluctuating fields, viz., the total number density and the composition.
The investigated binary liquid (with ρ_b=0.8σ^3) was demonstrated <cit.> to be
sufficiently far away from its liquid–vapor critical point and thus the density field
ρ(z) acts as a secondary field <cit.>, which does not exhibit critical enhancement
(see, e.g., the bulk structure factors in Figs. 4 and 5 of Ref. ).

In order to further elucidate the lateral structure of the layers near the surface, we have calculated the lateral structure factor S_ρρ(|k⃗|) of the density for each layer separately, by restricting the calculation to quasi-twodimensional slabs encompassing one layer each and such that the wave numbers point parallel to the wall surfaces, for details see Ref. .
Upon approaching the surface, layer by layer, the apparent compressibility ∝ S_ρρ(0) is reduced and the degree of local ordering increases, as follows from the growth and the sharpening of the first peak of S_ρρ(|k⃗|) (<ref>).
However, also the first fluid layer, which is in direct contact with the wall, does not exhibit crystallisation but remains in a fluid state.




Next, we investigate the dependence of the OP profile Φ(z) of the critical mixture on the pore width D for fixed temperature T=T_c.
For strong surface attraction (ε_w=8ε), the contact value Φ(z → 0) at the surface does not
exhibit any prominent D dependence [<ref>(a)]. 
As D is increased, the profile becomes deeper (|Φ_m| increases) and broadens: the thickness of the
surface adsorption layer (e.g., the distance to the surface at which Φ(z) changes its sign) increases.

In order to test the scaling property of these critical OP profiles, we have plotted the combination
φ_0^-1 (D/ξ_0^+)^β/νΦ(z,T,D) as a function of the scaling variable z̅ = z/D
for a range of pore widths D that is spread out by a factor of 3. 
<Ref>(b) demonstrates convincingly that this rescaling of the OP profiles of <ref>(a)  yields data collapse onto the scaling function
P_2(ẑ, 0); due to T=T_c, we have D̂ = D/ξ_b(T_c) = 0.
Deviations from this universal master curve occur for distances z ≲ 0.8σ, which corresponds to the non-universal microscopic regime.
We note that, in order to avoid spurious finite-size corrections to the scaling, we found it necessary to also fix the aspect ratio L/D=3.3 of the pore geometry; the ratio L/D enters the scaling form of the OP profile [<ref>] as an additional finite-size scaling variable (see, e.g., Refs.  and ).




The scaling function P_2 of the OP profile Φ(z;T,D) depends on two scaling variables, z̅ = z/D and D̂ = D/ξ_b(T) [<ref>].
For elevated temperatures, T > T_c, a similar data collapse as for the critical mixture can be obtained by considering a range of combinations of pore width D
and temperature T such that D̂ = D/ξ_b(T) = (D/ξ_0^+)ϵ^ν is kept constant.
Indeed, the rescaled data for the OP profiles with D̂ = 5.9 and for strong surface attraction (ε_w=8ε) collapse onto the master curve P_2(z̅, D̂ = 5.9) similarly well as in <ref>(b) for T=T_c.
However, repeating this analysis for weak surface attraction (ε_w=0.8ε), the quality of the data collapse is considerably worse [<ref>(b)]. In this situation, the OP profiles depend also on the strength ε_w of the surface field, which enters the scaling function in <ref> as another scaling variable, in addition to z/D, D/ξ_b, and L/D (see, e.g., Ref.  for details).



For a more systematic investigation of the dependence of the OP profile on the strength ε_w of the surface attraction, we consider the OP value close to the surface.
Within the mesoscopic GL theory employing a surface field h_1 <cit.>, it was predicted that Φ(z → 0)=Φ_0 displays a crossover from the scaling Φ_0 ∼ h_1 for weak fields, h_1 → 0,
to Φ_0 ∼ h_1^1/2 for strong fields, h_1 →∞.
Since there is no direct mapping between the surface field h_1 of GL theory and the surface interaction parameters (ε_w, w_A) used in the MD simulations, we check if the scaling of Φ_0 with ε_w is analogous to what has been predicted within GL theory for the dependence on h_1;
to this end, the mean-field exponents should be replaced by their counterparts in the 3D Ising universality class.
However, the contact value Φ(z→ 0) appears to be not suited for this task since universality of the surface adsorption profiles is expected to hold only for z ≫σ.
Hence, we define the surface value Φ_0 as an integral over the particle layer second-closest to the surface:

    Φ_0 = 1/0.5σ∫_σ^1.5σΦ(z)   z  .

Using this definition of Φ_0, the data from MD simulations of the critical mixture (T=T_c, D=20σ, and L=100σ)
are described the behaviour Φ_0(ε_w → 0) ∼ε_w^0.55 for weak surface interaction strengths, varying from ε_w=e-2ε to ≈ 0.2 ε [<ref>(a)].
Despite these low values of ε_w, the observed scaling is at variance with the expectation from perturbation theory for Ising and GL models that Φ_0 scales linearly with the surface field <cit.>.
For the the strong-surface field behaviour, one needs to account for the bound |Φ_0| ≤ 1 due to the definition of Φ(z), similarly as above for the excess adsorption Γ. Thus, we consider 1-Φ_0 in order to test the approach of Φ_0 to its saturation value.
Fitting a power law to the data points for the two largest surface interactions simulated (ε_w = 4ε and 8ε) suggests that
1-Φ_0(ε_w →∞) ∼ε_w^-0.57 [<ref>(b)].
We note that this value of the exponent is larger than the standard surface exponent <cit.> Δ_1 ≈ 0.46 and that, to the best of our knowledge, a theoretical prediction for the strong-surface field behaviour of Φ_0 under the simulated conditions is not available.




§ SUMMARY AND CONCLUSIONS


Understanding the surface critical behaviour of a fluid confined to a slit pore is demanding due to the presence of multiple length scales: there are the molecular size σ and the pore width D, the temperature dependence implies the bulk correlation length ξ_b(T),
the strength of the surface interaction ε_w adds another implicit length, and, in simulations, the finite lateral extent L of the slit pore cannot be ignored.
Here, we have used large-scale, off-lattice MD simulations to explore the scaling of the excess adsorption profiles Φ(z) of a binary liquid with locally conserved mass and concentration;
the liquid is kept confined to a slit whose walls have selective surface adsorption preferences.
For the demixing transition, the excess adsorption plays the role of the OP field, which develops critical fluctuations near the liquid–liquid critical point.
Relying on the corresponding scaling form [<ref>],
we have shown that the data for laterally averaged OP profile Φ(z;T, D) for a range of temperatures T and pore widths D collapse onto a master curve upon suitable rescaling (<ref>).
This master curve is given by the scaling function P_2 in the limit of strong surface adsorption and for fixed ratios D/ξ_b and L/D.
The obtained OP profiles behave monotonically near the surfaces, also in the presence of a pronounced layering in the number density profiles [<ref>(b)].
Furthermore, our data suggest that OP profiles obtained within the MD simulations are better described by the regime of strong surface attraction, even for low values of ε_w, which raises a question on the precise mapping between the particle model used in the MD simulations and GL theory (<ref>).
Overall, our results are in qualitative agreement with mean-field predictions from field-theoretical calculations <cit.>.

Whereas our study is based on the generic and somewhat artificial Lennard-Jones pair interaction,
we anticipate that our findings for critical adsorption are universal and can be transferred mutatis mutandis to real-world binary liquids, including water–lutidine mixtures.
In particular,
understanding the influence of surfaces on a confined fluid is important for the study of critical Casimir forces <cit.>. We hope this work will motivate future work on the non-equilibrium dynamics of near-critical confined fluids and on the dynamic response to a temperature quench <cit.>.


 We thank Siegfried Dietrich and Markus Gross for helpful discussions.
 Financial support by Freie Universität Berlin via a research stipend under the Future Development Concept: “International Network University” (S.R.)
 and by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 523950429 (F.H.) is gratefully acknowledged.






 Paper Title: Asymptotics of the overlap distribution of branching Brownian motion at high temperature.



Quantum Simulation via Stochastic Combination of Unitaries
    Prineha Narang
    August 5, 2024
==========================================================



§ ABSTRACT 
    At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity.
    We investigate the precise decay rate of the probability to obtain an overlap greater than a, for some a>0, in the whole subcritical phase of inverse temperatures β∈ [0,β_c).
    Moreover, we study this probability both conditionally on the branching Brownian motion and non-conditionally.
    Two sub-phases of inverse temperatures appear, but surprisingly the threshold is not the same in both cases.



	 
	



§ INTRODUCTION


The (binary) branching Brownian motion (BBM) is a continuous-time branching Markov process constructed as follows.
At time t = 0, a single particle starts a standard Brownian motion in  from the origin.
After a random time following an exponential distribution with parameter 1, it splits in two, or equivalently, it dies while giving birth to two children.
These new particles then repeats the same process, independently of each other.
More precisely, the children perform independent Brownian motions from the position of their parent at its death and, after independent exponential times, they in turn split in two.
For a formal construction, see <cit.>.

Branching Brownian motion has been widely studied over the last fifty years, initially for its link with reaction-diffusion equations <cit.> which motivated a precise study of its extremes <cit.>.
Other important motivations for this model come from physics.
In spin glass theory, the study of disordered systems with a hierarchical structure of correlation for the energies dates back to the generalized random energy model introduced by Derrida and Gardner <cit.> and includes the BBM (see e.g. Bovier's book <cit.>).
The BBM can also be seen as an infinite dimensional limit of directed polymers <cit.> and is also related to diffractive scattering in high-energy particle physics <cit.>.
See Section <ref> for more details.
In all these frameworks coming from physics, understanding the overlap distribution of particles chosen according to the Gibbs measure plays an important role, and this paper is set in this context, by investigating the probability to have an atypical overlap in the high temperature phase.




 §.§ Definitions and previous results


From a statistical physics perspective, for some fixed time t ≥ 0, one can identify the positions of particles alive at time t with the energy levels of a physical system.
Each particle can then be thought of as a configuration of the system.
Let (t) be the set of particles alive at time t and X_u(t) the position of particle u at time t.
The Gibbs measure of the BBM at time t and at inverse temperature β≥ 0 is the (random) probability measure _β, t on (t) that assigns to each particle u a weight proportional to ^β X_u(t).
By rescaling the associated partition function, we obtain the so-called additive martingale of the BBM at inverse temperature β, introduced by McKean <cit.>,

    W_t(β) = ^-ψ(β)t∑_u ∈(t)^β X_u(t),

where ψ(β) = 1 + β^2/2.
Since it is a positive martingale, it converges to a random variable W_∞(β).
This limit plays a key role in the study of the long-term behavior of the BBM, as illustrated in Theorem <ref> and Theorem <ref> below.
A phase transition occurs: by <cit.>, if β≥√(2) then W_∞(β) = 0 , if β < √(2) then W_∞(β) > 0 a.s.
In the latter case, the additive martingale is uniformly integrable, or equivalently, it converges in L^1.
In the the critical case β = √(2), it is more suitable to consider the derivative martingale, introduced by Lalley and Sellke <cit.>,

    Z_t = ∑_u ∈(t) (√(2)t - X_u(t)) ^√(2)X_u(t) - 2t.

The authors showed that it converges to a random variable Z_∞ and that Z_∞ > 0 The critical additive martingale is related to the derivative martingale through the following convergence

    √(t) W_t(√(2)) √(2/π) Z_∞,   in probability,

proved for the branching random walk by Aïdékon and Shi <cit.>.
For the BBM, this is a direct consequence of <cit.>.


In this paper, we are interested in the behavior of the overlap between particles chosen according to the Gibbs measure.
The overlap between two particles u, v ∈(t), denoted q_t(u, v) ∈ [0,1], is defined as 
the last time where u and v have had a common ancestor alive, rescaled by a factor of t.
In other words, it is the proportion of time where u and v have shared a common trajectory.
Then, the overlap distribution at time t ≥ 0 and inverse temperature β is the following random probability measure on [0,1]:

    ν_β, t = _β, t^⊗ 2(q_t(u, v) ∈·) = 1/W_t(β)^2∑_u, v ∈(t)^β(X_u(t) + X_v(t)) - 2 ψ(β) tδ_q_t(u, v),

so that, for a ∈ [0,1], ν_β, t([a,1]) is the conditional probability, given the BBM, that two particles chosen independently according to _β,t have an overlap at least a.
This quantity was introduced by Derrida and Spohn <cit.> by analogy with the overlap between two configurations of spin glasses.
We are also interested in the mean overlap distribution [ν_β, t], which is a deterministic probability measure on [0,1] such that [ν_β, t([a,1])] is now the (non-conditional) probability that two particles chosen independently according to _β,t have an overlap at least a.


The overlap distribution satisfies the following phase transition:

    ν_β, tδ_0,  
    		   if β≤√(2), 
    
    		(1 - π_β) δ_0 + π_βδ_1, 
    		   if β > √(2),

for the weak topology, where π_β is a random variable with values in (0,1), mean 1-√(2)/β and a law which depends explicitely on β.
This result, conjectured in <cit.>, has been proved for the branching random walk by Mallein <cit.> and for the BBM by Bonnefont <cit.>. Also note that this result was proved earlier for the mean overlap distribution in <cit.>.
In this paper, we focus on the subcritical phase β < √(2) and obtain the precise decay rate of the overlap distribution on (0,1], as well as its limit, once rescaled by this rate.
We investigate separately the typical overlap distribution in Theorem <ref> and the mean overlap distribution in Theorem <ref>.
Each theorem reveals two main regimes, though the threshold differs between them.



Theorem <ref> involves a new probability measure _β, introduced by Chauvin and Rouault <cit.>.
If we denote by (_t)_t ≥ 0 the natural filtration of the BBM, we can define it through the Radon–Nikodym derivatives

    _̣β̣|̣_̣_̣ṭ/|̣_̣_̣ṭ = W_t(β),

for t ≥ 0.
Under _β, one of the particles, called spine, performs a Brownian motion with drift β and splits at rate 2 while the behavior of the other particles remains unchanged, see <ref> for details.
We denote by __β the associated expectation.

Throughout this paper, the letters t and s denote some points in time [0, ∞).
The letters C and c denote constants, deterministic and time-invariant quantities, which may vary from one line to another.
We set = {0,1,2,…} and ^* = ∖{0}.
We use the Bachmann–Landau notations o, O, ∼, and Hardy's notation ≍ with their usual meaning.
More precisely, for f _+ → and g _+ →_+^*, we say 
that f(t) = o(g(t)) as t →∞ if lim_t→∞ f(t)/g(t) = 0, 
that f(t) = O(g(t)) as t →∞ if lim sup_t→∞f(t)/g(t) < ∞, 
that f(t) ∼ g(t) as t →∞ if lim_t→∞ f(t)/g(t) = 1, 
and that f(t) ≍ g(t) as t →∞ if 0 < lim inf_t→∞ f(t)/g(t) ≤lim sup_t→∞ f(t)/g(t) < ∞.



 §.§ Main results


In the following theorems, we describe the asymptotic behavior at high temperature of the overlap distribution and its expectation.


    Let 0 ≤β < √(2) and 0 < a < 1.
    
        
  *  If 0 ≤β < √(2)/2, then
        
    ^(1 - β^2)atν_β, t([a, 1]) W_∞(2β)/W_∞(β)^2W_∞(β)^2,

        
  *  If β = √(2)/2, then
        
    √(at)^at/2ν_β, t([a, 1]) √(2/π)Z_∞/W_∞(β)^2W_∞(β)^2,   in probability.

        
  *  If √(2)/2 < β < √(2), then
        
    (at)^3 β/√(2)^(√(2) - β)^2 atν_β, t([a, 1]) C_β Z_∞^√(2)β/W_∞(β)^2 S_2β,   in distribution,

        where S_2β is a non-degenerate (√(2)/2β)-stable random variable independent of the BBM and C_β > 0 is a constant that depends only on β.
    



    Let 0 ≤β < √(2) and 0 < a < 1.
    As t →∞,
    
        
  *  if β = 0, then
        
    ν_β, t([a, 1])∼ 2at ^-at,

        
  *  if 0 < β < √(2/3), then W_t(β) converges -and _2β-to some positive and finite variable W_∞(β), and
        
    ν_β, t([a, 1])∼__2β[ 1/W_∞(β)^2] W_∞(β)^2^-(1 - β^2)at,

        
  *  if β = √(2/3), then
        
    [ ν_β, t([a, 1]) ] ≍ t^-1/2^-at/3,

        
  *  if √(2/3) < β < √(2), then
        
    [ ν_β, t([a, 1]) ] ≍ t^-3/2^-(2 - β^2)^2at/8β^2.

    


The exponents appearing in the exponential decay of the overlap distribution in Theorems <ref> and <ref> are represented in Figure <ref>.



 
    If 0 ≤β < √(2)/2, the additive martingale W_t(2β) is uniformly integrable and therefore the measure _2β is uniformly continuous w.r.t.  with Radon–Nykodim derivative W_∞(2β).    
    In particular, if 0 < β < √(2)/2, we can rewrite Theorem <ref>.<ref> as
    
    ^(1 - β^2)atν_β, t([a, 1])W_∞(2β)/W_∞(β)^2W_∞(β)^2.

    Note that this limit is the expectation of the limit in Theorem <ref>.<ref>.
    This no longer holds if √(2)/2 ≤β < √(2/3) since the right-hand side of (<ref>) becomes zero.
    The probability measure _2β then appears as the appropriate framework to extend this formula.





 §.§ Comments, related literature and heuristics





  
Motivations.
As mentioned earlier, our motivation for studying the overlap comes from the physics literature.
In mean-field spin glass models such as the Sherrington–Kirkpatrick model, the overlap between two configurations is defined as an explicit function of the correlation between the energies of these configurations, and the overlap distribution plays a key role in the interpretation of Parisi's formula for the free energy <cit.> and of the ultrametric structure of the Gibbs measure in the infinite-volume limit <cit.>.
To understand the free energy and the overlap distribution on more tractable examples, Derrida and Gardner <cit.> introduced the Generalized Random Energy Model (GREM), where the ultrametric structure is directly implemented in the definition of the Hamiltonian at finite volume: more precisely, the energies have a hierarchical structure of correlations with a fixed number of levels, whereas the BBM corresponds to a number of levels growing like the number of particles.
As indicated above, the overlap distribution for the BBM has been first studied in physics by Derrida and Spohn <cit.>, motivated by both spin glass and polymers theories. 
Indeed, the BBM can also be seen as an infinite limit of the directed polymer model and, in this framework, the overlap distribution describes the proportion of time where two polymers chosen independently according to the Gibbs measure are overlapping.

More recently, Derrida and Mottishaw <cit.> investigated the finite-size corrections for the mean overlap distribution in the critical and supercritical phases β≥√(2): to be more precise than the known convergence [ν_β,t] →√(2)/βδ_0 + (1-√(2)/β) δ_1 as t →∞, they describe how the mass of [ν_β,t] between 0 and 1 vanishes. 
More precisely, they conjecture that, for any 0<a<b<1,

    [ν_β,t([a,b])] 
        ∼1/β√(2 π t)∫_a^b x̣/x^3/2 (1-x)^(1+2·1_β>√(2))/2.

This prediction is based both on numerical simulations and on an approximation by the GREM, for which Derrida and Mottishaw <cit.> calculated the finite-size corrections of the overlap as well.
In the case β>√(2), the asymptotic (<ref>) also has an interpretation in high-energy particle physics, uncovered by Mueller and Munier <cit.>: using a link between the BBM and the scattering of high-energy hadrons dating back to <cit.>, they show a link between the statistics of rapidity gaps in diffractive electron-nucleus scattering and statistics of the extremal particles of the BBM, including their overlap distribution, which is well-predicted by (<ref>) because supercritical Gibbs measures are supported by extremal particles.


In this paper, we study this question in the subcritical phase β < √(2) instead, and investigate moreover the behavior of the typical overlap distribution.
Observe that the exponent appearing in the exponential decay of the mean overlap distribution is vanishing as β approaches √(2) (see Figure <ref>), which is consistent with the fact that the decay is expected to be polynomial only for β≥√(2).
This is also the case for the typical overlap distribution, so we also expect a polynomial decay for β≥√(2) in that case.
However, we expect the typical overlap distribution to decay faster than the mean overlap distribution when β≥√(2), as it is the case for β∈ [√(2)/2,√(2)), which means that [ν_β,t([a,b])] is governed by a rare event (called “fluctuation” in <cit.>).





  
Typical overlap distribution.
Most heuristics can be deduced from the following rewriting of the overlap distribution (see (<ref>) for details)

    ν_β, t([a, 1]) = ^(β^2 - 1) at/W_t(β)^2∑_w ∈(at)^2β X_w(at) - ψ(2β)at W_t - at^(w, at)(β)^2,

where (W_s^(w, at)(β))_s ≥ 0 is the additive martingale of the BBM starting from particle w at time at.
Note that, in the sum of (<ref>), these additive martingales are tilted by Gibbs weights at inverse temperature 2β and time at.
Consequently, the particles u,v∈(t) that mainly contribute to the overlap distribution (see the expression in (<ref>)) are those contributing to W_t-at^(w, at)(β), with w itself being among the particles contributing to the Gibbs measure _2β, at, or equivalently, to the additive martingale W_at(2β).
The description of these particles' behaviors follows from this.

There is a first regime β∈ [0, √(2)/2) for which the Gibbs measure _2β, at is subcritical, in the sense that it is supported by an exponential number of particles.
These ones perform a Brownian motion with drift 2β (see <cit.> for a weaker version of this property and <cit.> for the case of the branching random walk).
So do the particles that contribute to the overlap distribution, up to time at, after which their trajectories relax to a drift β (see Figure <ref>).
The large amount of such particles induces a law of large numbers that allows to approximate (<ref>) with

    ^(β^2 - 1) at/W_t(β)^2∑_w ∈(at)^2β X_w(at) - ψ(2β)atW_∞(β)^2 = ^(β^2 - 1) at/W_t(β)^2 W_at(2β) W_∞(β)^2.

Theorem <ref>.<ref> then follows from the convergence of the additive martingales.



At β = √(2)/2, the Gibbs measure _2β, at is critical.
It is supported by a number of order ^C√(t) of particles, having a drift √(2) altered by an upper barrier of slope √(2) (see <cit.> for the branching random walk).
The same holds up to time at for the overlap distribution.
Theorem <ref>.<ref> can be seen as a consequence of the approximation (<ref>) and the convergence (<ref>).

In the remaining regime β∈ (√(2)/2, √(2)), the Gibbs measure _2β, at is supported by the extremal process, particles at a bounded distance from the top of the BBM.
Roughly, their trajectories are Brownian bridges conditioned to stay under a barrier of slope √(2) (see <cit.> for the branching random walk).
This again induces the behavior of the particles that contribute to the overlap distribution (see Figure <ref>).
The α-stable limit with α = √(2)/2β∈ (1/2, 1) is reminiscent of the limit appearing for the renormalized supercritical additive martingale, obtained by Barral, Rhodes and Vargas <cit.> for the branching random walk (see <cit.> for the BBM case): for β > β_c,

    t^3β/2√(2)^(β/√(2)-1)^2t W_t(β)
         Z_∞^β/√(2) S_β,   in distribution,

where S_β is a non-degenerate (√(2)/β)-stable random variable independent of Z_∞.
Here, it is not exactly W_at(2β) which appears in (<ref>) because of the additional factors W_t - at^(u, at)(β)^2, but we show that they do not affect the stable nature of the limit.


The transition at β = √(2)/2 described above is similar to the one appearing in the fluctuations of the additive martingales, obtained by Iksanov, Kolesko and Meiners <cit.> for the branching random walk (see <cit.> for the BBM case).
This fluctuations are obtained using the following decomposition

    W_∞(β) - W_t(β) 
        = ∑_u∈(t)^β X_u(t) - ψ(β) t (W_∞^(u)(β) - 1),

where the main difference with (<ref>) is that this is a sum of centered variables, whereas the terms in (<ref>) are positive.
However, similarly to the case of the overlap, for β∈ (0,√(2)/2), the sum in (<ref>) is dominated by particles contributing to _2β,t (particles u∈(t) such that X_u(t) = 2β t + O(√(t))) and the limit, after rescaling, is normally distributed with variance c W_∞(2β).
Furthermore, for β∈ (√(2)/2,√(2)), the sum is dominated by extremal particles and, as noted in <cit.>, the limit is α-stable with α = √(2)/β∈ (1, 2).
In the later regime, we apply methods similar to those used in <cit.>.



  
Mean overlap distribution.
It is surprising that, in terms of the mean overlap distribution, the threshold at β = √(2)/2 disappears, and this new threshold at β = √(2/3) appears. We do not know other phenomena for the BBM where this threshold plays a role.
Moreover, let us emphasize here that the proof of Theorem <ref> is more involved than the one of Theorem <ref>, even in the case β∈ (0,√(2)/2), partly due to the fact that 1/W_∞(β) is not integrable for any β∈ [0,√(2)), so the denominator in (<ref>) has to be handled carefully.

We now give some crude heuristics to explain the appearance of the threshold at β = √(2/3). 
In the rewriting (<ref>) of ν_β, t([a, 1]), we see that a particle w ∈(at) contributes to the numerator proportionally to ^2β X_w(at).
By Girsanov theorem, the event dominating in [^2β X_w(at)] is the one where (X_w(s))_s∈[0,at] behaves like a Brownian motion with drift 2β. There are typically such particles w in the BBM if and only if β < √(2)/2 and this is the main reason why the behavior of ν_β, t([a, 1]) in the case β∈ (0,√(2)/2) is simply obtained by taking the expectation of the typical behavior (see Remark <ref>).
As soon as β≥√(2)/2, ν_β, t([a, 1]) starts being dominated by a rare event where some particle w∈(at) goes to a position much higher than the typical maximum of the BBM.
However, this position is not necessarily 2β at + O(√(t)) as predicted by Girsanov theorem. 
Indeed, a particle w ∈(at) also contributes to the denominator in (<ref>): its contribution to W_t(β) equals ^β X_w(at) - ψ(β)at W_t - at^(w, at)(β).
If X_w(at) ≪ψ(β)at/β, then ^β X_w(at) - ψ(β)at≪ 1 and the contribution of w in W_t(β) is negligible (W_t(β) is typically of order 1), so only the numerator dictates the optimal behavior of the particle w. 
On the other hand, if X_w(at) ≫ψ(β)at/β, then W_t(β) is dominated by the contribution of particle w and then the numerator and the denominator compensate each other: forcing w to reach a higher position does not increase significantly ν_β, t([a, 1]) because it saturates at 1.
Therefore, particles w∈(at) contributing mostly to ν_β, t([a, 1]) are at a position roughly v(β) at, where v(β) is given by

    v(β) = 2β∧ψ(β)/β 
        = 
            2 β   if β∈ [0,√(2/3)], 
    1/β + β/2   if β∈ [√(2/3), √(2)).

See Figure <ref> for the graph of function v.
Hence, the threshold at β = √(2/3) corresponds to the inverse temperature above which, in mean, particles contributing mostly to the numerator in (<ref>) also contributes significantly to the denominator.
Note that we have cheated a little in the above heuristic: we took for granted that, in order to make ^2β X_w(at) W_t - at^(w, at)(β)^2 large, one has to make ^2β X_w(at) large, but the best strategy could involve making W_t - at^(w, at)(β)^2 large as well.
However, using the tail (W_t - at^(w, at)(β) ≥ x) ≤ C x^-2/β^2 (see Lemma <ref>), one can check that the best strategy to obtain ^2β X_w(at) W_t - at^(w, at)(β)^2 ≃^2β v(β) at is indeed to make only X_w(at) large.




As a consequence of this discussion, one can describe the behavior of particles contributing mostly to the mean overlap distribution.
If β∈ (0, √(2/3)), then contributing particles have a drift v(β) = 2β until time at and then a drift β (see Figure <ref>).
If β∈ [√(2/3),√(2)), we explained that contributing particles have a drift v(β) = 1/β + β/2 until time at, but one can be more precise than that. 
If the contributing particle has, at some time s ∈ [0,at], an ancestor at v(β) s + x, then one can check that the descendants at time t of this ancestor have a contribution of order ^β x in W_t(β), so any such event with a large x results in a small value of ν_β, t([a, 1]).
Hence, contributing particles are roughly constrained below the straight barrier s ↦ v(β)s.
Therefore, if β = √(2/3), contributing particles move like a Brownian motion with drift v(β) conditioned to stay below the barrier until time at, and then like a Brownian motion with drift β.
If β∈ (√(2/3), √(2)), the factor ^2β X_w(at) in the numerator of (<ref>) additionally forces particles to end up close to the barrier at time at, hence contributing particles move like a Brownian motion with drift v(β) conditioned to stay below the barrier until time at while reaching a position v(β) at + O(1) at time at, and then like a Brownian motion with drift β (see Figure <ref>).
Note that these constraints explain the polynomial corrections t^-1/2 and t^-3/2 appearing in Theorem <ref>.<ref> and <ref>.






 §.§ Organization of the paper


Section <ref> contains a formal definition of the model, which is then used to introduce properly some change of measures, and results concerning the additive martingales, the extremal process of the BBM and ballot theorems for Brownian motions.
In Section <ref>, we prove Theorem <ref>, with one subsection dedicated to each part of the statement (in the same order).
Finally, Section <ref> contains the proof of Theorem <ref>, again with one subsection dedicated to each part of the statement (but this time in the order <ref>, <ref>, <ref> and <ref>).



§ PRELIMINARY RESULTS





 §.§ The model


Following the formalism of Chauvin and Rouault <cit.>, we consider the BBM to be a binary tree endowed with random lifetimes and trajectory data.
In addition, it will be useful to distinguish one line of descent, called the spine.
Let us specify.

We consider the set of Ulam–Harris–Neveu labels = ⋃_i ≥ 0{1, 2}^i, with the convention {1, 2}^0 = {∅}.
For a label u ∈, we denote by |u| its length, with the convention |∅| = 0.
For two labels u, v ∈, we denote by uv their concatenation, with the convention u ∅ = ∅ u = u.
We say that u is an ancestor of v and that v is a descendant of u if there exists w ∈ such that v = uw.
In this case, we write u ≤ v.
If in addition u ≠ v, then we write u < v.
We denote by u ∧ v the most recent common ancestor of two particles u and v, u ∧ v = u_1 … u_i_0 with i_0 = sup{i ≥ 1 : u_1 … u_i = v_1 … v_i} if u_1 = v_1, and u ∧ v = ∅ otherwise.

The probability space on which we work is Ω the set of tuples ((σ_u)_u ∈, (Y_u)_u ∈, ξ), where, for each u ∈, σ_u ≥ 0 and Y_u : [0, ∞) → is a continuous function, and ξ = {∅, ξ_1, ξ_1ξ_2, ξ_1ξ_2ξ_3, …}, with (ξ_k)_k≥ 1∈{1,2}^^*, is an infinite line of descent of .
We denote by b_u = ∑_v < uσ_v the birthtime of a particle u, and d_u = ∑_v ≤ uσ_v its deathtime.
For t ≥ 0, we define

    (t) = {u ∈ : b_u ≤ t < d_u}  and  ([0, t]) = {u ∈ : b_u ≤ t}.

We denote by ξ(t) the only particle in (t) ∩ξ, called spine.
We define inductively the position at time t of a particle u ∈(t)

    X_u(t) = ∑_v < u X_v(d_v) + Y_u(t - b_u).

We extend the notion of position for a particle u ∈(t) to the whole interval [0, t]: for every s ∈ [0, t], we set X_u(s) = X_v(s), where v is the ancestor of u alive at time s.
To avoid redundancy, we denote the spinal position X_ξ(t) = X_ξ(t)(t).

For t ≥ 0, let _t be the σ-algebra containing all the information until time t except the spine,

    _t = σ( (u, d_u ∧ t, X_u|_[0, d_u ∧ t]) : u ∈([0, t]) )   and  _∞ = σ( ⋃_t ≥ 0_t ),

where s ∧ t denotes the minimum between s and t.
Let us define a finer filtration by adding the information about the spine,

    _t = σ( _t, ξ(t) )   and  _∞ = σ( ⋃_t ≥ 0_t ).

We define the BBM law as the unique probability measure  on (Ω, _∞) for which

    
  * the trajectories Y_u are standard Brownian motions,
    
  * the lifetimes σ_u are exponential variables with rate 1,
    
  * the spinal coordinates ξ_1, ξ_2, … are with (ξ_1 = 1) = (ξ_1 = 2) = 1/2,
    
  * (Y_u)_u ∈, (σ_u)_u ∈, ξ are independent.

We denote by  the associated expectation.


    For all t ≥ 0 and u ∈(t), ( ξ(t) = u | _t ) = 2^-|u|.


In order to state a branching property, we define, for t ≥ 0, a shift operator Θ_ξ, t : Ω→Ω by Θ_ξ, t(σ, Y, ξ) = (σ', Y', ξ'), where, if u = ξ(t),

    
  *  σ'_v = 
        d_u - t    if  v = ∅, 
σ_uv   otherwise,
    
  *  Y'_v(s) = 
        X_u(t + s) - X_u(t)    if  v = ∅, 

        Y_uv(s)    otherwise,
    
  * ξ' = {∅, ξ_|u| + 1, ξ_|u| + 1ξ_|u| + 2, …}.

We also define, for u ∈ and t ≥ 0, Ω_u, t = {(σ, Y, ξ) ∈Ω : u ∈(t)} and Θ_u, t : Ω_u, t→Ω by Θ_u, t(σ, Y, ξ) = (σ', Y', ξ') satisfying <ref>, <ref> and ξ' = {∅, 1, 11, …}.
A consequence of <cit.> is the following branching property.
If Y and Y_u, u ∈, are non-negative variables on Ω such that Y is _∞-measurable and Y_u, u ∈, are _∞-measurable, then for every t ≥ 0,

    [ Y ∘Θ_ξ, t×∏_u ∈(t) 
     u ≠ξ(t) Y_u ∘Θ_u, t| _t ] = [Y] ∏_u ∈(t) 
     u ≠ξ(t)[Y_u].

This means that, at time t, all the alive particles start independent BBMs shifted in time, space, label, and that the information about the spine is kept in the BBM started from ξ(t).



 §.§ Change of measure



In this section, we introduce some changes of measure that allow a useful interpretation of the process in term of spinal decomposition.
This approach dates back to Kahane and Peyrière <cit.> for the branching random walk and to Chauvin and Rouault <cit.> for the BBM, and it has been widely used since, see <cit.> and <cit.> for applications to the study of the additive martingales. 
The framework of these techniques has been extended in <cit.> and <cit.> to more general branching processes and the presentation below follows roughly the lines of <cit.>.

Let β≥ 0 and ψ(β) = 1 + β^2/2.
The process (2^|ξ(t)|^β X_ξ(t) - ψ(β)t)_t ≥ 0 is a positive (_t)_t ≥ 0-martingale with mean 1.
By Kolmogorov's extension theorem, we can define a probability measure _β on _∞ by

    _̣β̣|̣_̣_̣ṭ/|̣_̣_̣ṭ = 2^|ξ(t)|^β X_ξ(t) - ψ(β)t.



    The additive martingale defined in (<ref>) is the projection of the above martingale onto _t,
    
    W_t(β) = [ 2^|ξ(t)|^β X_ξ(t) - ψ(β)t| _t ].

    Therefore, the change of measure (<ref>) is consistent with (<ref>).


In <cit.>, Chauvin and Rouault describe the effect of this change of measure on the BBM.
Under _β,

    
  * the spine's motion is a Brownian motion with drift β,
    
  * the spine particle splits at rate 2,
    
  * at each splitting event of the spine particle, the label of the new spine particle is chosen uniformly among the two children,
    
  * at this splitting event, the other child starts an independent standard BBM without spine.

By <cit.>, a remarkable and useful property is that

    _β( ξ(t) = u | _t ) = 1_u ∈(t)^β X_u(t) - ψ(β)t/W_t(β).

As mentioned in Remark <ref>, if β < β_c, then the uniform integrability of (W_t(β))_t≥ 0 implies that the measure _β|__∞ is uniformly continuous w.r.t. |__∞ with Radon–Nykodim derivative W_∞(β) (note however that _β and  on the full σ-field _∞ are singular).
On the other hand, if β≥β_c, then the convergence of W_t(β) towards zero implies that the measures _β|__∞ and __∞ are singular.

In this paper, we will also use a slightly different change of measure, that turns out to be more suited to the study of the overlap distribution.
For β≥ 0 and t ≥ 0, we define _β, t on _∞ by

    _̣β̣,̣ ̣ṭ/ = 2^|ξ(t)|^β X_ξ(t) - ψ(β) t.



    Let Y_t, Y_ξ and Y_u, u ∈, be non-negative variables on Ω such that Y_t is _t-measurable, Y_ξ is _∞-measurable, and Y_u, u ∈, are _∞-measurable.
    Then,
    
    __β, t[ Y_t × Y_ξ∘Θ_ξ, t×∏_u ∈(t) 
     u ≠ξ(t) Y_u ∘Θ_u, t]    = [ 2^|ξ(t)|^β X_ξ(t) - ψ(β)t× Y_t ×[ Y_ξ∘Θ_ξ, t×∏_u ∈(t) 
     u ≠ξ(t) Y_u ∘Θ_u, t| _t ] ] 
       = __β[ Y_t ×[ Y_ξ] ×∏_u ∈(t) 
     u ≠ξ(t)[ Y_u ] ],

    by the branching property (<ref>) and the definition of _β.
    This means that, under _β, t,
    
        
  * between 0 and t, the BBM has the same behavior as under _β,
        
  * at time t, the spine particle starts a standard BBM, shifted in time, space and label,
        
  * at time t, the non-spine particles start standard BBMs without spine, shifted in time, space and label, independent of each other and of the spine-BBM.
    


Similarly to (<ref>), we have the following property.


    For all u ∈ and s, t ≥ 0,
    
    _β, t(ξ(t) = u | _t + s) = 1_u ∈(t)^β X_u(t) - ψ(β)t/W_t(β).




    Let A ∈_t+s.
    By definition (<ref>),
    
    __β, t[ 1_A 1_ξ(t) = u] = [ 1_A 1_ξ(t) = u 2^|ξ(t)|^β X_ξ(t) - ψ(β) t] = [ 1_A 1_ξ(t) = u 2^|u|^β X_u(t) - ψ(β) t].

    But u is the spinal particle at time t if and only if u is alive at time t and admits the spinal particle among its descendants at time t + s.
    This consideration together with Remark <ref> allow us to rewrite (<ref>) as
    
    [ 1_A 1_u ∈(t)(∑_v ∈(t+s) 
     v ≥ u1_ξ(t+s) = v) 2^|u|^β X_u(t) - ψ(β) t] = [ 1_A 1_u ∈(t)( ∑_v ∈(t+s) 
     v ≥ u 2^-|v|) 2^|u|^β X_u(t) - ψ(β) t].

    Note that {v ∈(t+s) : v ≥ u} is a binary tree and that, for its particles v, the number |v| - |u| counts the fission times from u to v.
    Therefore,
    
    ∑_v ∈(t+s) 
     v ≥ u 2^-(|v| - |u|) = 1.

    We can then rewrite (<ref>) as
    
    [ 1_A 1_u ∈(t)^β X_u(t) - ψ(β) t] = __β, t[ 1_A 1_u ∈(t)^β X_u(t) - ψ(β) t/W_t(β)],

    using here that _β,t|__∞ has density W_t(β) w.r.t. |__∞.
    This concludes the proof.




 §.§ Moments and tails of the additive martingale


In what follows, it will be convenient to use the following rewriting of the additive martingale (<ref>).
For t ≥ s ≥ 0,

    W_t(β) = ∑_u ∈(s)^β X_u(s) - ψ(β)s W_t - s^(u, s)(β),

where, for r ≥ 0,

    W_r^(u, s)(β) = W_r ∘Θ_u, s = ∑_v ∈(s + r) 
     v ≥ u^β (X_v(s + r) - X_u(s)) - ψ(β)r.

The above process is the additive martingale of a BBM shifted in time, space and label.
It converges to some limit W_∞^(u, s)(β) and hence we can extend (<ref>) to the case t = ∞.
By the branching property (<ref>), conditionally on _s, the processes (W_r^(u, s))_r ∈[0, ∞] for u ∈(s) are copies of (W_r)_r ∈ [0, ∞].


    Let β∈ [0, √(2)).
    For any p ∈ (1, 2/β^2), the additive martingale at inverse temperature β is bounded in L^p and hence converges in L^p.
    Moreover, if p ∈ (1, 2/β^2) ∩ (1, 2], then there exists C>0 such that, for any 0 ≤ s ≤ t ≤∞,
    
    | W_t(β) - W_s(β) |^p ≤ C ^-(p - 1)(1 - pβ^2/2)s.



The proof of this lemma relies partially on the following inequality, which is due to Von Bahr and Esseen <cit.> and is used repetitively throughout the paper. The application of this type of inequalities to branching processes dates back to Neveu <cit.> and Biggins <cit.>.


Let p ∈ [1,2] and X_1,…,X_n be centered independent real random variables in L^p. Then,

    ∑_k=1^n X_k^p≤ 2 ∑_k=1^n X_k^p.




    Let p ∈ (1, 2/β^2).
    Equation (<ref>) holds for t = ∞ and s = 1.
    Therefore, by <cit.>, W_∞(β) is in L^p if and only if
    
    1 > [ #(1) ] [ ( ^β X_ξ(1) - ψ(β))^p ] = ^1 + p^2 β^2/2 - pψ(β).

    This inequality holds since ψ(β) = 1 + β^2/2 and p ∈ (1, 2/β^2).
    Hence W_∞(β) ∈ L^p.
    But, recalling that (W_t(β))_t≥ 0 is uniformly integrable for β < √(2), we have W_t(β) = [ W_∞(β) | _t ].
    Jensen's inequality for conditional expectations implies that the additive martingale is bounded in L^p.

    Now let p ∈ (1, 2/β^2) ∩ [1, 2] and 0 ≤ s < t ≤∞.
    We rewrite
    
    W_t(β) - W_s(β) = ∑_u ∈(s)^β X_u(s) - ψ(β)s (W_t - s^(u, s) - 1).

    Conditionally on _s, the random variables (W_t - s^(u, s) - 1) for u ∈(s) are centered and independent.
    Therefore, by Lemma <ref> and the branching property (<ref>),
    
    |W_t(β) - W_s(β)|^p_s≤∑_u ∈(s)^p β X_u(s) - p ψ(β)s|W_t-s(β) - 1|^p = W_s(pβ) ^-(pψ(β) - ψ(pβ))s|W_t-s(β) - 1|^p.

    Note that
    
    pψ(β) - ψ(pβ) = (p - 1)(1 - pβ^2/2),

    which is positive, by choice of p.
    This concludes since W_s(pβ) has mean 1 and the additive martingale at β is bounded in L^p.



    Let β∈ [0, √(2)) and fix a number b such that
    
    0 < b < √(ψ(β)^2 + 2β^2) - ψ(β)/β^2.

    There exists K > 0 such that, for every t ∈ [0, ∞] and every x ≥ 0,
    
    W_t(β) ≤ x≤ K x^b.




    Equation (<ref>) holds for t = ∞ and s = 1.
    Therefore, by <cit.>, W_∞(β) ≤ x = O(x^-b) as x → 0 for any b>0 such that
    
    [ ( ^β X_ξ(1) - ψ(β))^-b] < ∞  and  [ ( ^β X_ξ(1) - ψ(β))^-b1_#(1) = 1] = ^(b^2 β^2/2 + b ψ(β) - 1) < 1,

    which is equivalent to (<ref>).
    As a consequence, W_∞(β)^-b < ∞ for any b satisfying (<ref>).
    Then, for any t ∈ [0, ∞], by Markov's inequality and Jensen's inequality (using that W_t(β) = [ W_∞(β) | _t ]),    
    
    W_t(β) ≤ x≤ x^b W_t(β)^-b≤ x^b W_∞(β)^-b,

    which concludes the proof.




 §.§ The extremal process


We first recall some important results concerning the extremes of the BBM. For t>0, let

    m(t) = √(2) t - 3/2 √(2)log t.

The convergence of the maximal position has been proved by Bramson <cit.> and the limit has been identified by Lalley and Sellke <cit.>, who introduced the derivative martingale (<ref>) and showed that

    max_u ∈(t) X_u(t) - m(t) 1/√(2) (G + log(CZ_∞)),   in distribution,

where G is a standard Gumbel variable.
As a consequence, for any δ > 0, the probability of the following event converges to 1 as t →∞,

    Λ_δ, t = {max_u ∈(t) X_u(t) ≤ m(t) + δlog t }.


Later, Arguin, Bovier and Kistler
<cit.> and Aïdékon, Berestycki, Brunet and Shi <cit.> independently established the convergence in distribution
of the extremal process,

    _t = ∑_u ∈(t)δ_X_u(t) - m(t).

This convergence holds in the space of Radon measures on  endowed with the vague topology.
They obtained the following limit,

    _∞ = ∑_i, j ≥ 1δ_p_i + Δ_ij + log(C Z_∞)/√(2),

where

    
  * = ∑_iδ_p_i is a Poisson point process on  with intensity measure √(2)^-√(2)xx̣, independent of Z_∞,
    
  * _i = ∑_j δ_Δ_ij are copies of a point process = ∑_j δ_Δ_j such that max = 0 and are independent of  and Z_∞.

In other words, the extremal process converges in distribution to a randomly shifted decorated Poisson point process, a Poisson point process  with a shift depending on Z_∞, where each point is replaced by an independent copy of a decoration .
One can deduce (see <cit.> for details) that for any continuous function with compact support f : → [0, ∞),

    lim_s →∞lim_t →∞^-_t(f)_s = ^-_∞(f)_∞,   ,

where we emphasize that in _∞, only Z_∞ is _∞-measurable, whereas  and (_i)_i≥1 are independent of _∞.
Cortines, Hartung and Louidor <cit.> studied the level sets of the decoration: in their Proposition 1.5, they obtained the following equivalent as x →∞,

    ([-x, 0]) ∼ C ^√(2) x.

In what follows, we write X d= Y to indicate that X and Y have same distribution.


    If β > √(2) and (ξ_k)_k≥1 denotes the points of _∞, then ∑_j ^βΔ_j < ∞, ∑_k ^βξ_k < ∞ and
    
    ∑_k ≥ 1^βξ_kd= C (Z_∞)^β/√(2) S_β,

    where S_β = ∑_i ^β p_i is a non-degenerate √(2)/β-stable random variable independent of Z_∞.



    As done in <cit.>, we rewrite
    
    ∑_j ≥ 1^βΔ_j = ∫_-∞^0 ^β x(x̣) = ∫_0^∞β^-β y([-y, 0]) ỵ.

    Applying the Fubini-Tonelli theorem and (<ref>), we see that the above quantity is finite.
    Now,
    
    ∑_k ≥ 1^βξ_k = (C Z_∞)^β/√(2)∑_i ≥ 1^β(p_i + X_i),

    where X_i = 1/βlog∑_j ^βΔ_ij.
    The random variables X_i for i ≥ 1 are , independent of  and satisfy
    
    ^√(2) X_1 = (∑_j ≥ 1^βΔ_j)^√(2)/β≤∑_j ≥ 1^βΔ_j < ∞.

    By <cit.>,
    
    ∑_i ≥ 1δ_p_i + X_id=∑_i ≥ 1δ_p_i + 1/√(2)log^√(2) X_1.

    By (<ref>) and (<ref>), we have ∑_k ^βξ_k < ∞ if and only if ∑_i ^β p_i < ∞ Computing the expectations, we see that, with probability 1, we have ([0, ∞)) < ∞ and ∑_i ^β p_i1_p_i < 0 < ∞, so ∑_i ^β p_i < ∞.
    Besides, since Z_∞ is independent of  and (_i)_i ≥ 1, (<ref>) and (<ref>) yield
    
    ∑_k ≥ 1^βξ_kd=(C Z_∞^√(2) X_1)^β/√(2)∑_i ≥ 1^β p_i.

    It remains to show that ∑_i ^β p_i is stable.
    Let λ > 0.
    Applying successively the exponential formula for Poisson point processes (see <cit.>) and the change of variable y = ^β x, we obtain
    
    exp(-λ∑_i ≥ 1^β p_i) = exp(-∫_ (1 - ^-λ^β x) √(2)^-√(2) xx̣) = exp(-√(2)/β∫_0^∞ (1 - ^-λ y) y^-1-√(2)/βỵ).

    We recognize the Laplace transform of a non-degenerate √(2)/β-stable law (see <cit.>).


We now define two point processes on ^2 by

    _t^* = ∑_u ∈(t)δ_(X_u(t) - m(t), W_∞^(u, t)(β))  and  _∞^* = ∑_k ≥ 1δ_(ξ_k, W_k),

where W_k for k ≥ 1 are copies of W_∞(β), independent of _∞ = ∑_kδ_ξ_k and of the BBM.
Let f : ^2 → be a bounded continuous function.
Assume that there exists a ∈ such that f(x, y) = 0 as soon as x ≤ a.
Under these conditions and for the branching random walk, Iksanov, Kolesko and Meiners showed in <cit.> that _t^*(f) converges in distribution to _∞^*(f) as t →∞.
Their result easily extends to BBM (see <cit.> for details).
Furthermore, thanks to (<ref>), one can adapt their proof to obtain the following joint convergence (see <cit.> for details), for any β≥ 0,

    (_t^*(f), W_t(β))  (_∞^*(f), W_∞(β)),   in distribution.




 §.§ Brownian estimates


Let (B_t)_t ≥ 0 be a standard Brownian motion on , α∈ (0, 1/2), and define, f_t(s) = s^α∧ (t - s)^α for t ≥ s ≥ 0.
By <cit.>, there exist C, c > 0 such that, for all t ≥ 0, x, y ∈ [1, √(t)],

    c x/(t + 1)^1/2≤sup_s ∈ [0, t] (B_s + f_t(s)) ≤ x≤C x/(t + 1)^1/2,

and

    c x y/(t + 1)^3/2≤sup_s ∈ [0, t] (B_s + f_t(s)) ≤ x, B_t ∈ [x - y, x - y + 1]≤C x y/(t + 1)^3/2.

Besides, the following bound will be useful.


    Fix δ > 0.
    There exists C_δ > 0 such that, for all x, y ≥ 0 and t ≥ 1,
    
    sup_s ∈ [0, t] B_s ≤ x, B_t ∈ [x - y, x - y + δ]≤C_δ x y/t^3/2( ^-(x - y)^2/2t + ^-(x - y + δ)^2/2t).




    By <cit.>, sup_s ∈ [0, t] B_s ≤ xB_t≤ 2 x (x - B_t)^+/t, where we write a^+ = max(a, 0).
    It follows that the left-hand side of (<ref>) is bounded by
    
    2 x (x - B_t)^+/t1_{B_t ∈ [x - y, x - y + δ]}≤2 x y/tB_t ∈ [x - y, x - y + δ].

    Next, we can use the bounds
    
    B_t ∈ [x - y, x - y + δ]≤
                C δ^-(x - y)^2/2t/√(t)   if  x - y ≥ 0, 
    
                C δ^-(x - y + δ)^2/2t/√(t)   if  x - y + δ≤ 0, 
    
                C/√(t)   otherwise.

    Each of these terms is bounded by the right-hand side of (<ref>) since t ≥ 1.




§ PROOF OF THEOREM <REF>




Let β∈ [0, √(2)) and a ∈ (0, 1).
Recall that ψ(β) = 1 + β^2/2 and that the additive martingale W_t(β) is defined in (<ref>).
In what follows, it will be convenient to use the following rewriting of the overlap distribution defined in (<ref>),

    ν_β, t([a, 1]) = 1/W_t(β)^2∑_u, v ∈(t) 
     u ∧ v ≥ at^β(X_u(t) + X_v(t)) - 2 ψ(β) t = ^(β^2 - 1) at/W_t(β)^2∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_t - at^(u, at)(β)^2,

where we recall that

    W_t^(u, s)(β) = W_t(β) ∘Θ_u, s = ∑_v ∈(s+t) 
     v ≥ u^β (X_v(s + t) - X_u(s)) - ψ(β)t.




 §.§ Upper temperature


The aim of this section is to show Theorem <ref>.<ref> which covers the case β∈ [0,√(2)/2).
Define

    F_t = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_t - at^(u, at)(β)^2,

so that ^(1 - β^2)atν_β, t([a, 1]) = F_t W_t(β)^-2, by (<ref>).
It suffices to show that

    F_t  F_∞ = W_∞(2β) W_∞(β)^2,




  §.§.§ Replacement along a partition of time


Let us fix the following partition of the time [0, ∞),

    T = ⋃_n ≥ 0(1/n+1) ∩ [n, n+1].

Here, we justify that, along this partition, we can replace F_t with

    F̅_t = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_∞^(u, at)(β)^2.

We use first branching property at time at, and then factorize the square and apply Cauchy–Schwarz inequality:

    |F_t - F̅_t|_at   ≤ W_at(2β) |W_t - at(β)^2 - W_∞(β)^2| 
       ≤ W_at(2β) 
        ( W_t - at(β)^2 + W_∞(β)^2)^1/2(W_t - at(β) - W_∞(β))^2^1/2
       ≤ C W_at(2β) ^-(2ψ(β) - ψ(2β))s

using Lemma <ref> in the last inequality.
By (<ref>), it follows that |F_t - F̅_t| ≤^-(1 - β^2)(t-at).
Hence, by the Borel–Cantelli lemma,

    |F_t - F̅_t|  0,




  §.§.§ Almost sure convergence along a partition of time


Here, we show that F_t converges to F_∞ along the partition T.
In view of the previous section, it suffices to show that the following process converges to 0 along the partition T,

    Δ_t = F̅_t - W_at(2β) W_∞(β)^2 = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at(W_∞^(u, at)(β)^2 - W_∞(β)^2).

Let us fix p ∈ [1, 2] such that p < 1/2β^2.
By Lemma <ref>,

    |Δ_t|^p_at≤ W_at(p2β) |W_∞(β)^2 - W_∞(β)^2|^p ^-(pψ(2β) - ψ(p2β))at.

The variable W_∞(β)^2 is in L^p because 2p<2/β^2, hence so is W_∞(β)^2 - W_∞(β)^2.
Hence,

    |Δ_t|^p ≤ C ^-(pψ(2β) - ψ(p2β))at.

By (<ref>) and by the Borel–Cantelli lemma,

    Δ_t  0,




  §.§.§ Extension of the almost sure convergence to continuum


Let us denote (t_n)_n ≥ 0 the sequence of the elements of T in increasing order.
Here, we show that

    sup_t ∈ [t_n, t_n+1] |F_t - F_t_n|  0,

We have

    sup_t ∈ [t_n, t_n+1] |F_t - F_t_n| ≤∑_u ∈(at_n)^2β X_u(at_n) - ψ(2β)at_n S_n^(u),

where

    S_n^(u) = sup_t ∈ [t_n, t_n+1]|∑_v ∈(at)
    v ≥ u^2β(X_v(at) - X_u(at_n)) - ψ(2β)(at - at_n) W_t - at^(v, at)(β)^2 - W_t_n - at_n^(u, at_n)(β)^2|.

Note that, by the branching property, for any u ∈(at_n),

    S_n^(u)_at_n = S_n,

where

    S_n = sup_s ∈ [0, t_n+1 - t_n]|∑_v ∈(as)^2β X_v(as) - ψ(2β)as W_s + t_n - a(s + t_n)^(v, as)(β)^2 - W_t_n - at_n(β)^2|.



    For any p ∈ [1, 1/β^2), the process S_n converges in L^p to 0.



    We first prove that, on the event Ω' where W_t(β) converges as t →∞, the process S_n converges to 0.
    Let us fix a realization ω∈Ω' and denote by τ(ω) the first time of branching of the BBM.
    There exists some rank n_0(ω) such that, for any n ≥ n_0(ω), we have t_n+1 - t_n < τ(ω) and then
    
    S_n(ω)    = sup_s ∈ [0, t_n+1 - t_n]|^2β X_∅(as)(ω) - ψ(2β)as W_s + t_n - a(s + t_n)^(∅, as)(β)^2(ω) - W_t_n - at_n(β)^2(ω)| 
       = sup_s ∈ [0, t_n+1 - t_n]|^(2ψ(β) - ψ(2β))as W_s + t_n - at_n(β)^2(ω) - W_t_n - at_n(β)^2(ω)|.

    Since W_t(β)(ω) converges as t →∞, the sequence S_n(ω) converges to 0. 
    Since the event Ω' has probability 1, we deduce that the process S_n converges to 0.
    
    In order to obtain a domination, we bound
    
    S_n ≤sup_s ∈ [0, 1]∑_v ∈(as)^2β X_v(as) - ψ(2β)as W_s + t_n - a(s + t_n)^(v, as)(β)^2 + W_t_n - at_n(β)^2.

    Note that
    
    ∑_v ∈(as)^2β X_v(as) - ψ(2β)as W_s + t_n - a(s + t_n)^(v, as)(β)^2 ≤^(2ψ(β) - ψ(2β))as W_s + t_n - at_n(β)^2.

    Hence,
    
    S_n ≤(sup_s ∈ [0, 1]^(2ψ(β) - ψ(2β))as + 1) sup_s ∈ [0, 1] W_s + t_n - at_n(β)^2.

    By Lemma <ref>, the process W_t(β) is bounded in L^2p for any p ∈ (1, 1/β^2).
    By Doob's maximal inequality, the domination in (<ref>) is in L^p.
    Thus, by dominated convergence, the process S_n converges in L^p to 0.


Let us come back to (<ref>).
By (<ref>), we have

    ∑_u ∈(at_n)^2β X_u(at_n) - ψ(2β)at_nS_n^(u)_at_n = W_at_n(2β) S_n.

By Lemma <ref> and since the additive martingale converges , the above process converges to 0 as n →∞.
To obtain (<ref>), it is then sufficient to show that

    V_n = ∑_u ∈(at_n)^2β X_u(at_n) - ψ(2β)at_n(S_n^(u) - S_n^(u)_at_n)  0,

Let us fix p ∈ (1, 2] such that p < 1/2β^2.
By Lemma <ref>,

    V_n^p_at_n≤ W_at_n(p2β) |S_n - S_n|^p ^-(pψ(2β) - ψ(p2β))at_n.

By Lemma <ref>, the process S_n - S_n converges in L^p to 0.
In particular,

    V_n^p≤ C ^-(pψ(2β) - ψ(p2β))at_n.

By (<ref>) and by the Borel-Cantelli lemma, the convergence (<ref>) holds, which concludes the proof of Theorem <ref>.<ref>.



 §.§ Intermediate temperature


Let us fix β = √(2)/2 and show Theorem <ref>.<ref>.
By (<ref>), we can rewrite

    √(at)^at/2ν_β, t([a, 1]) = √(at) (G_t + R_t) W_t(β)^-2,

where

    G_t = ∑_u ∈(at)^√(2) X_u(at) - 2at W_∞^(u, at)(β)^2   and   R_t = ∑_u ∈(at)^√(2) X_u(at) - 2at(W_t - at^(u, at)(β)^2 - W_∞^(u, at)(β)^2).

Since the additive martingale converges , it suffices to obtain the convergence in probability of √(at)(G_t + R_t) toward √(2/π) Z_∞W_∞(β)^2.
Let us justify that we can ignore the term R_t.
By the triangle inequality and the branching property,

    √(at)|R_t|_at≤√(at) W_at(√(2)) |W_t - at(β)^2 - W_∞(β)^2|.

By (<ref>) and Lemma <ref>, the above process converges in probability to 0 as t →∞, so does √(at) R_t.

It remains to study G_t.
Let us fix θ∈.
We use the conditional characteristic function

    exp( θ√(at) G_t )_at = ∏_u ∈(at)φ(λ^(u, at)),

where

    λ^(u, at) = θ√(at)^√(2) X_u(at) - 2at,   φ(λ) = ^λ W_∞(β)^2 = 1 + λW_∞(β)^2 + r_1(λ),

and r_1(λ) = o(λ) as λ→ 0.
In order to use the above expansion, we need to restrict to an event with high probability where max_u ∈(at)λ^(u, at) converges to 0 as t →∞.
We can choose the event Λ = Λ_δ, at defined in (<ref>), with an arbitrary δ∈ (0,1/√(2)).
On this event, for t large enough and for all u ∈(at), we have φ(λ^(u, at)) ∈∖ (-∞, 0].
This allows us to use the principal complex logarithm  and to write

    exp( θ√(at) G_t )_at1_Λ   = exp( ∑_u ∈(at)(1 + λ^(u, at)W_∞(β)^2 + r_1(λ^(u, at))) ) 1_Λ
       = exp( θ√(at) W_at(√(2)) W_∞(β)^2 + ∑_u ∈(at) r_2(λ^(u, at)) ) 1_Λ,

where r_2(λ) = o(λ) as λ→ 0.
By choice of Λ, with probability 1, we have

    ∑_u ∈(at) r_2(λ^(u, at)) 1_Λ = o ( ∑_u ∈(at)λ^(u, at)) 1_Λ = o ( √(at) W_at(√(2)) ) 1_Λ.

By (<ref>) and (<ref>), with probability 1, we have

    exp( θ( √(at) G_t - √(at) W_at(√(2)) W_∞(β)^2) ) _at1_Λ = exp( o ( √(at) W_at(√(2)) ) ) 1_Λ.

By (<ref>), the above process converges in probability to 1.
This being true for any θ∈, we deduce that √(at) G_t - √(2/π) Z_∞W_∞(β)^2 converges in probability to 0 as t →∞, which concludes the proof of Theorem <ref>.<ref>.




 §.§ Lower temperature


Let β∈ (√(2)/2, √(2)).
Here we follow the techniques from <cit.>, where Iksanov, Kolesko and Meiners computed the fluctuations of the additive martingales of the branching random walk, using the convergence of the extremal process _t.
The main differences are that we have to deal with this convergence jointly with the additive martingale and that we identify a stable distribution at the limit.
Recall that m(t) = √(2)t - 3 · 2^-3/2log t.
We rewrite

    (at)^3 β/√(2)^(√(2) - β)^2 atν_β,t([a, 1]) = H_t W_t(β)^-2,

where

    H_t = ∑_u ∈(at)^2 β (X_u(at) - m(at)) W_t - at^(u, at)(β)^2.

Let ℓ∈, which has here to be thought as a large negative number.
We will use the following continuous approximation of 1_[ℓ, ∞),

    χ_ℓ^+(x) = 
            0    if  x ≤ℓ, 
    
            x - ℓ   if ℓ≤ x ≤ℓ+1, 
    
            1    if  x ≥ℓ+1,

as well as χ_ℓ^- = 1 - χ_ℓ^+.
We decompose

    H_t    = ∑_u ∈(at)^2 β (X_u(at) - m(at))(W_t - at^(u, at)(β)^2 - W_∞^(u, at)(β)^2) 
          + ∑_u ∈(at)^2 β (X_u(at) - m(at)) W_∞^(u, at)(β)^2 χ_ℓ^+(X_u(at) - m(at)) 
          + ∑_u ∈(at)^2 β (X_u(at) - m(at)) W_∞^(u, at)(β)^2 χ_ℓ^-(X_u(at) - m(at)) 
       = R_t, ℓ^1 + H_t, ℓ + R_t, ℓ^2.

Below, Lemma <ref> and Lemma <ref> state that R_t, ℓ^1 and R_t, ℓ^2 are negligible as t →∞ and ℓ→ -∞.
Lemma <ref> gives a first characterization of the limit distribution.


    For all ℓ∈, R_t, ℓ^1 converges in probability to 0 as t →∞.



    We consider p ∈ (0,1] such that √(2)/2β < p < 1/β^2 which is possible because β∈ (√(2)/2, √(2)).
    Using the subadditivity of x ↦ x^p on [0, ∞), we have
    
    |R_t, ℓ^1|^p_at≤|W_t - at(β)^2 - W_∞(β)^2|^p ∑_u ∈(at)^p2β(X_u(at) - m(at)).

    Since p <1/β^2, the expectation on the right-hand side tends to 0 (see Lemma <ref>).
    By (<ref>), the sum on the right-hand side converges in distribution to a finite limit because p 2β > √(2).
    Therefore, by Slutsky's theorem, the above quantity converges in distribution to 0 as t →∞.
    Then, R_t, ℓ^1 converges in probability to 0 as t →∞.



    For any δ > 0, lim_ℓ→ -∞lim sup_t →∞(R_t,ℓ^2 > δ) = 0.



    Note that if a process (X_t, ℓ)_t ≥ 0, ℓ∈ takes its values in [0, ∞), then the following two statements are equivalent
    
        
  *  for any δ > 0, lim_ℓ→ -∞lim sup_t →∞X_t, ℓ≥δ = 0,
        
  *  lim_ℓ→ -∞lim sup_t →∞X_t, ℓ∧ 1 = 0.
    
    We consider p ∈ (0,1] such that √(2)/2β < p < 1/β^2.
    By Jensen's inequality,
    
    (R_t, ℓ^2)^p ∧ 1≤(R_t, ℓ^2)^p_at∧ 1.

    But, using the subadditivity of x ↦ x^p on [0, ∞),
    
    (R_t, ℓ^2)^p_at   ≤W_∞(β)^2p∑_u ∈(at)^p2 β(X_u(at) - m(at))χ_ℓ^-(X_u(at) - m(at))^p 
            = C _at(^p2 β xχ_ℓ^-(x)^p),

    using that W_∞(β) ∈ L^2p because p < 1/β^2 (see Lemma <ref>).
    Since <ref> and <ref> are equivalent, it suffices to obtain that, for any δ > 0,
    
    lim_ℓ→ -∞lim sup_t →∞_at(^p2 β xχ_ℓ^-(x)^p) ≥δ = 0.

    This is proved by Bonnefont in <cit.> (this requires p2β > √(2), which is ensured by our choice of p).



    We have
    
    (at)^3 β/√(2)^(√(2) - β)^2 atν_β, t([a, 1])  X_over = 1/W_∞(β)^2∑_k ≥ 1^2 βξ_k W_k^2,   in distribution,

    where W_k, k ≥ 1, are copies of W_∞(β), independent of W_∞(β) and _∞.



    We first check that X_over is well-defined by showing
    
    ∑_k ≥ 0^2 βξ_k W_k^2 < ∞, 
              almost surely.

    To see this, fix p ∈ (0,1] such that √(2)/2β < p < 1/β^2. 
    Then, by subadditivity,
    
    (∑_k ≥ 1^2 βξ_k W_k^2)^p_∞ 
            = ∑_k ≥ 1^p2 βξ_kW_∞(β)^2p,

    which is finite, by Lemma <ref> and Lemma <ref>, proving (<ref>).

    Now, in order to prove the lemma, by Lemma <ref>, Lemma <ref> and <cit.>, it is sufficient to show that, for all ℓ∈,
    
    H_t, ℓ/W_t(β)^21/W_∞(β)^2∑_k ≥ 1^2 βξ_kχ_ℓ^+(ξ_k) W_k^2,   in distribution,

    and that
    
    1/W_∞(β)^2∑_k ≥ 1^2 βξ_kχ_ℓ^+(ξ_k) W_k^2  X_over,   in distribution.

    Note that (<ref>) is a driect consequence of (<ref>).
    To prove (<ref>), we introduce the test function f(x,y) = ^2β xχ_ℓ^+(x) y^2.
    By (<ref>)
    [Note that f is not bounded as required in (<ref>). However, one can first apply (<ref>) to f ∧ M for some M> 0, and then let M →∞, using that ^*(f) < ∞ by (<ref>).], 
    (_at^*(f), W_at(β)) converges in distribution to (_∞^*(f), W_∞(β)).
    By Slutsky's theorem, the same is true for (_at^*(f), W_t(β)) and hence we get (<ref>).


We are now able to prove Theorem <ref>.<ref>.
By definition of _∞, we can rewrite

    X_over = (C Z_∞)^2β/√(2)/W_∞(β)^2∑_i≥ 1^2β(p_i + X_i),

where X_i = 1/2βlog∑_j ^2 βΔ_ij W_ij^2 and where W_ij are copies of W_∞(β), independent of _∞, , (_i)_i ≥ 1.
Now, since the variables X_i for i ≥ 1 are and independent of , it follows from <cit.> that

    ∑_i ≥ 1δ_p_i + X_id=∑_i ≥ 1δ_p_i + 1/√(2)log^√(2) X_1,

under the condition that [^√(2) X_1] < ∞, which we check as follows: taking p ∈ (√(2)/β, 2/β^2) ∩ [1, 2], using first Jensen's inequality and then subadditivity of x ↦ x^p/2 on [0, ∞), we obtain

    ^√(2) X_1^β p /√(2)≤^β p X_1
        = (∑_j ≥ 1^2 βΔ_1j W_1j^2)^p/2≤∑_j ≥ 1^p βΔ_1jW_∞(β)^p
        < ∞,

where the last quantity is finite by Lemma <ref> and Lemma <ref>.
By (<ref>) and (<ref>),

    X_overd=(C Z_∞^√(2) X_1)^2β/√(2)/W_∞(β)^2∑_i≥ 1^2β p_i.

Finally, by Lemma <ref>, the sum S_2β = ∑_i ^2β p_i is a non-degenerate (√(2)/2β)-stable random variable.
This remark together with Lemma <ref> and (<ref>) conclude the proof of Theorem <ref>.<ref>.




§ PROOF OF THEOREM <REF>



In this section, the key tools will be the changes of measure (<ref>) and (<ref>) at inverse temperature 2β, as well as the associated spinal decomposition.
We denote by Π the point process whose points s_i for i ≥ 1 are the ordered spinal branching times.
Recall that Π is a homogeneous Poisson point process on [0,∞) with intensity 1 under  and 2 under _2β.
We also write s_0 = 0.
In what follows, it will be convenient to decompose the additive martingale along the spine,

    W_t(β) = ∑_i ≥ 1^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) 1_s_i ≤ t + ^β X_ξ(t) - ψ(β)t,

where, if d_u ∧ v denotes the deathtime of the last common ancestor of u and v,

    W_s^(i)(β) = ∑_u ∈(s_i + s) 
     d_u ∧ξ(s_i + s) = s_i^β (X_u(s_i + s) - X_u(s_i)) - ψ(β)s.

Under both probability measures  and _2β, given Π, the processes (W_s^(i)(β))_s ≥ 0 for i ≥ 1 are the additive martingales of independent BBMs without spine.
Therefore, they converge -and _2β-to copies of W_∞(β), that we denote by W_∞^(i)(β).
This allows us to define

    W(β) = ∑_i ≥ 1^β X_ξ(s_i) - ψ(β)s_i W_∞^(i)(β),

which is the limit of W_t(β) under _2β as showed in the following lemma. Note that we distinguish W(β) and W_∞(β) for now, but, once this lemma is proved, we can afterwards say that they are the same quantity, see Remark <ref>.
We also emphasize here that the convergence of W_t(β) is classically studied under _β (see e.g. <cit.>), but we need it here under _2β.


    If β∈ [0, √(2/3)), the random variable W(β) is _2β-positive and finite.
    If β∈ [√(2/3), √(2)), it is _2β-infinite.
    In both cases, it is the _2β-limit of the process W_t(β) as t →∞.



    Let β∈ [0,√(2)).
    Under _2β,
    
    W(β) = ∑_i ≥ 1^β B_s_i + (3β^2/2 - 1) s_i W_∞^(i)(β),

    where B_s = X_ξ(s) - 2 β s is a standard Brownian motion.
    Note that 3β^2/2 - 1 ≥ 0 if and only if β≥√(2/3).
    In particular, if β∈ [√(2/3), √(2)), the (non-negative) terms of the series defining W(β) do not converge to 0 and then W(β) = ∞, _2β-Besides, by Fatou's lemma,
    
    lim inf_t →∞ W_t(β) ≥∑_i ≥ 1lim inf_t →∞^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) 1_s_i ≤ t,

    and this lower bound is _2β-equal to W(β) = ∞.

    Now let β∈ [0, √(2/3)).
    We have
    
    __2β[ W(β) | Π, X_ξ] = ∑_i ≥ 1^β B_s_i - (1 - 3β^2/2) s_i,

    and, _2β-, this series converges since
    
    √(^β B_s_n - (1 - 3β^2/2) s_n)^-(1 - 3β^2/2)/2 < 1.

    Thus, W(β) is _2β-finite.
    Furthermore, it is _2β-positive since the terms of the series (<ref>) are _2β-positive.
    
    It remains to control W(β) - W_t(β).
    We rewrite it
    
    ∑_i ≥ 1 
     s_i ≤ t^β X_ξ(s_i) - ψ(β)s_i (W_∞^(i)(β) - W_t - s_i^(i)(β)) + ∑_i ≥ 1 
     s_i > t^β X_ξ(s_i) - ψ(β)s_i W_∞^(i)(β) - ^β X_ξ(t) - ψ(β)t = T_1(t) + T_2(t) + T_3(t).

    The term T_3(t) converges _2β-to 0 since the exponent β X_ξ(s) - ψ(β)s is a Brownian motion with drift -1 + 3β^2/2 < 0 (and variance β^2).
    The term T_2(t) converges _2β-to 0 since it is the remainder of a series which converges _2β-To control the term T_1(t), we first show that it converges along the sequence of spinal branching times.
    By Lemma <ref>, for any p ∈ [1, 2] and n ≥ 1,
    
    __2β[ |T_1(s_n)|^p  | Π, X_ξ] ≤∑_i = 1^n ^p β X_ξ(s_i) - p ψ(β) s_i f(s_n - s_i),

    where, for s ≥ 0, f(s) = [|W_∞(β) - W_s(β)|^p].
    By Lemma <ref>, we have f(s) ≤ C ^-(p - 1)(1 - pβ^2/2)s. Hence, by (<ref>), the above quantity is bounded by
    
    C ^-(p - 1)(1 - pβ^2/2)s_n∑_i ≥ 1^p β X_ξ(s_i) - ψ(p β) s_i,

    which is _2β-summable in n as soon as p ∈ (1, 2/β^2) to ensure that the exponential prefactor vanishes, and β^2 (2p-p^2/2) < 1 to ensure that the sum is _2β-finite by proceeding as in (<ref>) (such a choice of p exists because β < √(2/3)).
    By the Borel–Cantelli lemma, T_1(s_n) converges _2β-to 0 as n →∞.
    Now let us control
    
    sup_t ∈ (s_n, s_n+1) |T_1(t) - T_1(s_n)| ≤∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_isup_t ∈ (s_n, s_n+1)| W_t - s_i^(i)(β) - W_s_n - s_i^(i)(β) |_= S_n, i.

    Similarly to (<ref>), we proceed by showing that
    
    ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i__2β[ S_n, i| Π]  0,   _2β-,

    and
    
    ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i( S_n, i - __2β[ S_n, i| Π] )  0,   _2β-

    By applying the Cauchy-Schwarz inequality and Doob's maximal inequality, we obtain, for any p ∈ (1,2],
    
    __2β[ S_n, i| Π] 
            ≤__2β[ (S_n, i)^p | Π]^1/p≤p/p-1 f(s_n - s_i, s_n+1 - s_i)^1/p,

    where, for s ≤ t,
    
    f(s, t) = [ | W_t(β) - W_s(β) |^p ] ≤ C ^-(p - 1)(1 - pβ^2/2)s,

    by Lemma <ref>.
    In particular, since p ∈ (1, 2/β^2), each term of the sum in (<ref>) converges _2β-to 0 as n →∞.
    Then, by dominated convergence with domination
    
    ^β X_ξ(s_i) - ψ(β)s_i__2β[ S_n, i| Π] ≤ C ^β X_ξ(s_i) - ψ(β)s_i,

    we obtain (<ref>).
    As for the convergence (<ref>), it can be deduced from Lemma <ref> and (<ref>), by proceeding as in the proof that T_1(s_n) converges _2β-to 0 above.
    Finally, the bound in (<ref>) converges _2β-to 0 as n →∞ and the term T_1(t) converges _2β-to 0 as t →∞.


 
    In the case 0 ≤β < √(2)/2, a consequence of Lemma <ref> is that W(β) = W_∞(β) -and _2β-since the measures  and _2β are equivalent.
    On the other hand, if β∈ [√(2)/2,√(2)), then  and _2β are mutually singular so we can assume that W_∞(β) = W(β) _2β-. 
    Moreover, one can check that W(β) = W_∞(β) -stays true for β∈ [√(2)/2,√(2)) by following an argument similar to the proof of Lemma <ref> in the case β∈ (0,√(2/3)).
    This ensures that there is no notation conflict if we write W_∞(β) instead of W(β) under  or _2β, which we will do from now on.




 §.§ Upper temperature


Let us fix β∈ (0, √(2/3)) and show Theorem <ref>.<ref>. 
Note that the claim concerning the convergence of W_t(β) has already been proved in Lemma <ref> (see also Remark <ref>), so we focus on the convergence of the rescaled overlap distribution.
We define

    F_t = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_t - at^(u, at)(β)^2,

so that ^(1 - β^2)atν_β, t([a, 1]) = F_t W_t(β)^-2, by (<ref>).
We have

    ^(1 - β^2)atν_β, t([a, 1])   = [ F_t W_t(β)^-2 - F_t W_at(β)^-2] + [ F_t W_at(β)^-2] 
       = [ F_t W_t(β)^-2 - F_t W_at(β)^-2] + __2β[ W_at(β)^-2] [ W_t-at(β)^2 ],

using the branching property and the definition of _2β in the second term.
Then, by Lemma <ref> and Lemma <ref> below, together with Lemma <ref>,

    ^(1 - β^2)atν_β, t([a, 1])__2β[ W_∞^-2] W_∞(β)^2,

which is Theorem <ref>.<ref>.


    For every β∈ (0, √(2/3)),
    
    __2β[W_t(β)^-2] __2β[W_∞(β)^-2] ∈ (0, ∞).



 
    The convergence in this lemma stays true for any β∈ [0, √(2)), but for β∉ (0, √(2/3)) the limit is not in (0,∞) anymore:
    
        
  * If β = 0, the fixed point equation (<ref>) with t=∞ and s=1, together with <cit.> implies that __0[W_∞(0)^-2] =  [W_∞(0)^-1] = ∞.
        Then, by Fatou's lemma, __0[W_t(0)^-2] →∞ as t →∞.
        
  * If β∈ [√(2/3), √(2)), Lemma <ref> states that W_t(β) →∞, _2β-, and Lemma <ref> below implies that __2β[W_t(β)^-2] → 0 as t →∞.
    



    Let β∈ (0, √(2/3)) and define F_t as in (<ref>).
    Then,
    
    [ F_t W_t(β)^-2 - F_t W_at(β)^-2]  0.



The remainder of this section is dedicated to the proofs of Lemma <ref> and Lemma <ref>


    Let X_i for i ≥ 1 be independent random variables and a_i, i ≥ 1, be positive coefficients.
    Assume that there exist b, K > 0 such that, for each i ≥ 1 and every x > 0,
    
    X_i ≤ x≤ K x^b.

    Then, for every p ∈ [1, ∞) such that p/b ∉, there exists C = C(b,K,p) >0 such that, letting n = ⌈ p/b ⌉,
    
    ( ∑_i = 1^n a_i X_i )^-p≤ C a_1^-b⋯ a_n - 1^-b a_n^-p + b(n - 1).




    Note that b(n - 1) < p < bn because p/b ∉.
    We can assume that a_1 ≥ a_2 ≥⋯≥ a_n since the general case follows from
    
    a_i^-b a_n^-p + b(n - 1)≤ a_n^-b a_i^-p + b(n - 1).

    For each j = 1, …, n,
    
    ∑_i = 1^n a_i X_i ≤ x≤a_1 X_1 ≤ x⋯a_j X_j ≤ x≤ K^j a_1^-b⋯ a_j^-b x^bj.

    In particular, we can use the following bound
    
    ∑_i = 1^n a_i X_i ≤ x≤
                1    if  x ≥ a_1, 
    
                K^n - 1 a_1^-b⋯ a_n-1^-b x^b(n-1)   if  a_n ≤ x < a_1, 
    
                K^n a_1^-b⋯ a_n^-b x^bn   if  x < a_n.

    This yields
    
    ( ∑_i = 1^n a_i X_i )^-p   = ∫_0^∞∑_i = 1^n a_i X_i ≤ x^-1/px̣
       ≤∫_0^a_1^-px̣ + ∫_a_1^-p^a_n^-p K^n - 1 x^-b(n-1)/p a_1^-b⋯ a_n-1^-bx̣ + ∫_a_n^-p^∞ K^n x^-bn/p a_1^-b⋯ a_n^-bx̣
       ≤ a_1^-p + C a_1^-b⋯ a_n - 1^-b a_n^-p + b(n-1),

    using b(n-1)/p<1 in the second integral and bn/p >1 in the third one.
    To conclude, it suffices to note that a_1^-p≤ a_1^-b… a_n-1^-b a_n^-p + b(n-1).




    Let β∈ [0, √(2)) and p ∈ [1, ∞).
    There exist n_0≥ 1 and C > 0 such that, for any n ≥ n_0, any I = {i_1, …, i_n}⊂{1, 2, …} with i_1 < … < i_n and any t ∈ [0, ∞], on the event where s_i_n≤ t, we have
    
    __2β[ ( ∑_i ∈ I^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-p| Π] ≤ C ^(p^2β^2/2 + p(1 - 3β^2/2))s_i_n.




    Let b_max = (√(ψ(β)^2 + 2β^2) - ψ(β))/β^2 (recall it appears in the condition (<ref>) of Lemma <ref>).
    We choose n_0 > p/b_max.
    Now note that it is enough to prove (<ref>) for n=n_0: indeed, adding elements to I can only decrease the left-hand side of (<ref>) and increase the right-hand side (note that p^2β^2/2 + p(1 - 3β^2/2) ≥ 0).    
    So let n = n_0.
    Since n > p/b_max, there exists b ∈ (0,b_max) such that n = ⌈ p/b ⌉ and p/b ∉.
    Then, by Lemma <ref> and Lemma <ref>,
    
    __2β[ ( ∑_i ∈ I^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-p| Π, X_ξ]    ≤ C ( ∏_i ∈ I ∖{i_n}^β X_ξ(s_i) - ψ(β)s_i)^-b( ^β X_ξ(s_i_n) - ψ(β)s_i_n)^-p + b(n-1)
       = C ∏_j = 1^i_n( ^β (X_ξ(s_j) - X_ξ(s_j-1)) - ψ(β)(s_j - s_j-1))^-α_j,

    with α_j = p - b #(I ∩{1, …, j-1}).
    Recall that, under _2β, the spine trajectory X_ξ is a Brownian motion with drift 2β, independent of Π.
    Then,
    
    __2β[ ( ∑_i ∈ I^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-p| Π]    ≤ C ∏_j = 1^i_n^(α_j^2 β^2/2 + α_j (1 - 3β^2/2))(s_j - s_j - 1)
       ≤ C ∏_j = 1^i_n^(p^2 β^2/2 + p (1 - 3β^2/2))(s_j - s_j - 1),

    using in the last inequality that α_j ∈ [0,p] and that the function x ∈ [0,p] ↦ x^2 β^2/2 + x (1 - 3β^2/2) reaches its maximum at x=p, because β <√(2) and p≥ 1.
    This gives the desired inequality by telescoping.



    For every β∈ (0, √(2)), the process (W_t(β)^-2)_t ≥ 0 is bounded in L^p(_2β) for some p > 1.




    Let β∈ (0, √(2)) and p > 1, which will be chosen close enough to 1 depending on β later on.
    Let n ≥ 1 be given by Lemma <ref> applied to β and p.
    Let t ≥ 0 and set A = A_n, t = {s_n ≤ t}.
    On the one hand, 
    
    __2β[ W_t(β)^-2p1_A ] ≤__2β[ ( ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-2p1_A ] ≤ C __2β[ ^(2p^2β^2 + 2p(1 - 3β^2/2)) s_n],

    by Lemma <ref>.
    If we choose p close enough to 1, we have 2p^2β^2 + 2p(1 - 3β^2/2) < 2 and then the above bound is finite (recall s_n is Γ(n,2) distributed under _2β).
    On the other hand, keeping only the contribution of the spine to W_t(β), we have
    
    __2β[ W_t(β)^-2p1_A^c] ≤__2β[ ^-2p β X_ξ(t) + 2p ψ(β)t] _2β(s_n > t) 
            ≤^(2 - 3β^2 + 2pβ^2)pt C t^n-1^-2t.

    If we choose p close enough to 1, we have 2 - 3β^2 + 2pβ^2 < 0 and the above quantity converges to 0 as t →∞.
    Combining the two previous bounds shows the result.



    By Lemma <ref>, for every β∈ (0, √(2/3)), the random variable W_∞(β) is _2β-positive and finite.
    In particular, __2β[W_∞(β)^-2] > 0.
    Besides, W_t(β)^-2 converges _2β-to 
    W_∞(β)^-2 as t →∞ and is uniformly integrable for _2β, by Lemma <ref>.
    This implies that
    
    __2β[W_t(β)^-2] __2β[W_∞(β)^-2] < ∞,

    which concludes the proof.


It remains to prove Lemma <ref>.
We will use the following concentration inequality.


    Let β∈ [0, 1) and p ∈ [1, 2]. There exists C>0 such that, for any 0 ≤ s ≤ t ≤∞,
    
    | W_t(β) - W_s(β)/W_s(β)|^p≤ C ^-(1 - β^2)ps/2.




    We have, using decomposition (<ref>),
    
    (W_t(β) - W_s(β))^2 
        = ∑_u ∈(s)^2β X_u(s) - 2ψ(β)s (W_t - s^(u, s)(β) - 1)^2 + ∑_u, v ∈(s) 
     u ≠ v^β(X_u(s) + X_v(s)) - 2ψ(β)s (W_t - s^(u, s)(β) - 1)(W_t - s^(v, s)(β) - 1),

    where, conditionally on _s, the variables W_t - s^(u,s)(β) for u ∈(s) are independent with mean 1 and are bounded in L^2() because β <1 (see Lemma <ref>).
    Then, by applying successively Hölder's inequality and taking a conditional expectation given _s, we obtain
    
    | W_t(β) - W_s(β)/W_s(β)|^p≤(W_t(β) - W_s(β))^2/W_s(β)^2^p/2≤ C ^-(1 - β^2)s W_s(2β)/W_s(β)^2^p/2,

    since ψ(2β) - 2ψ(β) = -(1 - β^2) and since the additive martingale is bounded in L^2.
    The result follows from Lemma <ref>, which implies that [W_s(2β) W_s(β)^-2] = __2β[W_s(β)^-2] converges to a finite limit and hence is bounded.



    Let β∈ (0, √(2/3)).
    In order to take advantage of Lemma <ref>, we will work on the event A = A_n, a't = {|ξ(a't)| ≥ n}, where a' ∈ (0, a) is “close enough” to a and n ≥ 1 is “large enough”.
    However, in view of applying Lemma <ref>, it will be convenient to deal with a slightly smaller event that is measurable w.r.t. _∞.
    Let us define A̅ = A̅_n, a't = {∀ u ∈(a't), |u| ≥ n}.
    
    First, let us show that
    
    [ (F_t W_at(β)^-2 - F_t W_t(β)^-2) 1_A̅^c]  0.

    By conditioning on _at, we obtain
    
    [F_t W_at(β)^-21_A̅^c] = [W_at(2β) W_at(β)^-21_A̅^c] = __2β[W_at(β)^-21_A̅^c].

    Note that 1_A̅^c converges to 0 as t →∞.
    In addition, a direct consequence of Lemma <ref>, Lemma <ref> and Scheffé's lemma is that W_at(β)^-2 converges in L^1(_2β).
    In particular, this process is uniformly integrable, and so is W_at(β)^-21_A̅^c. Hence, the right-hand side of (<ref>) converges to 0 as t →∞.
    Besides,
    
    [F_t W_t(β)^-21_A̅^c] = ^(1 - β^2)at[ ν_β, t([a, 1]) 1_A̅^c] ≤^(1 - β^2)at(A̅^c).

    Applying the union bound, the change of measure (<ref>) and Lemma <ref>, we obtain
    
    (A̅^c) ≤∑_u∈(a't) 1_{|u| ≤ n} = __0[ ∑_u ∈(a't)1_{|u| ≤ n}/W_a't(0)] = e^a't_0(|ξ(a't)| ≤ n) ≤ C t^n e^-a' t,

    using in the last inequality that, under _0, the spinal generation |ξ(s)| has Poisson distribution with parameter 2s.
    Then,
    
    [F_t W_t(β)^-21_A̅^c] ≤ C t^n ^(1 - β^2)at - a't.

    By choice of a', this bound converges to 0 as t →∞.
    Hence (<ref>).
    
    Now, let us show that
    
    [(F_t W_at(β)^-2 - F_t W_t(β)^-2) 1_A̅]  0.

    Recalling that _2β, at|__∞ has Radon–Nykodim derivative W_at(2β) w.r.t. |__∞, the left-hand side of  (<ref>) equals
    
    __2β, at[ F_t/W_at(2β) (W_t(β) - W_at(β)) ( 1/W_t W_at(β)^2 + 1/W_t(β)^2 W_at(β)) 1_A̅] 
    
            = __2β, at[ W_t - at^(ξ(at), at)(β)^2 (W_t(β) - W_at(β)) ( 1/W_t(β) W_at(β)^2 + 1/W_t(β)^2 W_at(β)) 1_A̅],

    where we applied Lemma <ref> to rewrite
    
    F_t/W_at(2β) = ∑_u ∈(at)_2β, at( ξ(at) = u | _t ) W_t - at^(u, at)(β)^2.

    Let us abbreviate W_s^(ξ(at), at)(β) = W_s^(ξ, at)(β).
    Decomposing the process |W_t(β) - W_at(β)| along the spine and applying the triangle inequality, we bound it by
    
    ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i |W_t - s_i^(i)(β) - W_at - s_i^(i)(β)| + | ∑_i > n 
     s_i ≤ at^β X_ξ(s_i) - ψ(β)s_i (W_t - s_i^(i)(β) - W_at - s_i^(i)(β)) | 
    
            + ^β X_ξ(at) - ψ(β)at |W_t - at^(ξ, at)(β) - 1| = T_1(t) + T_2(t) + T_3(t).

    Besides,
    
    1/W_t(β) W_at(β)^2 + 1/W_t(β)^2 W_at(β)≤1/W_at(β)^3 + 2/W_t(β)^2 W_at(β).

    Therefore, and since A̅⊂ A, to obtain (<ref>), it suffices to show that, for i = 1, 2 and γ = 1, a,
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_i(t)/W_γ t(β)^2 W_at(β)1_A ]  0.


    Control of T_1.
    Let us show (<ref>) for i = 1.
    Let γ = 1  or  a.
    We have
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_1(t)/W_γ t(β)^2 W_at(β)1_A ] ≤[ W_t - at(β)^2 ] 
    ×∑_i = 1^n __2β[ ^β X_ξ(s_i) - ψ(β)s_i |W_t - s_i^(i)(β) - W_at - s_i^(i)(β)|/W_at(β)( ∑_j = 1 
     j ≠ i^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ].

    Let us denote by V_1^(i)(t) the process appearing in the last expectation.
    Since the additive martingale is bounded in L^2(), it suffices to get the convergence in mean of V_1^(i)(t) to 0 as t →∞, for i = 1, …, n.
    Using that W_at(β) ≥^β X_ξ(s_i) - ψ(β)s_i W_at - s_i^(i)(β) and then applying Lemma <ref>, we obtain
    
    __2 β[ V_1^(i)(t) ]    ≤__2 β[ |W_t - s_i^(i)(β) - W_at - s_i^(i)(β)|/W_at - s_i^(i)(β)( ∑_j = 1 
     j ≠ i^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ] 
       ≤ C __2 β[ ^-(1-β^2)(at-s_i)/2( ∑_j = 1 
     j ≠ i^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ].

    We can assume n ≥ n_0+1 where n_0 is given by Lemma <ref> applied to β and p = 2.
    This yields
    
    __2 β[ V_1^(i)(t) ] ≤ C __2 β[ ^-(1-β^2)(at-s_i)/2^(2 - β^2) s_n1_A ].

    By definition of the event A, we have s_i ≤ a't < at.
    The above expectation is then bounded by
    
    C ^-(1-β^2)(a-a')t/2__2 β[ ^(2 - β^2) s_n]  0,

    since s_n is Γ(n, 2) distributed under _2β and 2-β^2 < 2.
    Hence (<ref>) for i = 1.

    Control of T_2.
    Let us show (<ref>) for i = 2.
    Let γ = 1  or  a.
    We have
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_2(t)/W_γ t(β)^2 W_at(β)1_A ] ≤[ W_t - at(β)^2 ] 
    ×__2β[ | ∑_i > n 
     s_i ≤ at^β X_ξ(s_i) - ψ(β)s_i (W_t - s_i^(i)(β) - W_at - s_i^(i)(β))/W_at(β)| ( ∑_j = 1^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ].

    Let us denote by V_2(t) the process appearing in the last expectation and show that it converges in mean to 0 as t →∞.
    To do so, we show that it converges _2β-to 0 and that it is bounded in L^p(_2β) for some p > 1.
    
    By Lemma <ref>, the processes W_at(β) and W_γ t - s_j^(j)(β) for j ≥ 1 converge _2β-to some positive limits as t →∞.
    Then, the _2β-convergence of V_2(t) to 0 will follow from
    
    ∑_i ≥ 1^β X_ξ(s_i) - ψ(β)s_i (W_t - s_i^(i)(β) - W_at - s_i^(i)(β)) 1_s_i ≤ at 0,   _2β-

    Each term of the above series converges _2β-to 0 and is dominated by 2 ^β X_ξ(s_i) - ψ(β)s_isup_t ≥ 0 W_t^(i)(β).
    By Doob's maximal inequality, this domination is summable.
    Hence (<ref>).
    
    Now fix p ∈ (1, 2) such that α_p = 2p^2β^2 + 2p(1 - 3β^2/2) < 2.
    We can assume n ≥ n_0 where n_0 is given by Lemma <ref> applied to β and 2p.
    Using that W_at(β) ≥^β X_ξ(s_i) - ψ(β)s_i W_at - s_i^(i)(β), applying Lemmas <ref> together with Lemma <ref>, and finally Lemma <ref>, we obtain
    
    __2 β[ V_2(t)^p ]    ≤__2 β[ | ∑_i > n 
     s_i ≤ atW_t - s_i^(i)(β) - W_at - s_i^(i)(β)/W_at - s_i^(i)(β)|^p ( ∑_j = 1^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-2p1_A ] 
       ≤ C __2 β[ ∑_i > n 
     s_i ≤ at__2 β[ | W_t - s_i^(i)(β) - W_at - s_i^(i)(β)/W_at - s_i^(i)(β)|^p | Π] ^α_p s_n1_A ] 
       ≤ C __2 β[ ∑_i > n 
     s_i ≤ at^-(1 - β^2)p(at - s_i)/2^α_p s_n1_A ].

    We then decompose the sum over the spinal branching times located in [0, a't] and those in (a't, at].
    First,
    
    __2 β[ ∑_i > n 
     s_i ≤ a't^-(1 - β^2)p(at - s_i)/2^α_p s_n1_A ] ≤^-(1 - β^2)p(a - a')t/2__2 β[ #(Π∩ [0, a't]) ^α_p s_n].

    Note that #(Π∩ [0, a't]) has Poisson distribution with parameter 2a't and then its moment of order k ≥ 1 is an O(t^k).
    Thanks to Hölder's inequality, we see that the last expectation factor is an O(t) since α_p < 2.
    In particular, the above quantity is bounded.
    Now, by independence,
    
    __2 β[ ∑_i > n 
     a't < s_i ≤ at^-(1 - β^2)p(at - s_i)/2^α_p s_n1_A ]    = __2 β[ ∑_a't < s_i ≤ at^-(1 - β^2)p(at - s_i)/2] __2 β[ ^α_p s_n1_A ] 
       ≤( ∫_a't^at^-(1 - β^2)p(at - s)/2 2ṣ) __2 β[ ^α_p s_n],

    which is also bounded since α_p < 2.
    Finally, the process V_2(t) is bounded in L^p(_2β) and then its convergence to 0 implies its convergence in mean.
    Hence (<ref>) for i = 2.

    Control of T_3.
    Let us show (<ref>) for i = 3.
    Let γ = 1  or  a.
    We have
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_3(t)/W_γ t(β)^2 W_at(β)1_A ] 
    ≤| W_t - at(β)^3 - W_t - at(β)^2 | __2β[ ^β X_ξ(at) - ψ(β) at/W_at(β)( ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i W_γ t - s_i^(i)(β) )^-21_A ].

    Let us denote by V_3(t) the process appearing in the last expectation.
    Since the additive martingale is bounded in L^3(), it suffices to show that V_3(t) converges in mean to 0 as t →∞.
    But it converges _2β-to 0 since
    
        
  * the processes W_at(β) and W_γ t - s_i^(i)(β) for i ≥ 1 converge to some positive limits as t →∞, by Lemma <ref>,
        
  * β X_ξ(s) - ψ(β)s = β B_s - (1 - 3β^2/2)s, where B_s is a standard Brownian motion, and 1 - 3β^2/2>0.
    
    In addition, the process V_3(t) is bounded in L^p(_2β).
    Indeed, noting that ^β X_ξ(at) - ψ(β) at≤ W_at(β),
    
    __2β[ V_3(t)^p ] ≤__2β[ ( ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i W_γ t - s_i^(i)(β) )^-2p],

    and we see that the right-hand side is bounded thanks to Lemma <ref>.
    Finally, the process V_3(t) converges in mean to 0.
    Hence (<ref>) for i = 3.
    This concludes the proof of Lemma <ref>.




 §.§ Lower temperature


We aim here at proving Theorem <ref>.<ref>, which covers the case β∈ (√(2/3), √(2)).
We tackle separately the lower bound and the upper bound.



  §.§.§ Lower bound


Our goal is to find a constant c > 0 such that, for every t ≥ 1,

    ν_β, t([a, 1])≥ c t^-3/2^-(2 - β^2)^2 at/8β^2.

It will be useful for the proof of Theorem <ref>.<ref> to note that the arguments below work for β = √(2/3) as well, so we only assume β∈ [√(2/3), √(2)).

We can restrict ourselves to the event Γ = Γ_at = {sup_s ∈ [0, at] (X_ξ(s) - v(β) s + f_at(s)) ≤ 1}, where v(β) = 1/β + β/2, f_t(s) = s^α∧ (t-s)^α and α is an arbitrary number in (0, 1/2).
By (<ref>), the definition of _2β,at, and then Lemma <ref>,

    ν_β, t([a, 1])   = ^(β^2 - 1) at__2β, at[ 1/W_t(β)^2∑_u ∈(at)^2β X_u(at) - ψ(2β)at/W_at(2β) W_t - at^(u, at)(β)^2 ] 
       = ^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-2] ≥^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_Γ].

Then, to prove (<ref>), it suffices to get

    __2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_Γ] ≥ c t^-3/2^-(2β - v(β))^2at/2.

Decomposing W_t(β) along the spine, we can rewrite the left-hand side of (<ref>) as

    __2β, at[ W_t - at^(ξ, at)(β)^2 (U_t + V_t W_t - at^(ξ, at)(β))^-21_Γ],

where

    U_t = ∑_i ≥ 1 
     s_i ≤ at^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β)   and   V_t = ^β X_ξ(at) - ψ(β)at.

For all U, V, W > 0,

    W^2 (U + V W)^-2≥ W^2 (V + V W)^-2∧ W^2 (U + U W)^-2≥ W^2 (1 + W)^-2 (U + V)^-2.

Thus, the left-hand side of (<ref>) admits the following lower bound,

    [ W_t - at(β)^2 (1 + W_t - at(β))^-2] __2β[ (U_t + V_t)^-21_Γ] ≥[ W_∞(β)^2 (1 + W_∞(β))^-2] __2β[ (U_t + V_t)^-21_Γ],

by Fatou's lemma.
Since W_∞(β) > 0 , to prove (<ref>), it suffices to get

    __2β[ (U_t + V_t)^-21_Γ] ≥ c t^-3/2^-(2β - v(β))^2at/2.

On the event Γ, since β v(β) = ψ(β),

    U_t ≤^β∑_i ≥ 1 
     s_i ≤ at^-β f_at(s_i) W_t - s_i^(i)(β)   and   V_t ≤^β.

Then,

    __2β[ (U_t + V_t)^-21_Γ] ≥^-2 β__2β[ ( ∑_i ≥ 1 
     s_i ≤ at^-β f_at(s_i) W_t - s_i^(i)(β) + 1 )^-2] _2β( Γ).

By Jensen's inequality and recalling the s_i's are the points of a PPP(2 ṣ),

    __2β[ ( ∑_i ≥ 1 
     s_i ≤ at^-β f_at(s_i) W_t - s_i^(i)(β) + 1 )^-2] ≥( ∫_0^at^-β f_at(s) 2 ṣ + 1 )^-2≥ c > 0.

Besides, recalling that, under _2β, (X_ξ(s))_s≥ 0 is a Brownian motion with drift 2β, it follows from Girsanov theorem that

    _2β( Γ) 
        = [ ^(2β - v(β))B_at-(2β - v(β))^2 at/21_{sup_s ∈ [0, at] (B_s + f_at(s)) ≤ 1}],

where (B_s)_s≥ 0 denotes a standard Brownian motion.
We then restrict ourselves to the event { B_at∈ [0,1] } and use that 2β - v(β) ≥ 0 to get

    _2β( Γ) 
        ≥^-(2β - v(β))^2 at/2( sup_s ∈ [0, at] (B_s + f_at(s)) ≤ 1, B_at∈ [0, 1] ) 
        ≥ c ^-(2β - v(β))^2 at/2 t^-3/2,

by (<ref>).
By (<ref>), (<ref>) and (<ref>), the inequality (<ref>) holds.
Hence, it proves (<ref>) and the lower bound in Theorem <ref>.<ref>.



  §.§.§ Upper bound


Our goal is to show that

    ν_β, t([a, 1]) = O(t^-3/2^-(2 - β^2)^2 at/8β^2).

We work with β∈ [√(2/3),√(2)), except in the last step where we specify that β>√(2/3), so that the beginning of the argument can be used to prove the upper bound in Theorem <ref>.<ref> as well.

Consider an arbitrary integer n ≥ 1 and define A̅ = A̅_2n, at = {∀ u ∈(at), |u| ≥ 2n}.
As seen in (<ref>), we have

    ν_β, t([a, 1]) 1_A̅^c≤A̅^c≤ C t^2n-1^-at = o(t^-3/2^-(2 - β^2)^2 at/8β^2),

using that (2 - β^2)^2/8β^2 ≤ 1/3 < 1 for β∈ [√(2/3), √(2)).
Concerning the contribution of the event A̅, we can rewrite, similarly to (<ref>),

    ν_β, t([a, 1]) 1_A̅ = ^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_A̅] ≤^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_A ],

where A = A_2n, at = {|ξ(at)| ≥ 2n}.
By (<ref>) and (<ref>), to prove (<ref>), it suffices to get

    __2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_A ] = O(t^-3/2^-(2β - v(β))^2 at/2),

where v(β) = 1/β + β/2.
Then we decompose W_t(β) = U_t + V_t W_t - at^(ξ, at)(β) with U_t and V_t defined in (<ref>). We also change measures, first using that _2β, at has density ^2β X_ξ(at)-2β^2at w.r.t. _0, at, and then applying Girsanov theorem under _0, at to trade the factor ^v(β)X_ξ(at)-v(β)^2at/2 for a drift v(β) added to the spine trajectory, so that U_t and V_t become (recall ψ(β) = β v(β))

    U_t' = ∑_i ≥ 1 
     s_i ≤ at^β X_ξ(s_i) W_t - s_i^(i)(β)   and   V_t' = ^β X_ξ(at).

Therefore, we get that (<ref>) is equivalent to

    __0, at[ ^(2β - v(β)) X_ξ(at) W_t - at^(ξ, at)(β)^2 (U_t' + V_t' W_t - at^(ξ, at)(β))^-21_A ] = O(t^-3/2).


For all U, V, W > 0 and any α∈ [0, 2],

    W^2 (U + V W)^-2≤ W^2 U^-α (V W)^-2 + α = U^-α V^-2 + α W^α.

Thus, the left-hand side of (<ref>) is bounded by

    __0, at[ ^(2β - v(β))X_ξ(at) (U_t')^-α (V_t')^-2 + α W_t - at^(ξ, at)(β)^α1_A ] = [ W_t - at(β)^α] __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A ].

Let α∈ [0, 2] ∩ [1, 2/β^2) so that the additive martingale is bounded in L^α() by Lemma <ref>.
To prove (<ref>), it suffices to get

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A ] = O(t^-3/2).

Now, decompose the process over the following events, indexed by integers k, ℓ≥ 0,

    E_k, ℓ = E_k, ℓ(X_ξ(s), 0 ≤ s ≤ at) = {sup_[0, at] X_ξ∈ [k, k+1], sup_[0, at] X_ξ - X_ξ(at) ∈ [ℓ, ℓ+1] }.

Fix k, ℓ and denote s_k, n the n-th point of Π after time τ_k = inf{s ≥ 0 : X_ξ(s) = k}.
Let us first treat the case s_k, n≤ at.
On the event E_k, ℓ∩{s_k, n≤ at},

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α| Π, X_ξ] 
    ≤ C ^-v(β)k - (αβ - v(β))ℓ__0[ ( ∑_i ≥ 1 
    τ_k ≤ s_i ≤ s_k, n^β(X_ξ(s_i) - X_ξ(τ_k)) W_t - s_i^(i)(β) )^-α| Π, X_ξ] 
    ≤ C ^-v(β)k - (αβ - v(β))ℓ^αβsup_[τ_k, s_k, n] (X_ξ(τ_k) - X_ξ)__0[ ( ∑_i ≥ 1 
    τ_k ≤ s_i ≤ s_k, n W_t - s_i^(i)(β) )^-α| Π, X_ξ].

By Lemma <ref> and Lemma <ref>, we can choose n so that the last conditional expectation is bounded, which yields

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A 1_E_k, ℓ∩{s_k, n≤ at}] 
    ≤ C ^-v(β)k - (αβ - v(β))ℓ__0[ ^αβsup_[τ_k, s_k, n] (X_ξ(τ_k) - X_ξ)1_E_k, ℓ∩{s_k, n≤ at}].

By using time reversal invariance of Brownian motion and Poisson point process, we can treat the case s_k, n > at similarly.
Indeed, on the event A, if s_k, n > at, then there are at least n spinal branching times between 0 and τ_k.
Define s_i' = at - s_i, Π' the corresponding point process and X_ξ'(s) = X_ξ(at - s) - X_ξ(at). Denote s_ℓ, n' the n-th point of Π' after time τ_ℓ' = inf{s ≥ 0 : X_ξ'(s) = ℓ} if it exists, and set s_ℓ, n' = +∞ otherwise.
Note that, on E_k,ℓ, we have τ_k ≤ at-τ_ℓ', so on the event E_k,ℓ∩ A ∩{ s_k, n > at }, there are at least n point of Π' after τ_ℓ', so we get

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A 1_E_k, ℓ∩{s_k, n > at}] 
       ≤__0[ ^-(αβ - v(β)) X_ξ'(at)( ∑_i ≥ 1 
    τ_ℓ' ≤ s_i' ≤ s_ℓ, n'^β(X_ξ'(s_i') - X_ξ'(at)) W_t - (at - s_i')^(i)(β) )^-α1_E_ℓ, k(X_ξ'(s), 0 ≤ s ≤ at) ∩{s_ℓ, n' ≤ at}] 
       = __0[ ^-(αβ - v(β)) X_ξ(at)( ∑_i = 1^n ^β(X_ξ(s_ℓ, i) - X_ξ(at)) W_t - (at - s_ℓ, i)^(i)(β) )^-α1_E_ℓ, k(X_ξ(s), 0 ≤ s ≤ at) ∩{s_ℓ, n≤ at}] 
       ≤ C ^-v(β)k -(αβ - v(β))ℓ__0[ ^αβsup_[τ_ℓ, s_ℓ, n] (X_ξ(τ_ℓ) - X_ξ)1_E_ℓ, k∩{s_ℓ, n≤ at}],

by Lemma <ref> and Lemma <ref> as before.
Using α≤ 2 if β<1 and α < 2/β^2 otherwise, we have αβ < 2, which allows us to apply Lemma <ref> below.
This together with (<ref>) and (<ref>) lead to

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A ] ≤ C t^-3/2∑_k = 0^∞∑_ℓ = 0^∞^-v(β)k - (αβ - v(β))ℓ (k + ℓ + 1) ^- c ℓ^2/t.

We finally assume that β∈ (√(2/3),√(2)), so that v(β)/β < 2 ∧ 2/β^2 and we can thus choose α > v(β)/β. This ensures that the summand in (<ref>) decreases exponentially fast in ℓ, and be can simply bound ^- c ℓ^2/t≤ 1.
Then, the double sum in (<ref>) is a O(1), which proves (<ref>) and therefore (<ref>).
This concludes the proof of Theorem <ref>.<ref>.


    For any γ∈ [0,2), there exist C, c > 0 such that, for all t ≥ 1 and k, ℓ≥ 0,
    
    __0[ ^γsup_[τ_k, s_k, n] (X_ξ(τ_k) - X_ξ)1_E_k, ℓ(X_ξ(s), 0 ≤ s ≤ t)1_{s_k, n≤ t}] ≤ C (k + ℓ + 1) t^-3/2^-c k^2/t - c ℓ^2/t.




    The exponential factor in the bound (<ref>) has not been used to prove (<ref>), because we bounded the factor ^- c ℓ^2/t in (<ref>) by 1.
    However, it will play a key role in the proof of the upper bound for Theorem <ref>.<ref>, which is also based on (<ref>).


To prove Lemma <ref>, we will need the following intermediate lemma.


    Let σ_n be a random variable with distribution Γ(n, 2) and (B_t)_t ≥ 0 be a standard Brownian motion independent of σ_n.
    For any γ < 2, there exist C, c > 0 such that, for any t > 0,
    
    [ ^γsup_[0, σ_n] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1], σ_n ≤ t}] ≤
                C (t + 1)^-3/2   if ℓ = 0  or  1, 
    
                C ℓ t^-3/2^-c ℓ^2/t   if ℓ≥ 2.




    First consider ℓ = 0 or 1.
    The proof of the case ℓ≥ 2 will be similar although more technical.
    We have
    
    [ ^γsup_[0, σ_n] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1], σ_n ≤ t}] = C ∫_0^t [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ.

    Bounding the indicator function by 1 and using [e^γsup_[0,s] B] ≤ 2 ^γ^2s/2, we obtain
    
    ∫_t/2^t [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ   ≤ C ∫_t/2^t s^n - 1^-(2 - γ^2/2)sṣ
       ≤ C t^n ^-(2 - γ^2/2)t/2.

    Concerning the integration over s ∈ [0, t/2], we use Markov property at time s and Lemma <ref> (recall ℓ = 0 or 1) to get 
    
    [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] 
               ≤[ ^γsup_[0, s] B1_{B_s ≥ -1}( C (B_s + 2) (t - s)^-3/2∧ 1 ) ] 
       ≤ C [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 2) (t - s + 1)^-3/2].

    Note that (t - s + 1)^-3/2≤ C(t + 1)^-3/2 since s ≤ t/2.
    Moreover, (B_s, sup_[0, s] B) has the following density (see <cit.> or <cit.>),
    
    (x, y) ↦√(2/π) s^-3/2 (2y - x) ^-(2y - x)^2/2s1_x ≤ y, 0  ≤ yx̣ỵ.

    We deduce that, up to a multiplicative constant, (<ref>) is bounded by
    
    1/(t + 1)^3/2 s^3/2∫_0^∞ (y + 1)^2 ^γ y∫_-1^y ^-(2y - x)^2/2sx̣ỵ≤C/(t + 1)^3/2 s∫_-∞^∞ (y + 1)^2 ^γ y - y^2/2sỵ = P(s) ^γ^2 s/2/(t + 1)^3/2√(s),

    for some polynomial P.
    By integrating this against s^n-1^-2s1_s ∈ [0, t/2]ṣ, we obtain
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C (t + 1)^-3/2.

    By (<ref>) and (<ref>), the bound (<ref>) holds for ℓ = 0 or 1.

    Now consider ℓ≥ 2.
    By Hölder's inequality, for all p, q > 1 such that 1/p + 1/q = 1,
    
    [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] ≤[ ^p γsup_[0, s] B]^1/pB_t ≥ℓ - 1^1/q≤ C ^p γ^2 s/2^-(ℓ - 1)^2/2qt.

    Taking p close enough to 1 so that pγ^2 < 4 and integrating (<ref>) against s^n-1^-2s1_s ∈ [t/2, t]ṣ, we obtain
    
    ∫_t/2^t [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ^-c t - c ℓ^2/t.

    The integration over s ∈ [0, t/2] is more intricate.
    If t ≤ 2, then (<ref>) implies that
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ^-cℓ^2/t≤ C ℓ (t/2)^-3/2^-cℓ^2/t.

    Now assume that t > 2 so that t - s > 1.
    By using Markov property and Lemma <ref>,
    
    [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] 
    ≤ C [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 1) ℓ (t - s)^-3/2( ^-(B_s - ℓ - 1)^2/2(t-s) + ^-(B_s - ℓ + 1)^2/2(t-s)) ].

    Let ℓ' = ℓ± 1.
    Since (B_s, sup_[0, s] B) has density (<ref>),
    
    [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 1) ^-(B_s - ℓ')^2/2(t-s)] 
    ≤C/s^3/2∫_0^∞ (y + 1)^2 ^γ y∫_-1^y ^-(x - ℓ')^2/2(t-s) - (2y - x)^2/2sx̣ỵ.

    The function x ↦√(t/2 π s(t - s))^-(x - ℓ')^2/2(t-s) - (2y - x)^2/2s is the density at time s of a Brownian bridge from (0, 2y) to (t, ℓ').
    It achieves its maximum at x_max = 2y (t-s)/t + ℓ' s/t, which is larger than y since ℓ' > 0 and s ≤ t/2.
    We deduce that (<ref>) is bounded by
    
    C/s∫_0^∞ (y + 1)^2 ^γ y - (y - ℓ')^2/2(t-s) - y^2/2sỵ = C/s^γ^2 s (t-s)/2t + γℓ' s/t - ℓ'^2/2t∫_0^∞ (y + 1)^2 ^-(y - y_max)^2 t/2s(t-s)ỵ,

    where y_max = ℓ' s/t + γ s(t-s)/t.
    Subsequently,
    
    [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 1) ^-(B_s - ℓ')^2/2(t-s)]    ≤ C (y_max + 1)^2 + s(t-s)/t/√(s)^γ^2 s (t-s)/2t + γℓ' s/t - ℓ'^2/2t
       ≤P(s)/√(s)^γ^2 s/2 + γℓ' s/t - cℓ'^2/t,

    for some polynomial P, where we have used that (ℓ' s/t)^2 ^-ℓ'^2/2t≤ s (ℓ'^2/t) ^-ℓ'^2/2t≤ C s ^-c ℓ'^2/t.
    Hence,
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{B_s ≥ -1}(B_s + 1) ℓ/(t - s)^3/2^-(B_s - ℓ')^2/2(t-s)] s^n - 1^-2sṣ
    ≤C ℓ/t^3/2^-c ℓ'^2/t∫_0^t/2P(s) s^n - 1/√(s)^(-2 + γ^2/2 + γℓ'/t)sṣ,

    Let ε∈ (0, 1).
    
        
  * If t ≥ (1 + ε) 2 γ/4 - γ^2ℓ', then
        
    -2 + γ^2/2 + γℓ'/t≤-4 + γ^2/2 + 4 - γ^2/2 (1 + ε) < 0.

        In this case, (<ref>) is bounded by C ℓ t^-3/2^-c ℓ'^2/t.
        
  * If t < (1 + ε) 2 γ/4 - γ^2ℓ', then we can directly use Hölder's inequality (<ref>) to bound
        
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ^-cℓ^2/t≤ C ^-ct^-cℓ^2/t,

        up to a change of the constant c.
    
    In both cases, there exist C, c > 0 such that, for all t > 2,
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ℓ t^-3/2^-cℓ^2/t.

    By (<ref>), (<ref>) and (<ref>), the bound (<ref>) holds for ℓ≥ 2.



    If k = 0, then τ_k = 0 and (<ref>) is a direct consequence of Lemma <ref>, because X_ξ under _0 has the same distribution as -B.
    Assume k ≥ 1.
    In order to use Markov property, define σ_n = s_k, n - τ_k and
    
    F_ℓ(s) = ^γsup_[s, s + σ_n] (X_ξ(s) - X_ξ)1_{inf_[s, t] (X_ξ(s) - X_ξ) ≥ -1, X_ξ(s) - X_ξ(t) ∈ [ℓ - 1, ℓ + 1], σ_n ≤ t - s}.

    The left-hand side of (<ref>) is bounded by
    
    __0[ F_ℓ(τ_k) 1_{sup_[0, τ_k] X_ξ≤ k + 1, τ_k ≤ t}] = ∫_0^t __0[ F_ℓ(s) ] _0 ( sup_[0, τ_k] X_ξ≤ k + 1, τ_k ∈ṣ).

    If ℓ = 0 or 1, then by Lemma <ref> and <cit.>, up to a multiplicative constant, (<ref>) is bounded by
    
    ∫_0^t k ^-k^2/2s/(t - s + 1)^3/2 s^3/2ṣ≤2^3/2 k/t^3/2( ∫_0^t/2^-k^2/2s/s^3/2ṣ + ∫_t/2^t ^-k^2/2s/(t - s + 1)^3/2ṣ),

    which is bounded by C k t^-3/2^-k^2/2t since
    
    ∫_0^t^-k^2/2s/s^3/2ṣ≤^-k^2/4t∫_0^∞^-k^2/4s/s^3/2ṣ 
            = C ^-k^2/4t/k  and  ∫_0^∞ṣ/(s + 1)^3/2 < ∞.

    If ℓ≥ 2, then by Lemma <ref> and <cit.>, up to a multiplicative constant, (<ref>) is bounded by
    
    ∫_0^t ℓ^-c ℓ^2/(t-s) k ^-k^2/2s/(t-s)^3/2 s^3/2ṣ≤2^3/2 k ℓ/t^3/2( ^-c ℓ^2/t∫_0^t/2^-k^2/2s/s^3/2ṣ + ^-k^2/2t∫_t/2^t ^-c ℓ^2/(t-s)/(t - s)^3/2ṣ).

    We can use (<ref>) to bound this by C (k + ℓ + 1) t^-3/2^-c k^2/t - c ℓ^2/t, which concludes.




 §.§ Intermediate temperature


Let β = √(2/3).
In this subsection, we prove Theorem <ref>.<ref>. The ideas are closed to the one used in the case β∈ (√(2/3),√(2)) in the previous subsection, only the end of the argument has to be adapted.



  §.§.§ Lower bound


Let α be an arbitrary number in (0, 1/2).
The arguments used to prove (<ref>) also hold for β = √(2/3), but the fact of restricting ourselves to the event { B_at∈ [0,1] } in the last step (<ref>) of the proof is now too crude: there the factor ^(2β - v(β))B_at prevents B_at from being large in the negatives, whereas here this factor does not play a role because v(β)=2β.
Instead, we stop the reasoning at (<ref>) to get 

    ν_β, t([a, 1])≥ c ^-at/3sup_s ∈ [0, at] (B_s + s^α∧ (at - s)^α) ≤ 1.

By (<ref>), this quantity is larger than c ^-at/3 t^-1/2, which concludes the proof of the lower bound of Theorem <ref>.<ref>.
Note that, here, we could have replaced s^α∧ (at - s)^α with s^α since the resulting probabilities are of the same order.



  §.§.§ Upper bound


Our goal is to show that ν_β, t([a, 1]) = O(t^-1/2^-at/3).
We follow the proof of (<ref>) up to (<ref>) (recall it has been done in the case β∈ [√(2/3),√(2))) and choosing α = 2 which satisfies the condition α∈ [0, 2] ∩ [1, 2/β^2).
Noting that 2β = v(β) when β = √(2/3), (<ref>) becomes

    __0[ (U_t')^-21_A ] ≤ C t^-3/2∑_k = 0^∞∑_ℓ = 0^∞^-2 β k (k + ℓ + 1) ^-c k^2/t - c ℓ^2/t 
        = O(t^-1/2).

This implies ν_β, t([a, 1]) = O(t^-1/2^-at/3) in the same way as (<ref>) implies (<ref>).
This concludes the proof of Theorem <ref>.<ref>.




 §.§ Infinite temperature


We consider here the case β = 0 and prove Theorem <ref>.<ref>. 
Using the first expression in (<ref>) with β=0, and then that #(t) is geometrically distributed with parameter ^-t (see the first example in Section III.5 of <cit.>), we can write

    ν_0, t([a, 1]) 
    = 1/(#(t))^2∑_u∈(at) (#{ v ∈(t) : v ≥ u})^2
    = ∑_i=1^M N_i^2/(∑_i=1^M N_i)^2,

where M is geometrically distributed with parameter q = ^-at and (N_i)_i≥ 1 is a sequence of independent geometric r.v. with parameter p = ^-(1-a)t, independent of M.
Using the representation x^-2 = ∫_0^∞ u ^-uxụ for x > 0 together with Fubini's theorem, we get

    [ ν_0, t([a, 1]) ]
    = ∫_0^∞[ ∑_i=1^M N_i^2 exp( - u ∑_i=1^M N_i) ] u ụ.

Noting that 

    [ ∑_i=1^M N_i^2 exp( - u ∑_i=1^M N_i) | M ]
    = M ·[ N_1^2 ^-uN_1] ·[ ^-uN_1]^M-1.

and [M s^M-1] = q(1-(1-q)s)^-2 for any s ∈ [0,1/(1-q)), we get

    [ ν_0, t([a, 1]) ]
    = ∫_0^∞q u [ N_1^2 ^-uN_1]/(1-(1-q)[^-uN_1])^2ụ
    = pq ∫_0^∞u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ,

using [^-uN_1] = p^-u/(1-(1-p)^-u) and [N^2 ^-uN_1] = p^-u(1+(1-p)^-u)/(1-(1-p)^-u)^3.
We now split the integral on the right-hand side of (<ref>) in three pieces, the main part being the one from pq to p.
Recall q = ^-at and p = ^-(1-a)t.

For the part u ∈ [0,pq], we use 1-(1-r)^-u≥ r ^-1 with r =p and r=pq to get

    ∫_0^pqu ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ≤C/p (pq)^2∫_0^pq u ụ≤C/p.

For the part u ∈ [p,∞), we use 1-(1-r)^-u≥ 1-^-u≥ (1-^-1)(u∧1) with r =p and r=pq to get

    ∫_p^∞u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ≤∫_p^∞Cu ^-u/(u∧1)^3ụ≤∫_p^1 C/u^2ụ
    + ∫_1^∞ Cu ^-uụ≤C/p.

Finally, we consider the part u ∈ [pq,p]. We have the following asymptotic behavior, as t→∞, uniformly in u ∈ [pq,p],

    u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2 
    = u (1+O(u)) (2+O(p))/(p+O(u+p^2))(u+O(pq+u^2))^2 
    = 2/pu( 1 + O ( p + u/p + pq/u) ),

which yields 

    ∫_pq^p
    u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ
    = 2/p( log1/q + O ( p log1/q + 1 ) ).

Combining (<ref>), (<ref>), (<ref>) together with (<ref>), we get 
[ν_0, t([a, 1])] ∼ 2q log1/q, which concludes the proof of Theorem <ref>.<ref> recalling that q = ^-at.




§ ACKNOWLEDGEMENT


We warmly thank Pascal Maillard and Bastien Mallein for helpful discussions throughout the project. We also acknowledge support from the Réseau thématique Mathématiques et Physique.

abbrv




 Paper Title: Dephasing-assisted transport in a tight-binding chain with a linear potential.


1

Dephasing-assisted transport with a linear potential]Dephasing-assisted transport in a tight-binding chain with a linear potential 

] 
 
 





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§ ABSTRACT 


§ 


An environment interacting with a quantum system can enhance transport through the suppression of quantum effects responsible for localization. In this paper, we study the interplay between bulk dephasing and a linear potential in a boundary-driven tight-binding chain. A linear potential induces Wannier-Stark localization in the absence of noise, while dephasing induces diffusive transport in the absence of a tilt. We derive an approximate expression for the steady-state current as a function of both dephasing and tilt which closely matches the exact solution for a wide range of parameters. From it, we find that the maximum current occurs for a dephasing rate equal to the period of Bloch oscillations in the Wannier-Stark localized system. We also find that the current displays a maximum as a function of the system size, provided that the total potential tilt across the chain remains constant. Our results can be verified in current experimental platforms and represents a step forward in analytical studies of environment-assisted transport.

 

§ KEYWORDS:
 Dephasing-assisted transport, Wannier-Stark localization





§ INTRODUCTION


The quantum features of an open system interacting with a macroscopic environment are inevitably destroyed in a process known as decoherence <cit.>. However, the notion that the environment is always detrimental for quantum processes – such as information processing and transport – has been challenged for more than a decade, prompted by investigations of quantum effects in biological systems <cit.>. It is now understood that the environment can assist energy transport in non-interacting quantum systems <cit.>, an effect which has been experimentally verified in quantum networks of photons <cit.>, trapped ions <cit.> and superconducting circuits <cit.>. Although several mechanisms for environment-assisted quantum transport have been proposed and debated <cit.>, a clear mechanism is at play in localized quantum systems – when destructive interference responsible for localization and transport suppression is destroyed by the environment, quantum transport is enhanced. This is indeed the expected impact of environmental coupling on Anderson and Wannier-Stark localization. In Anderson localized systems, quantum transport is suppressed as a consequence of lattice disorder <cit.>. On the other hand, Wannier-Stark localization occurs in the presence of a linear lattice potential such as an electric field <cit.>; in this case, coherent (Bloch) oscillations take place within the region of localization <cit.>. Despite the fact that Wannier-Stark localized systems subject to noise have been investigated <cit.>, the literature on this subject is still scarce.

Environment-assisted transport can also affect the non-equilibrium transport properties of many-body quantum systems. A common approach to evidence this effect is to drive the quantum system to a non-equilibrium steady state by high temperature reservoirs located at the boundaries <cit.>. The dephasing effect of the environment renders the transport diffusive, which has been shown to enhance the steady-state current in boundary-driven systems with disorder <cit.> or quasi-periodicity <cit.> and to modify the transport properties of systems with long-range hopping <cit.>. Moreover, Markovian boundary-driven non-interacting systems are amenable to analytical treatment even in the presence of dephasing, with exact and approximate solutions of steady-state quantities available for tight-binding chains <cit.>.

In this paper, we study dephasing-assisted transport in a boundary-driven, tight-binding chain with a linear potential. We make use of a steady-state ansatz elaborated in Refs. <cit.> to numerically access the relevant steady-state quantities for up to one thousand chain sites. By performing a minimal approximation, we derive an analytical expression for the steady-state current which matches the exact dynamics for a vast range of parameters. From it, we find that the current is maximized at a dephasing rate equal to the period of Bloch oscillations in a Wannier-Stark localized system. We also find that the current displays a maximum as a function of the system size, provided that the total potential tilt across the chain is kept constant. Our results present a significant contribution in the analytics of environment-assisted transport and can be experimentally verified in several platforms.

The paper is organized as follows. In Sec. <ref> we introduce the model and the steady-state ansatz used in our study. In Sec. <ref> we present and discuss our results for dephasing-assisted transport. In Sec. <ref> we present the conclusion and outlook of our work. Sec. <ref> is the supplementary material, containing technical details referenced throughout the main text. 



§ SETUP





 §.§ The model


We study a non-interacting, one-dimensional lattice of L sites with Hamiltonian given by the XX model

    H = J ∑_j=1^L-1  (σ_j^x σ^x_j+1 + σ_j^y σ^y_j+1) + ∑_j=1^Lϵ_j σ_j^z   .

Here, {σ^x_j, σ^y_j, σ^z_j }_j=1^L are Pauli matrices, J is the hopping term and ϵ_j is a local field which will be specified shortly. In order to induce a non-equilibrium state in the system, we couple the boundary sites to high-temperature reservoirs with
different chemical potentials; in addition, each site is also exposed to its own local reservoir which induces dephasing. All the reservoirs are assumed to be ideal and couple weakly to the system. The dynamics of the system's density operator ρ(t) at time t is given by the Lindblad master equation <cit.>

    d ρ(t)/dt = -i/ħ[H,ρ(t)] + ∑_α = l,r,dℒ^α[ρ(t)]   .

Here the superoperator ℒ^α describes dissipation induced by the reservoirs α = l,r,d corresponding to left boundary, right boundary and dephasing, respectively. It is a sum of local Lindblad jump operators of the form

    ℒ^α  [·] = ∑_j,β L^αβ_j · L_j^αβ† - 1/2{ L_j^αβ† L^αβ_j, ·}  ,

where j labels the site acted on by the reservoir α and β labels different jump operators on that same site. The effect of each boundary reservoir (α = l,r) is represented by two jump operators

    L^l ±_1 = √(Γ(1 ± f)/2)σ_1^± ,  L^r±_L = √(Γ(1 ∓ f)/2)σ_L^±  ,

where Γ is the coupling rate at the boundaries, f is the chemical potential bias and σ_j^± = (σ_j^x ± i σ_j^y)/2. For forward bias (0 ≤ f ≤ 1) excitations are mostly created on the first site and annihilated on the last site; the opposite reasoning applies for reverse bias (-1 ≤ f ≤ 0). In order to describe the effects of dephasing, we consider L jump operators acting on individual sites as if each of them were coupled to its own reservoir

    L_j^d = √(γ/2)σ^z_j   ,  j = 1, 2, ..., L   ,

where γ is the dephasing rate. 

The key observables in the system are the magnetization and its associated current. We denote by ⟨ A ⟩(t) ≡Tr[A ρ(t)] the expectation value of an observable A at time t. Differentiating with respect to time and using Eq. (<ref>) yields a continuity equation (see Sec. <ref> for more details)

    d ⟨σ^z_j ⟩(t)/dt    = ⟨ I_j-1⟩(t) - ⟨ I_j⟩(t) + δ_j,1Γ [f - ⟨σ^z_j ⟩(t)] - δ_j,LΓ [f + ⟨σ^z_j ⟩(t)]    ,

involving  the expectation value of the magnetization current ⟨ I_j ⟩(t)  flowing from j to j+1, where

    I_j = 2J/ħ (σ_j^xσ_j+1^y - σ_j^yσ_j+1^x)

is the corresponding current operator. Note that the dephasing reservoir does not change the average magnetization and therefore does not contribute explicitly to Eq. (<ref>), although it still affects the average magnetization and currents implicitly. Moreover, the boundary reservoirs do not contribute to the expression of the current in the bulk of the chain. 



When the steady state is reached, the left-hand side of Eqs. (<ref>) and (<ref>) vanishes. We denote by ⟨ A ⟩_∞≡Tr[A ρ(∞)] the expectation value of A at the steady state ρ(∞). From Eq. (<ref>) we obtain the following equations for the steady-state current

    ⟨ I_1 ⟩_∞    = Γ (f - ⟨σ^z_1 ⟩_∞)   , 
    ⟨ I_j-1⟩_∞    = ⟨ I_j⟩_∞  ,  j = 2, 3, …, L-1 
    ⟨ I_L-1⟩_∞    = -Γ (f + ⟨σ^z_L ⟩_∞)      .

This current is uniform across the chain, with its value dictated by the average magnetization at the boundaries; from now on, we remove the subscript and denote it simply by ⟨ I ⟩_∞. In the absence of any bias (f=0), the boundary reservoirs induce an infinite temperature (maximally-mixed) steady state ρ(∞) = 𝕀/2^L, in which the values of magnetization and current all vanish.



 §.§ Steady-state ansatz


In order to study the transport properties of the model at the steady state, both numerically and analytically, we make use of a perturbative ansatz characterized by an expansion in terms of potential bias f as <cit.>

    ρ    = 1/2^L[𝕀 + f (H + B) + 𝒪(f^2) ]   , 
    
        H     = ∑_r=1^L∑_j=1^L+1-r h^(r)_j H^(r)_j   , 
     
        B     = ∑_r=2^L∑_j=1^L+1-r b^(r)_j B^(r)_j   ,

where h_j^(r) and b_j^(r) are expansion coefficients, and the corresponding operators are given by

    H^(1)_j    = - σ^z_j , 
    
        H^(r)_j    = σ^x_j Z^(r-2)_j+1σ^x_j+r-1 + σ^y_j Z^(r-2)_j+1σ^y_j+r-1  ,  r ≥ 2 
    
        B^(r)_j     = σ^x_j Z^(r-2)_j+1σ^y_j+r-1 - σ^y_j Z^(r-2)_j+1σ^x_j+r-1  ,  r ≥ 2   .

These operators are thus strings of Pauli matrices starting at site j and having length r, where Z^(r-2)_k+1 = σ^z_k+1σ^z_k+2 ... σ^z_k+r-3σ^z_k+r-2 are strings of Pauli z matrices of length r-2. The defining characteristic of this ansatz is that all operators in the expansion are orthogonal according to the Hilbert-Schmidt inner product ⟨ A, B⟩≡Tr[A^†B]. This implies that the expectation values of H^(r)_j and B^(r)_j are exactly determined within first order in f. In particular, H_j^(1) and B_j^(2) are respectively proportional to the magnetization and current

    ⟨σ^z_j⟩_∞ = -f h^(1)_j  , ⟨ I_j ⟩_∞ = f 4J b^(2)_j/ħ  .

Once we determine the first order expansion coefficients h^(r)_j and b^(r)_j, we have access to the exact current and magnetization profile <cit.>. In order to obtain them, we insert Eq. (<ref>) into Eq. (<ref>) and set the left-hand side to zero as demanded at stationarity (see Refs. <cit.> and Sec. <ref> for more details). This gives rise to a set of equations for the coefficients

    ħΓ(1+h^(1)_1) - 4J b_1^(2) = 0   , 
       ħΓ(1-h^(1)_L) - 4J b_L-1^(2) = 0   , 
        b_j^(2) - b_j-1^(2) = 0   ,  j = 2,...,L-1   ,

for r = 1. Note that Eqs. (<ref>)–(<ref>) just express the fact that the current is uniform in the steady state, with its value dictated by the boundary reservoirs as discussed before. For r ≥ 2 we have

    J [ h_j^(r-1) - h_j+1^(r-1) + h_j^(r+1) - h_j-1^(r+1)] +  (ϵ_j - ϵ_j+r-1)h_j^(r) + ħ[ γ + Γ/4(δ_j,1 + δ_j+r-1,L) ] b^(r)_j = 0   , 
        J [b_j^(r-1) - b_j+1^(r-1)+ b_j^(r+1) - b_j-1^(r+1)] + (ϵ_j - ϵ_j+r-1)b_j^(r) - ħ[ γ + Γ/4(δ_j,1 + δ_j+r-1,L) ] h^(r)_j = 0   .

In total, Eqs. (<ref>)–(<ref>) are a closed set of L^2 coupled equations and will be the starting point of our study. This set of equations is equivalent to a Lyapunov-type equation for the correlation matrix in the presence of dephasing <cit.>. Instead of the exponential scaling with the system size, the quadratic scaling makes the problem computationally tractable. Moreover, as we show below, it also allow us to find analytical expressions for the current under some mild assumptions.



§ DEPHASING-ASSISTED TRANSPORT



We now discuss separately the effect of the tilt and dephasing on the transport properties of the model, while their interplay is addressed in the next section. The bias f of each reservoir at the boundaries induces transport along the chain. Intuitively, a similar effect is expected in the presence of an electric field which increases/decreases at each site at a rate U, or tilt between neighbour sites. If the tilt decreases from left to right, the local potentials are

    ϵ_j = - Uj   .

Setting the bias to f=1 drives excitations in the same direction (see Fig. <ref>). Note that changing the sign of f inverts the sign of the current and magnetization, while changing the sign of U affects neither <cit.>. 

Since the tilt U represents the energy difference between nearest neighbours in the chain, the total tilt V=U(L-1) represents the difference in energy across the whole chain. This quantity diverges in the thermodynamic limit, so that a physically meaningful analysis requires fixing the total tilt V and allowing U to decrease with the system size L. As discussed in Secs. <ref> and <ref>, this procedure introduces a non-trivial dependence of the current with system size in the presence of dephasing: while for fixed U the current is expected to decrease with the system size L, this will no longer be the case for fixed V.



Such a linear potential is known to induce Wannier-Stark localization in the absence of dephasing, with U/ħ being the frequency of Bloch oscillations <cit.>.
In the absence of dephasing γ = 0 and for non-vanishing tilt V>0, the steady state magnetization profile shows the formation of domain walls <cit.> – regions of constant magnetization ⟨σ^z_j ⟩_∞≃± f starting at the boundaries and extending inwards as shown in Fig. <ref>(a). According to Eqs. (<ref>) or (<ref>), this implies a vanishing current, which characterizes the localized system. Indeed we observe in Fig. <ref>(b) that, as V increases, the current displays an exponential decay with the system size ⟨ I ⟩_∞∼ e^-L/L_0 with L_0 being the localization length. Note that L_0 here is different from the localization region of a single particle in a Wannier-Stark localized system given by J/U <cit.>. We did not find any direct relationship between the two.



The results in the absence of tilt U = V = 0 are shown in Figs. <ref> and <ref>(a), respectively for the magnetization and the current. We distinguish two regimes: γ = 0 and γ > 0. The first case corresponds to ballistic transport, characterized by a small (but non-zero) magnetization at the boundaries and zero bulk magnetization (inset of Fig. <ref>), together with a size-independent current (see later Fig. <ref>(a)). The second case corresponds to diffusive transport, characterized by an emerging linear magnetization profile as the dephasing rate increases, accompanied by a decreasing current in accordance with Eqs. (<ref>) or (<ref>). For very large dephasing or system sizes the current scales as ⟨ I ⟩_∞∼ 1/(γ L) signaling diffusive transport. The exact expression for the current in the absence of tilt is given by Eq. (<ref>) (with U=V=0) as we discuss in Sec. <ref>.



 §.§ Exact current for small system sizes


Having shown the degrading effect of the tilt and dephasing separately on the spin transport across the system, we now study their interplay which gives rise to dephasing-enhanced transport. In this section, we focus on transport for small system sizes L=2,3. An exact and compact solution for the current can be found from Eqs. (<ref>)–(<ref>) in these cases. For L = 2 we obtain

    ⟨ I ⟩_∞ = f 16 J^2 Γ (2 γ + Γ)/16 J^2 (2 γ + Γ) + 4 Γ U^2 + ħ^2Γ(2 γ + Γ)^2  .

Dephasing-assisted transport can be characterised through the maximum of Eq. (<ref>) with respect to the dephasing rate γ, yielding an optimal dephasing rate

    γ_max = U/ħ - Γ/2≥ 0   .

The behaviour of Eq. (<ref>) is shown in Fig. <ref>. Note that since the dephasing is always positive, the maximum only occurs if U/ħ≥Γ/2.

For L=3 we have a similar expression, namely

    ⟨ I ⟩_∞ = f 16 J^2 Γ (4 γ + Γ)/16 J^2 (4 γ + Γ) + 16 Γ U^2 + ħ^2Γ(4 γ + Γ)^2   ,

where the maximum occurs at

    γ_max = U/ħ - Γ/4≥ 0   .

The complexity of these expressions increases rapidly for L > 3, so the exact evaluation of the current and its maximum is too involved to be displayed here for larger systems. In the following section, we resort to an approximate solution for the current.



 §.§ Tridiagonal approximation 


In order to find an exact solution for systems of arbitrary size, we perform the tridiagonal approximation which consists of retaining strings of Pauli matrices up to r=2, that is, ignoring higher order coefficients in Eqs. (<ref>)–(<ref>) so that h^(r)_j,b^(r)_j = 0 for r ≥ 3. In the correlation matrix approach, this is equivalent to retaining only the diagonal and first off-diagonal elements <cit.>. Physically, this implies that spin and current correlations can be ignored within first order in f, which allows the derivation of an exact expression for the current as a function of all parameters as we now show. We only have to consider Eqs. (<ref>) and (<ref>) for r=2. Summing Eqs. (<ref>) from j=1 to j=L-1 we obtain

    J(h^(1)_1 - h^(1)_L)     + ∑_j=1^L-1 (ϵ_j - ϵ_j+1) h^(2)_j +  ħ b [ γ(L-1) + Γ/4] = 0   ,

where b ≡ b^(2)_j is the coefficient associated with the current, which is independent of j as a consequence of Eq. (<ref>). First, we can write h^(1)_1 - h^(1)_L as a function of b by using Eqs. (<ref>) and (<ref>), which yields

    h^(1)_1 - h^(1)_L = 8 J b/ħΓ - 2   .

Second, we can also express h^(2)_j as a function of b by using Eq. (<ref>), which leads to

    h^(2)_j =  (ϵ_j - ϵ_j+1) b/ħ[γ + (δ_j,1 + δ_j,L-1)Γ/4]  .

We can now substitute the previous two equations into Eq. (<ref>) and obtain an expression for b, namely

    b = 4J/2 ħγ(L-1) + ħΓ + 16J^2/ħΓ  + 2/ħ∑_j=1^L-1(ϵ_j - ϵ_j+1)^2/γ +(δ_j,1 + δ_j,L-1)Γ/4  .

The current is now obtained from Eq. (<ref>). For the local tilted potential in Eq. (<ref>) it becomes

    ⟨ I ⟩_∞ = f 16J^2 Γ/16J^2 + ħ^2 Γ[Γ + 2 γ(L-1)] + 4 Γ U^2 
        
          1/(2γ + Γ)     L = 2 
    
          (L-3)/(2γ) + 4/(4γ + Γ)     L ≥ 3
          .

We note that this expression is exact for L=2 and L=3, in which case we recover Eqs. (<ref>) and (<ref>), respectively. It is also exact in the absence of tilt U=V=0 <cit.>, which is verified in Fig. <ref>(a). In last three panels of Fig. <ref> we plot the exact solution and the tridiagonal approximation for non-zero values of total tilt V > 0. We observe that the approximation works remarkably well, deviating from the exact solution only for very small dephasing rates and system sizes; when γ = 0, the approximation fails dramatically and only the exact solution is shown. The latter case corresponds to a Wannier-Stark localized system which we already discussed. Moreover, we see that for fixed γ the current can now increase with the system size up to a critical size L_max where it is maximal, and only then decreases. In the same way, we see that for fixed L there is a critical dephasing rate γ_max at which the current is maximal. 





 §.§ Interplay between dephasing and tilt 


Since the validity of Eq. (<ref>) is established, we now use it to determine both L_max and γ_max. The maxima of Eq. (<ref>) as a function of L (for fixed V and γ) and γ (for fixed L and V) can be determined exactly. However, as before, the expressions are too lengthy to be displayed. More physically intuitive expressions are obtained by assuming that L ≫ 1 and L ≫ 8 γ / (4γ + Γ) in the denominator of Eq. (<ref>) which leads to the approximate maxima

    L_max   ≃V/ħγ  , 
    γ_max   ≃V/ħ L = U/ħ  .

These maxima characterize dephasing-assisted transport as we show in Fig. <ref>(a). 

Below the critical length (L_max≫ L ≫ 1), dephasing is not yet effective and an increase in L decreases the local tilt U = V/(L-1) and thus localization, which increases the current according to Eq. (<ref>). Above this length scale (L ≫ L_max≫ 1) dephasing becomes effective and diffusive transport clearly emerges ⟨ I ⟩_∞∼ 1/(γ L). Analogously, below the critical dephasing rate (γ_max≫γ) dephasing jump events happen at a much lower frequency than Bloch oscillations, so the current flowing remains hampered by localization. Increasing the rate of dephasing destroys localization and increases the current. When the dephasing rate passes this threshold (γ≫γ_max)  diffusive behaviour emerges. We illustrate this in Fig. <ref>(b) where we plot the (finite size) diffusion coefficient defined by ⟨ I ⟩_∞≡ f D/L as a function of dephasing rate for the largest system size we reached L = 1000.

The imprint of dephasing-enhanced transport is also evident in the magnetization profile, as shown in Fig. <ref>. Here, varying the dephasing strength towards its critical value (<ref>) decreases the magnetization gradient, rendering the magnetization profile almost uniform. This is in contrast with the results obtained in the absence of tilt, where an increasing dephasing rate tends to increase the magnetization gradient and decrease uniformity as shown in Fig. <ref>(a). Uniformity as a feature of environment-assisted transport has been pointed out for quantum systems subject to on-site disorder <cit.>; our results confirm that the same phenomenon appears in the presence of a linear potential inducing Wannier-Stark localization.





§ CONCLUSION AND OUTLOOK


In this paper, we have exploited an exact ansatz to obtain the steady state of a non-interacting spin chain subject to bulk dephasing, a linear potential and boundary-driving. This description has allowed us to unravel the delicate interplay between Wannier-Stark localization and dephasing-induced diffusive transport. We applied the tridiagonal approximation, where only on-site and nearest-neighbour correlation matrix elements are retained, to derive an approximate expression for the steady-state current as a function of both dephasing and tilt. This expression is found to closely match the exact solution for a wide range of parameters, and reveals that the maximum current occurs for a dephasing rate equal to the period of Bloch oscillations in the Wannier-Stark localized system. Fixing the total tilt across the system then revealed a maximum in the current as a function of system size. This evidences a critical system size beyond which Bloch oscillations are suppressed by dephasing.

Our work motivates future analytical and numerical analysis on the interplay between dephasing and a tilted potential in more complex and richer systems, e.g. those including interactions between neighboring sites <cit.>, time-periodic driving <cit.>, or more realistic (non-Markovian) boundary-driving schemes <cit.>. Furthermore, our results can be readily verified in current photonic <cit.>, ion-trap <cit.> and cold-atom <cit.> experimental platforms.



§ FUNDING


The authors declare financial support was received for the research, authorship, and/or publication of this article. S.L.J. acknowledges the financial support from a Marie Skłodowska-Curie Fellowship (Grant No. 101103884). S.R.C. gratefully acknowledge financial support from UK's Engineering and Physical Sciences Research Council (EPSRC) under grant EP/T028424/1 which is part of an EPSRC-SFI joint project QuamNESS funded also by the SFI under the Frontier For the Future Program. J.G. is supported by a SFI Royal Society University Research Fellowship.



§ ACKNOWLEDGEMENTS


The authors want to thank Gabriel T. Landi for useful discussions.



§ SUPPLEMENTARY MATERIAL





 §.§ Heisenberg equation



For completeness, we show here how to obtain Eq. (<ref>). Differentiating ⟨σ^z_j ⟩(t) ≡Tr[σ_j ρ(t)] with respect to time and using Eq. (<ref>) yields

    d ⟨σ^z_j ⟩(t)/dt = i/ħ Tr[ [H,σ^z_j] ρ(t) ] + ∑_α = l,r,dTr[ σ^z_j ℒ^α[ρ(t)] ]   .

Throughout this section, we make use of several relations for the Pauli matrices, which can all be deduced from their commutation and anti-commutation relations [σ^α,σ^β] = 2i ε_αβγσ^γ and {σ^α,σ^β} = 2 δ_αβ𝕀_2, respectively. Here α, β, γ∈{ x, y, z }, 𝕀_2 is the identity operator for a two-level system and ϵ_αβγ is the Levi-Civita symbol with the Einstein summation notation being implicit. The commutator in Eq. (<ref>) can then be easily computed

    i/ħ[H,σ^z_j]     = iJ/ħ∑_k=1^L-1[ σ_k^x σ^x_k+1 + σ_k^y σ^y_k+1,σ^z_j] 
        =  2J/ħ (σ_j-1^xσ_j^y - σ_j-1^yσ_j^x) - 2J/ħ (σ_j^xσ_j+1^y - σ_j^yσ_j+1^x) 
        = I_j-1 - I_j  ,

In the first line only the kinetic energy part of the Hamiltonian is non-vanishing. In the second line, only the pairs (k,k+1) which overlap with j survive. In the last line, we identified the current operator in Eq. (<ref>). Regarding the last term in Eq. (<ref>), we can make progress by considering the dual superoperator acting on σ^z_j defined by Tr[ σ^z_j ℒ^α[ρ(t)] ] = Tr[ ℒ^α†[σ^z_j] ρ(t) ] with

    ℒ^α†[·] = ∑_j,β L^αβ†_j · L_j^αβ - 1/2{ L_j^αβ† L^αβ_j, ·}  .

The dephasing superoperator is equal to its dual and reads explicitly

    ℒ^d[·] = γ/2∑_j=1^L ( σ_j^z ·σ_j^z - · ) = ℒ^d †[·]   .

It is straightforward to see that its action vanishes when applied to any Pauli z-matrix. Therefore dephasing does not contribute to Eq. (<ref>). For the reservoir at the left boundary, we have the dual superoperator

    ℒ^l †[·]     = Γ(1+f)/2 O_1^l+ †[·] + Γ(1-f)/2 O_1^l- †[·] 
    
        O_1^l±†[·]    ≡σ_1^∓·σ_1^± - 1/2{σ_1^∓σ_1^±,·}  ,

and a similar expression holds for the right reservoir in Eq. (<ref>). Applying these superoperators to the Pauli z-matrix yields after some manipulation

    ℒ^l †[σ^z_j]     = δ_j,1Γ(f - σ^z_1)   , 
    ℒ^r †[σ^z_j]     = - δ_j,LΓ(f + σ^z_L)   .

Using the last two expressions and Eq. (<ref>) in Eq. (<ref>) immediately leads to Eq. (<ref>).



 §.§ Steady-state ansatz



In order to derive Eqs. (<ref>)–(<ref>), we insert Eq. (<ref>) into Eq. (<ref>) and set the left-hand side to zero as demanded at steady state. This gives

    0     = 1/2^L∑_α = l,r,d,0[ ℒ^α[𝕀] + f ( ℒ^α[H] + ℒ^α[B])]   , 
    ℒ^α[H]     = ∑_r=1^L∑_j=1^L+1-r h^(r)_j ℒ^α[H^(r)_j]   , 
    ℒ^α[B]     = ∑_r=2^L∑_j=1^L+1-r b^(r)_j ℒ^α[B^(r)_j]   .

In the last expression we ignore all terms beyond first order in f and defined the superoperator for free evolution

    ℒ^0 [·] ≡ -i/ħ[H, · ]   .




  §.§.§ Action of the dephasing reservoir


We start by computing the action of the dephasing superoperator which reads explicitly

    ℒ^d[·] = γ/2∑_j=1^L ( σ_j^z ·σ_j^z - · ) ≡γ/2∑_j O^d_j[·]   .

It is straightforward to see that the action of this superoperator vanishes when applied to 𝕀 and σ^z. For r ≥ 2, we have to compute terms of the following form

    ℒ^d[σ^α_j Z^(r-2)_j+1σ^β_j+r-1]     = γ/2[O^d_j[σ^α_j] Z^(r-2)_j+1σ_j+r-1^β + σ^α_j Z^(r-2)_j+1 O^d_j+r-1 [σ_j+r-1^β] ]   ,

Note that the right-hand side has only two non-vanishing terms on the boundaries since the action on σ^z vanishes. We then have

    O^d_j[σ^α_j] = σ_j^z σ_j^ασ_j^z - σ_j^α = - 2 σ_j^α  (δ_α,y + δ_α,x)   .

Using this last expression and carrying through simple algebraic manipulation, the action of the dephasing is written as

    ℒ^d[𝕀]     = ℒ^d[H_j^(1)] = 0   , 
    ℒ^d[H_j^(r)]     = -2 γ H_j^(r) ,  r ≥ 2 
    ℒ^d[B_j^(r)]     = -2 γ B_j^(r) ,  r ≥ 2   .




  §.§.§ Action of the boundary reservoirs


We first illustrate the action for the left boundary reservoir. The superoperator is given by

    ℒ^l[·]     = Γ(1+f)/2 O_1^l+[·] + Γ(1-f)/2 O_1^l-[·]   , 
    
        O_1^l±[·]    ≡σ_1^±·σ_1^∓ - 1/2{σ_1^∓σ_1^±,·}  .

In opposition to dephasing, this superoperator does not preserve the identity

    O_1^l±[𝕀] = ±σ_1^z = ∓ H^(1)_1   .

As a consequence, its action on the Pauli matrices σ_j^z depends on j. Namely, we obtain

    O_1^l±[σ_j^z] = 
        
          - σ^z_1   ,     j = 1 
    ±σ_1^zσ^z_j   ,     j = 2,3...L   .

Inserting these results into Eq. (<ref>) we obtain the actions

    ℒ^l[𝕀]     = -Γ f H^(1)_1   , 
    ℒ^l[H^(1)_j]     = 
        
            - Γ H_1^(1)    j = 1 
    
            f Γ H_1^(1) H_j^(1)    j = 2,3,…,L   .

Inspecting the last expression together with Eq. (<ref>), we see that for j=2,3,..L the reservoir creates terms in the expansion which are proportional to f^2 instead of f – they represent higher order spin correlations which do not affect the steady state values of magnetization and current. We henceforth ignore these second order terms by considering the action of the reservoir on Pauli matrices located at the reservoir site. Proceeding with the calculations for r ≥ 2, we have to compute terms of the form

    ℒ^l[σ^α_j Z^(r-2)_j+1   σ^β_j+r-1] = δ_j,1{Γ(1+f)/2 O^l+_1[σ^α_1] + Γ(1-f)/2 O^l-_1[σ^α_1] } Z^(r-2)_j+1σ^β_j+r-1  .

It is easy to verify that

    O^l ±_1[σ^α_1] = -σ_1^α/2 , α = x,y   .

Using the last two expressions and carrying out the algebra, the results for the left reservoir to first order in f are

    ℒ^l[𝕀]     = - Γ f H^(1)_1   , 
    ℒ^l[H^(1)_j]     = -δ_j,1Γ H^(1)_j   , 
    ℒ^l[H^(r)_j]     = - δ_j,1Γ/2 H_j^(r) ,  r ≥ 2   , 
    ℒ^l[B^(r)_j]     = - δ_j,1Γ/2 B_j^(r) ,  r ≥ 2   .

Proceeding the same way for the right boundary reservoir we obtain the result

    ℒ^r[𝕀]     = Γ f H^(1)_L   ,
    ℒ^r[H^(1)_j]     = -δ_j,LΓ H^(1)_j   , 
    ℒ^r[H^(r)_j]     = - δ_j+r-1,LΓ/2 H_j^(r) ,  r ≥ 2   , 
    ℒ^r[B^(r)_j]     = - δ_j+r-1,LΓ/2 B_j^(r) ,  r ≥ 2   .




  §.§.§ Action of Hamiltonian


Regarding the Hamiltonian superoperator in Eq. (<ref>), it is easy to see that it preserves the identity ℒ^0[𝕀] = 0 while ℒ^0[H_j^(1)] yields the well-known result

    ℒ^0[H_j^(1)] = - 2J/ħ(B^(2)_j - B^(2)_j-1)   .

This result was already implicitly used in Eq. (<ref>) and simply expresses that the change in magnetization at a given site is due to differences between incoming and outgoing currents at that site. For r ≥ 2, it is useful to write the superoperator  
as a sum of two contributions ℒ^0≡ℒ^0_1 + ℒ^0_2, where

    ℒ^0_1·   ≡ -i J/ħ∑_j=1^L-1 { [σ_j^x σ_j+1^x,·] + [σ_j^y σ_j+1^y,·] }  , 
    ℒ^0_2·   ≡ -i/ħ∑_j=1^L ϵ_j [σ_j^z,·]   .

We compute only the action of ℒ^0_1 and the one for ℒ^0_2 is obtained in a similar manner. We first evaluate operators for r = 2 which contain no Pauli z-strings (see Eq. (<ref>) for the exact expressions). It involves computing terms of the form

    ∑_j=1^L-1 [σ_j^γσ_j+1^γ, 
     σ_k^ασ_k+1^β]     = σ^γ_k-1[σ_k^γ,σ_k^α]σ_k+1^β+ [σ_k^γσ_k+1^γ,σ_k^ασ_k+1^β] + σ^α_k[σ_k+1^γ,σ_k+1^β]σ_k+2^γ
        = 2i [ε_γαωσ_k-1^γσ_k^ωσ_k+1^β + ε_αβω(δ_γασ_k+1^ω - δ_γβσ_k^ω) + ε_γβωσ_k^ασ_k+1^ωσ_k+2^γ]   .

In the first line, the only non-vanishing terms are for those pairs (j,j+1) that partially or completely overlap with (k,k+1). In the second line, we used the standard properties of the Pauli matrices. The last expression is enough to compute the action of ℒ^0_1 on H^(2)_k and B^(2)_k which reads

    ℒ^0_1[H^(2)_j]     = -2J/ħ(B_j^(3) - B_j-1^(3))   , 
    ℒ^0_1[B^(2)_j]     = -4J/ħ(H_j+1^(1) - H_j^(1))-2J/ħ(H_j^(3) - H_j-1^(3))   .

For r ≥ 3, the operators in Eq. (<ref>) have Pauli-z strings. We need to compute terms of the form

    ∑_j=1^L-1   [σ_j^γσ_j+1^γ,σ_k^α Z^(r-2)_k+1σ_k+r-1^β] = σ_k-1^γ [σ_k^γ,σ_k^α] Z^(r-2)_k+1σ_k+r-1^β + [σ_k^γσ_k+1^γ,σ_k^ασ_k+1^z] Z^(r-2)_k+2σ_k+r-1^β
        + σ_k^α Z^(r-2)_k+1[σ^γ_k+r-1,σ^β_k+r-1]σ^γ_k+r + σ_k^α Z^(r-3)_k+1 [σ_k+r-2^γσ_k+r-1^γ,σ_k+r-2^zσ_k+r-1^β]   , 
        = 2i [ ε_γαωσ_k-1^γσ_k^ω Z^(r-2)_k+1σ_k+r-1^β + ε_γβωσ_k^α Z^(r-2)_k+1σ_k+r-1^ωσ_k+r^γ
        + (ε_γαωδ_γ zσ_k^ω + ε_γ z ωδ_γασ_k+1^ω) Z^(r-3)_k+2σ_k+r-1^β + σ_k^α Z^(r-3)_k+1 (ε_γ z ωδ_γβσ_k+r-2^ω + ε_γβ zδ_γ zσ_k+r-1^ω)]   .

Again, the only non-vanishing terms are for those pairs (j,j+1) that partially or completely overlap with the string (k,k+1,...k+r-1). No commutator appears containing pairs of σ_z since these also vanish. The last expression suffices to compute the action of ℒ^0_1 on H^(r)_k and B^(r)_k for r ≥ 3 and a similar reasoning holds for ℒ^0_2. We summarize the results below:

    ℒ^0[𝕀]     = 0   , 
    ℒ^0[H_j^(1)]     = - 2J/ħ(B^(2)_j - B^(2)_j-1)   , 
    ℒ^0[B_j^(2)]     = 4J/ħ(H_j^(1) - H_j+1^(1))+2J/ħ(H_j^(3) - H_j-1^(3)) + 2(ϵ_j - ϵ_j+1)/ħH_j^(2)  , 
    ℒ^0[H_j^(r)]     = - 2J/ħ[ (B_j^(r-1) - B_j+1^(r-1)) + (B_j^(r+1) - B_j-1^(r+1)) ] - 2(ϵ_j - ϵ_j+r-1)/ħB_j^(r) ,  r ≥ 3   , 
    ℒ^0[B_j^(r)]     = 2J/ħ[ (H_j^(r-1) - H_j+1^(r-1)) + (H_j^(r+1) - H_j-1^(r+1)) ] + 2(ϵ_j - ϵ_j+r-1)/ħH_j^(r) ,  r ≥ 3   .

With the action of all superoperators computed and using Eq. (<ref>) we obtain the set of Eqs. (<ref>)–(<ref>) for the coefficients used in the main text.

Frontiers-Harvard




 Paper Title: Modelling competition for space: Emergent inefficiency and inequality due to spatial self-organization among a group of crowd-avoiding agents.



Crowd avoidance model]Modelling competition for space: Emergent inefficiency and inequality due to spatial self-organization among a group of crowd-avoiding agents


annmaryettolil@gmail.com
Department of Physics, Cochin University of Science and Technology
Assumption College Autonomous, Changanassery


sasidevan@gmail.com
Department of Physics, Cochin University of Science and Technology


§ ABSTRACT 
Competition for a limited resource is the hallmark of many complex systems, and often, that resource turns out to be the physical space itself. In this work, we study a novel model designed to elucidate the dynamics and emergence in complex adaptive systems in which agents compete for some spatially spread resource. Specifically, in the model, the dynamics result from the agents trying to position themselves in the quest to avoid physical crowding experienced locally. We characterize in detail the dependence of the emergent behavior of the model on the population density of the system and the individual level agent traits such as the extent of space an agent considers as her neighborhood, the limit of occupation density one tolerates within that neighborhood, and the information accessibility of the agents about neighborhood occupancy. We show that the efficiency with which the agents utilize the physical space shows transitions at two values of densities. The first of these transitions demarcates efficient and inefficient phases of the system, and the second one signifies the density at which the inefficiency is maximum. We show that the variation of inefficiency with respect to the information accessible to the agents shows opposing behavior above and below this second transition density. We also look into the inequality of resource sharing in the model and show that although inefficiency can be a non-monotonic function of information depending upon the parameters of the model, inequality, in general, decreases with information. Our study sheds light on the role of competition, spatial constraints, and agent traits within complex adaptive systems, offering insights into their emergent behaviors. 


[
    V. Sasidevan
    August 5, 2024
==================





§ INTRODUCTION

From minute living cells to large-scale societies, complex systems are vastly diverse regarding the behavior of their constituent components and the nature of the interaction between them <cit.>. Driven by the interactions among the constituent components, these systems exhibit unique non-trivial properties at larger scales which is of great relevance in understanding the emergent behavior in a variety of natural phenomena <cit.>. Competition between the constituent components, aka agents, forms the hallmark of many complex systems <cit.>. More specifically, in many socio-economic and biological systems, the context of interaction among the agents is set by the competition for some limited resource <cit.>. There are numerous examples of complex systems that are driven by a group of agents in common want for a shared resource <cit.>

We see the emergence of intriguing spatio-temporal patterns in social systems, such as sharing of public spaces like theaters, traffic systems, streets or squares <cit.>; economic systems, such as the opening of shops in an area or the mining of fuel <cit.>; biological systems, such as animal foraging and in other similar spatial systems <cit.>. It is easily seen that the active pursuit to acquire sparsely populated regions in space among a group of agents leads to competition between the agents in many of these systems <cit.>. 
  
Models in which agents have spatial preferences and respond to the occupancy and behavior of agents in the neighborhood have been considered in the past. Models of cellular automata, traffic flow, swarming, flocking, opinion dynamics, spatial segregation models, etc. capture the collective behavior of a collection of agents in response to the presence and behavior of other neighboring agents <cit.>. A particular example is the Schelling's segregation model which shows how even weak individual preferences of the agents can lead to large-scale global segregation <cit.>. Researchers have also explored models that examine the competitive interactions of groups of agents on spatial structures and lattices <cit.>. Nevertheless, an exclusive model to trace spatial spreading among agents in the quest to avoid local crowding is absent in the literature, although it is a universal problem visible across many social, economic, and biological domains <cit.>.


In this paper, we introduce a simple setting in which the aspect that space itself could be the limited resource for which the agents compete is made explicit. In such competitions, often, the crowding is experienced locally. The motivation comes from the fact that,  in real space, the effect of crowding and its discomfort is predominantly felt at a local level rather than at a global scale. For instance, one might be sitting in one of the most crowded theaters, yet may feel a lesser effect of crowding if a few of her neighboring seats are vacant. Conversely, one might be in one of the least populated places and yet feel crowded if her immediate neighborhood is over-packed above her comfort level. As a result, the uncomfortable agents would seek to occupy less populated local regions in space.  This is in contrast to the typical setting considered for the competition of a scarce resource where all agents choose between the available options and the group that chose the least populated option wins <cit.>.

We define a game theoretic model that captures the behavior of spatial systems where adaptive agents compete for a spatially spread resource. The model's rules, the pay-off structure, and agent behavior reflect the motive of the agents to avoid local crowding above their comfort level in a spatial setting.  This framework captures how the adaptive behavior of agents in pursuit of sparsely populated regions relative to their comfort level results in the emergence of interesting macro-scale behavior. In a nutshell, the model consists of agents interacting in a defined spatial environment (e.g. a lattice) with fixed rules for playing and winning.  An agent wins by occupying a neighborhood with fewer agents than her comfort threshold; otherwise, she seeks relocation to nearby regions.

The ensuing dynamics within the model unveil intriguing collective properties at a global scale. We show that the agent density significantly influences the collective behavior.  Specifically, two distinct critical densities mark abrupt shifts in emergent behavior. Below a critical density, the system always reaches an absorbing state where all agents are winning. Above a higher density, termed the peak sustainable density, no states of the system exist wherein all agents win. We formulate an analytical method to calculate the peak sustainable density and use Monte Carlo simulations to analyze the emergent behavior of the system. We show that the spatial self-organization of the agents becomes maximally inefficient around the peak sustainable density.

We study in detail how agents' accessible information about local occupancy shapes global behavior. We show that the variation of global resource utilization with respect to the amount of information exhibits opposing trends on either sides of the peak sustainable density. We find that below the peak sustainable density, there is a non-trivial optimum value of the amount of information at which the global resource utilization is a maximum. In particular, we show that global resource utilization can be a non-monotonic function of the amount of information.  Intriguingly, possessing more information proves counterproductive for better global outcomes at these densities. We also probe how the inequality between the agents measured in terms of the difference in their total payoff obtained over the course of the game is affected by the density and information. Unlike inefficiency, we find that inequality, in general, shows a decreasing trend with increasing information. Thus, analyzing the macro-scale properties, we showcase how agents' crowd-avoidance and relocation lead to novel emergent behavior. Our results uncover scenarios where local crowd-avoidance prompts global crowding, and increased individual-level information yields diminished global efficiency - defying conventional expectations.


The paper is organized as follows. In Section <ref>, we introduce the model and define the terminologies and relevant macroscopic parameters of the system. We also present calculation of the peak sustainable density in the system. In Section <ref>, we present the simulation results regarding the macroscopic properties like the global inefficiency, inequality, and fraction of permanent winners in the system. We discuss in detail their dependence on the parameters of the model. Section <ref> contains the conclusion and discussion.



§ THE MODEL
 

In the model, agents engage in competition to secure sparsely populated neighborhoods, guided by their individual comfort thresholds. We consider the space as a lattice upon which the agents are distributed. For conceptual simplicity, we consider the 1D lattice consisting of  L sites where N agents are randomly distributed initially (N ≤ L). Each site can be occupied by only one agent at a time. Periodic boundary conditions are assumed to reduce edge effects. System's Density (ρ) is defined as the ratio of the total number of agents (N) to the number of sites (L). In the following, we introduce the main parameters of the model and discuss the dynamics. 

The neighborhood of a lattice site is constituted by the physically nearest sites of it. In one dimension, the simplest neighborhood consists of the two immediate neighboring sites to the left and right of a focal location. The size of the neighborhood (z) in this case is 2. Advancing this notion,  z=4 corresponds to the case where the neighborhood comprises the four nearest sites, two on each side of a site (for simplicity, we will consider only neighborhoods that are symmetric to the right and left of a site). The success of an agent is assumed to depend on the occupation density (ν) within her neighborhood. For the i^th site, the occupation density of its neighborhood (ν_i) is defined as the ratio of the number of agents occupying the neighborhood to the number of sites in the neighborhood.

The tolerance threshold (τ) is the limit of the occupation density within the neighborhood of an agent's location, up to which she is a winner (payoff = 1). When the occupation density crosses the tolerance threshold, an agent will turn a loser (payoff = 0) and seek options for relocation. For example, for a 1D lattice with z=2, τ can adopt values of 0 (indicating agents' intolerance to any neighborhood occupancy), 0.5 (indicating tolerance for up to 50% neighborhood occupancy), or 1 (indicating tolerance for 100% neighborhood occupancy). Clearly, for τ=1, all agents are winning all the time. While, in general, τ can vary among agents or even across sites, we focus on the simplest scenario where τ remains uniform across all agents and sites. 

The notion of neighborhood size and tolerance threshold encapsulates the concept of competition for space in the model. Together, they determine the level of tolerance of agents towards local crowding. The essence of the crowd avoidance rule is contained in the fact that when an agent chooses a region in space whose occupation density is less than or equal to the tolerance threshold, she is winning. 




In the model, an agent is assumed to have access to information about the occupancy of all lattice points up to ‘r’  sites on either side from her present location, defined as the Information radius. Alternatively, we consider the standardized information radius (r_s), which is the ratio of the information radius (r) to z/2, thus normalizing the information radius with respect to the neighborhood size. Again, for simplicity, the value of r is assumed to be the same for all agents. An agent uses information about the occupancy within her information radius to decide what to do in the subsequent round of the game.  Consequently, an agent's movement is assumed to be confined within this radius, given the absence of information beyond this limit. 

The adaptive behavior of the agents is ingrained in the mechanisms they adopt at the individual level to respond to the local environment. 
All agents follow a simple win-stay, lose-shift behavioral rule <cit.>. The winners do not move. i.e., they remain at their current locations for the next round of the game. Each loser attempts to relocate to any of the vacant sites within her information radius with equal probability. If no vacant sites exist within this radius, the losing agent remains stationary. We employ a sequential update scheme in which, at each step, an agent is chosen randomly to update its location. Fig. <ref> gives a schematic illustration of the decision-making process of an agent, and Fig. <ref> depicts typical examples of the game's dynamic progression across sequential Monte Carlo steps, where each row represents the arrangement of agents at one instance.  A Monte Carlo step (MC step) consists of one move made by a randomly selected losing agent. The states in which all agents are winners are the absorbing states of the system since no agents would move any further. Thus, the absorbing states can only be reached but not left, breaking the condition of detailed balance. The existence of absorbing states makes the overall dynamics a non-equilibrium process <cit.>.






The agents carry an accumulated payoff that contains their long-term winning history. For each agent, the payoff structure adds a score of one for a winning round and a score of zero for a losing round. Therefore, the cumulative payoff of an agent who is at the i^th site after t rounds can be written as 


    P(t) = 
    
        P(t-1)+1     if ν_i(t)≤τ
    
       
        P(t-1)     if ν_i(t) > τ


The cumulative payoff thus measures an individual's long-term wealth and success rate. All the agents start with a score of zero. The motive of each agent is to maximize their personal payoff through relocation and is assumed to be indifferent to the welfare of neither the other agents nor the system. 



 §.§ Emergent Macroscopic Properties 


The local crowd avoidance in the model leads to collective behavior at the global level. In this respect, an interesting question is whether the agents will achieve emergent coordination leading to optimal occupation of sites. Another aspect is how the winning chances are distributed in the population. To characterize the large-scale behavior of the system, we consider the emergent properties of global inefficiency, inequality, and fraction of frozen agents which are defined below.

 Global inefficiency (η) is defined as the normalized deviation of the number of losers from its minimum possible value averaged over several rounds in the steady state and for several initial conditions.  If N_l is the random variable representing the number of losers found in the system for a round of the game and N_l_min is its minimum possible value, we can define the global inefficiency per agent by, 
 
    η = ⟨⟨ N_l - N_l_min⟩⟩/N

 
Here, the double angular bracket indicates averaging over several rounds in the steady state and also over several initial conditions. N_l = N_l_min corresponds to the best possible arrangement of agents in the system for a given set of parameters. It is easy to see that N_l_min will be zero for very low densities and it will approach N as the density approaches one. Inefficiency η thus quantifies the degree of emergent organization in the system. In other words, it is a measure of the wastage of resources in the system at a global scale.

In the model, inequality between the cumulative payoffs of different agents may arise. Losers may remain losers repeatedly, and winners remain winners. We use the Gini coefficient to measure the level of inequality in the system <cit.>. If P_k denotes the wealth of the k^th agent and P denotes the mean wealth of all the N agents in a round, then the Gini coefficient for a round is given by, 


    g=∑_k=1^N∑_l=1^N| P_k-P_l|/2∑_k=1^N∑_l=1^NP_k
        =∑_k=1^N∑_l=1^N| P_k-P_l|/2N^2P
 

As in the case of η, the Gini coefficient G for a particular set of parameters is then found by averaging g over several rounds in the steady state and several initial conditions. The value of G ranges from 0 to 1, with 0 representing perfect equality and 1 indicating perfect inequality.

A different quantity to characterize the inequality is the fraction of frozen agents in the system (ϕ_w). i.e., agents who remain permanent winners and therefore remain stationary throughout the duration of the game. As we'll see, ϕ_w also characterises the order-disorder transition in the system.



 §.§ Peak Sustainable Density
 
We first address the following question: What is the maximum number of agents that can be packed or the maximum density that can be achieved in a 1D lattice such that the occupation densities inside the neighborhood of all the agents are less than their tolerance threshold (i.e., all agents are winning)? The answer to this can give us useful insights into the dynamics of the system for given values of z and τ. More importantly, this will give us the value of N_l_min, which is required to calculate global inefficiency (Eq. <ref>).

 We define peak sustainable density, ρ_p, as the maximum density of agents possible in the system with all the agents in a winning state and would like to determine it as a function of z and τ. Now, the densest packing of agents is attained when a maximum number of agents share each vacant site, and each agent accommodates the maximum possible number of agents in her neighborhood, still being a winner. To keep things manageable, we restrict our search to configurations with clusters of agents assuming equal size. A cluster of agents is defined as a maximal set of contiguous agents with vacant sites surrounding it. An exhaustive search for relatively small system sizes and neighborhood sizes confirms that in most cases, the densest packing is obtained for configurations with uniform-sized clusters of agents. In the following, we describe a systematic procedure for finding this dense packing, assuming that the clusters are of equal size. 
 




First, form clusters of agents on the lattice with a minimal but sufficient number of vacancies in between the clusters such that all the agents are winning. Within the neighborhood size z, each winning agent can afford a maximum of τ z number of agents. Hence, the biggest possible cluster of winning agents has a size of τ z +1. Obviously, the smallest possible cluster size is one. Now consider all possible configurations with cluster sizes varying from the maximum to the minimum possible value, dispersing vacancies (cluster gaps) in between, ensuring that all the agents have sufficient vacancies within their neighborhood to be winners. Among the different configurations that can be generated in this way, we can choose the one with the highest value of density (ρ). 


 To illustrate the above, consider, for example, a system with a neighborhood size, z=8 and tolerance threshold, τ=4/8. Each agent requires at least z- τ z=4 vacant sites in her neighborhood to be a winner. Fig. <ref> shows the various possibilities of forming clusters of winning agents with minimal but sufficient vacancies around the cluster. Clusters vary in size from the maximum possible value of five to the minimum possible value of one. The densest configuration is found to be the one with a cluster size of three and a cluster gap size of two. 

As the cluster size is incrementally reduced from the largest possible value, the neighborhood of the agents starts to extend into the nearby clusters and cluster gaps. If removing one agent from the cluster results in all the other agents gaining an additional vacant site in their neighborhood, then the cluster gap size too can be reduced accordingly. For example, in Fig. <ref>, when the cluster size is reduced from five to four, the cluster gap size can be reduced by one. On the other hand, if the removal of one agent results in the neighborhood of any one of the agents in the cluster extending into an occupied site in a nearby cluster, the cluster gap size can not be reduced. Notice that the cluster gap size remains stagnant at two when the cluster size is reduced from three to two in Fig. <ref>. 

Peak sustainable density is the highest value of density for the configurations generated in this manner. If the cluster size and cluster gap size for the densest configuration are denoted by a and b, respectively, then the peak sustainable density is given by,


    ρ = Cluster Size/Cluster Size + Gap Sizeρ_p = a/a+b




  §.§.§ General Expression for Peak Sustainable Density


We formulate a general expression for the peak sustainable density in terms of z and τ, following the method described above. Initially, when the biggest possible cluster of size τ z +1 is formed, the requirement of having z-τ z vacant sites in the neighborhood of each winning agent necessitates placing cluster gaps of size z-τ z on both sides of each cluster. Even so, the cluster gap size need not be greater than z/2, since the neighborhood of the agents extends only up to this limit towards either side of an agent. As a result, we have two scenarios that should be treated separately. 




  * For z-τ z > z/2, (i.e., τ < 1/2), the minimum cluster gap size needed for the biggest possible cluster with all agents winning is fixed at z/2. In such configurations, all the agents in a cluster can access the two neighboring cluster gaps on either side, even for the biggest possible cluster. 


  * For z-τ z ≤z/2, (i.e., τ≥1/2), the minimum cluster gap size required for all the agents in the biggest possible cluster to be winning  is z- τ z, a function of τ. In this case, a few agents of the biggest possible cluster can access only one of the neighboring cluster gaps. 



For τ < 1/2, we find that the densest possible packing with all agents winning is when the agents are packed as clusters of the largest possible size with a cluster gap of z/2 in between (see Appendix <ref> for details). Hence, the peak sustainable density for τ < 1/2 is given by


    ρ_p= 
        τ z+1/τ z+1+ z/2


For τ≥1/2, we see that there are two candidate arrangements of agents that will result in the highest density with all agents winning (See Appendix <ref> for details). The first possibility is when the agents are arranged as the biggest possible clusters with a cluster gap of size z-τ z in between. In this scenario, the neighborhoods of some agents within a cluster extend only to a single cluster gap. The packing density in this case is given by, 

    ρ =  τ z+1/z+1


The second possibility involves forming the largest possible cluster where the neighborhoods of agents at the cluster's edge span entirely across the two nearest cluster gaps, one on each side. For such a scenario, the packing density is given by,


    ρ = 
        z/2-j_max/(z/2-j_max)+⌈z-τ z/2⌉


where j_max is given by,


    j_max = ⌊z-τ z/2⌋-1


Here, ⌈ x ⌉ and ⌊ x ⌋ represent the ceil and floor functions, which, respectively, round a real number x up and down to the nearest integer. The peak sustainable density for τ≥1/2 is the highest value among the two fractions given by Eqs. (<ref>) and (<ref>).






Peak sustainable density values for a few values of z and τ determined using the method described above and verified by exhaustive search are listed in Table <ref>. 


 

  §.§.§ Guaranteed number of losers

 




In the model, the states of the system in which all the agents are winners and hence cease to have further dynamics are absorbing states. For low values of density, such absorbing states are aplenty. However, as the density increases, the configurations where everyone is a winner become fewer in number. Above the peak sustainable density (ρ_p), there are no such absorbing states, and a non-zero number of losers will always be present in the system. Thus, the peak sustainable density signals the change in the value of the minimum possible number of losers (N_l_min) from zero to non-zero.

Let x denote the number of extra agents present in the system above the number of agents corresponding to the peak sustainable density (ρ_p × L). i.e.,


    x = 
    
        0   if ρ≤ρ_p
    
       
        (ρ-ρ_p)L   if ρ > ρ_p


When such extra agents are present, the configurations with the maximum number of agents winning are achieved when the population is arranged in clusters and cluster gaps of size a and b, respectively, and the extra agents are filled in nearby cluster gaps. An example is shown in Fig. <ref>. Note that adding extra agents in this way will render some existing agents as well as the newly added ones losers. We can say that, in general, adding b number of extra agents into the system creates approximately (a+b) number of losers. Therefore, the expression for the minimum guaranteed number of losers in the system can be written as 


    N_l_min≈ (a+b) x/b
 
We use this value to calculate the global inefficiency in Eq. (<ref>).



§ SIMULATION RESULTS
 



We simulate the model using Monte Carlo methods and investigate its emergent behavior. For a given system with specific values of z and τ, two factors that play a major role in determining its dynamics and evolutionary behavior are density (ρ) and information radius (r). We study the steady-state statistical properties of the system, focusing on the relation of the emergent properties to these two parameters. Specifically, we focus on the macroscopic properties of the system, such as global inefficiency, the fraction of frozen agents, and the inequality measured by the Gini coefficient. 

The simulations start with a random distribution of agents on the lattice, and the results discussed in the subsequent sections use a system size of L=201, unless otherwise specified. The qualitative features of the system measured show negligible dependence on the system size L. As visible in Fig. <ref>, for lower densities, the inefficiency reaches the steady state value of zero in a few MC steps. At higher densities, the system stabilizes to the steady state value in approximately around 10^4 steps. Relaxation time diverges as we approach a critical density which is defined later.
We measure the macroscopic properties of the system over the final 10% of the total 10^4 Monte Carlo steps. The properties were then averaged over 10^3 random initial distribution of agents on the lattice.



 §.§ Population Density and Inefficiency
 

The population density is an important determinant of the evolution of the system and impacts the inefficiency and other macroscopic properties. Now, the inefficiency of a random arrangement of agents can be easily found to be

    η_rand  = ∑_i=0^z-τ z-1ziρ^(z-i)(1-ρ)^i-N_l_min/N

where the summation is over the number of vacant sites in the neighborhood of an agent and (z-τ z-1) is its maximum value such that the agent is a loser.





The adaptive behavior of the agents facilitates the reduction of inefficiency from its value of a random configuration. The variation of inefficiency with density for different values of the standardized information radius r_s = 2r/z and different τ values for a z=4 system are shown in Fig. <ref>. 

We define critical density (ρ_c) as the density below which a system can self-organize and achieve zero inefficiency. Below ρ_c, the system always reaches one of its absorbing states. ρ_c can be considered as the ‘carrying capacity’  of the system demarcating the transition from an efficient to an inefficient phase. The value of critical density depends on system parameters of neighborhood size, tolerance threshold, and information radius. 





From Fig. <ref>, it is clear that for certain systems, the value of critical density is non-zero and closer to ρ_p, and for certain other systems, its value tends towards zero. This is because, for a given τ, whenever r_s ≤ min{ 2 τ, 1}, a losing agent between two clusters of winning agents on either side, each of size r = τ z, can get trapped between them forever (two sample configurations where this happens for z=4, τ = 1/4, r_s=0.5 and z=4, τ = 2/4, r_s=1 are shown in Fig. <ref>). This means that we will get a non-zero value of η for a relatively low density of agents ρ for r_s ≤ min{2 τ, 1} as can be verified from Fig. <ref>. Although for r_s ≤ min{ 2 τ, 1} and at low densities, there are many configurations where the system eventually reaches the absorbing state, a handful of configurations exists where losing agents are trapped bounded by clusters of winning agents. At low densities, such configurations may only be realized over repeated random realizations. But such trapping states are guaranteed, even when the system size tends to infinity, and hence, theoretically, critical density tends towards zero for r_s ≤ min{ 2 τ, 1}.

For r_s > min{ 2 τ, 1}, the system has a non-zero value of ρ_c. In this case, we can see that the system exhibits three distinct behaviors in inefficiency with respect to its variation with density. For low values of density below ρ_c, the system's inefficiency is zero. This is because, for low densities, the number of vacant sites is large, and with r_s > min{ 2 τ, 1}, the competing agents can easily relocate and find sites whose neighborhoods are sparsely populated. The global inefficiency assumes a non-zero value at ρ_c and continues to rise to finally peak near ρ_p defined earlier (see Eqs. <ref>, <ref> & <ref>). We can see from Fig. <ref> that in this second range of density (ρ_c<ρ≲ρ_p), the inefficiency is non-zero but is lower than that of a random configuration. There are still absorbing states available in this range of ρ, but even with r_s >1, the system is set to unending dynamics and never achieves the highest efficiency possible.

The difference in the observed behavior for variation of inefficiency with density for r_s ≤ min{ 2 τ, 1} and r_s> min{ 2 τ, 1} is pronounced only when ρ≲ρ_p. As the density increases above ρ_p the inefficiency gets closer to that of a random configuration and shows a declining trend. The number of vacant sites is very small, and even a random arrangement of agents could distribute the few available vacant sites across the system. 

As ρ→ 1, most of the sites become occupied. The relevance of dynamics starts to fade at these densities, and the system's inefficiency remains equal to that of a random configuration. Hence, the very high-density range (ρ∼ 1) is also categorized as the inefficient phase of the system, although the wastage of resources compared to the best possible arrangement is low here.  





In Fig. <ref>(a), we show the variation of ρ_c with τ for r_s> min{ 2 τ, 1} for a z = 16 system. The critical density shows a linear growth with an increasing tolerance threshold for a constant value of z. Pearson's correlation coefficients, which measure the correlation between critical density and tolerance threshold, are approximately 0.999 for different r_s values. The corresponding p-values are close to zero (approximately 10^-19), indicating a robust linear correlation <cit.>. This behavior of ρ_c reflects how adaptive agents with larger tolerance thresholds can sustain systems of higher density at better efficiency. In Fig. <ref>(b), we plot the variation of ρ_c with r_s for r_s > min {2τ,1}. It is observed that, in general, ρ_c remains constant as r_s varies. 



 §.§ Information radius and its impact on the emergent behavior


The variation in the information radius is significant only for the density range above ρ_c. For ρ<ρ_c, the system evolves into one of its absorbing states with zero inefficiency. Above ρ_c, we observe two diametrically opposite responses for inefficiency with respect to variation in the information radius. The peak sustainable density approximately demarcates the transition in the response.





Fig. <ref>(a) shows the variation of inefficiency with respect to the standardized information radius r_s for a z=16 system at two different values of τ for ρ_c < ρ≤ρ_p. Contrary to the notion that the more information, the better could be the agent's performance and hence the self-organization of the system, lower inefficiency is observed at lower information radii in this density range. As the information radius gets larger, the randomness in the decision-making and arrangement process hinders the efficient arrangement of agents in the system. High efficiency of the system requires ordering at a very local level, which essentially happens when the information radius roughly matches the neighborhood size. In these systems, an optimum value of r_s exists at which the inefficiency is the least as evident from Fig. <ref>(a). 

The above result is in line with a few existing studies which show the irrelevance of high amount of input information in complex systems and how even highly detailed information can turn counter-productive for the system coordination and individual pay-offs <cit.>. Our model exhibiting better adaptation at lower values of information radius below peak sustainable density emphasizes the fact that better information, whether in quality or quantity, need not always result in better outcomes.

As the system's density rises above ρ_p, the variation of inefficiency with respect to the standardized information radius shows an inversion of behavior. The system exhibits significant inefficiency in the lower information range and performs worse than a random distribution, as illustrated in Fig. <ref>(b). The highest inefficiency is observed at r_s ∼ 1, paradoxically, which is the optimal information range for densities below the peak sustainable density. As the information radius increases further, the inefficiency of the system shows a slight decline. 

Another point to be noted is that the information radius (r) need not be larger than the neighborhood radius (z/2) for the system to self-organize. This is evident from the better global efficiency of the system observed even for r_s < 1 (Fig. <ref>(a)). In these systems, individual-level information and the corresponding behaviors can be thought to generate a collective intelligence which in turn drives the system to higher levels of coordination and adaptation <cit.>. 





Fig. <ref> plots the phase diagram of the model in the (ρ,r_s) plane and shows the impact of incremental changes in standardised information radius and density on the global inefficiency. The efficient phase is observed until ρ_c, up to which the system attains zero inefficiency. Above ρ_c, the system's inefficiency displays a highly dependent behavior on the input information and shows two contrasting responses to the input information (r_s) above and below ρ_p. In the efficient phase, the model consistently attains the absorbing state, while in the inefficient phase, a fluctuating steady state is obtained. 

The response of the system's inefficiency to the information radius depends on the tolerance threshold of the agents, as evident from Fig. <ref>. For changes in the tolerance threshold, the response of inefficiency to the standardized information radius alters its behavior. The region of inverted response to information, observed above ρ_p, gradually changes appearance before vanishing for the highest tolerance threshold. 







 §.§ Behaviour of Fraction of frozen agents phi and Gini Coefficient


Agents who are permanent winners and remain in their positions form an interesting group of agents whose behavior can give insights into the internal dynamics of the system. Their fraction in the population (ϕ_w) explains the critical density values. ϕ_w=1 signifies the attainment of the absorbing state, and hence ϕ_w can be used as an order parameter to indicate the transition in the system. As observed in Fig. <ref>, for r_s> min{ 2 τ, 1}, the fraction of permanent winners (ϕ_w) suddenly drops from 1 to 0  when the system transits from the efficient phase to the inefficient phase. 

As discussed in Section <ref>, for r_s ≤ min{τ z,1}, there are states where losing agents are permanently bound between immobile winners. Therefore, the curves corresponding to r_s ≤ min{τ z,1}, exhibit a decrease from 1 starting from ρ≈ 0. When the system remains completely static, for τ = z-1/z & r_s<1, the fractions of the permanent winners are the same as that of a random configuration. In this case, the fraction of the permanent winners is given by,


    ϕ_w=1-ρ^τ z+1


In Fig. <ref>(d), the cases for r_s<1 correspond to this scenario. 





 The inequality in agents' wealth quantified using the Gini coefficient (G) exhibits interesting characteristics when studied as a function of density and information radius. Fig. <ref> shows the variation of G with density for different standardized information radii (r_s). The Gini coefficient (G) changes from zero to a non-zero value at ρ_c. Thus, ρ_c marks the onset of not only inefficiency, but also inequality in a dynamic system. 
 
 The Gini coefficient decreases with an increase in the agents' information radius, as observed in Fig. <ref>. The observation holds for most density ranges, except for a narrow range immediately following ρ_c visible in Fig. <ref>. Thus, even when the role of information in reducing inefficiency remains variable, it is safe to generalize that a higher information radius almost always guarantees better equality in the system. 







§ CONCLUSION AND DISCUSSION


The local interactions and preferences of individual agents determine the large-scale statistical properties in many complex systems. In the past, researchers have looked into in detail how different agent preferences like alignment with neighbors (as in Voter models <cit.>), anti-alignment with neighbors (as in contrarian Voter models<cit.>), affinity to be among a particular type of agents (as in Schelling's model of racial segregation <cit.>), etc. affect global properties in a spatial setting. Strategic interaction between the neighboring agents is also considered extensively (As in spatial Prisoner's Dilemma <cit.>, Stag Hunt <cit.>, Hawk-dove <cit.> and Snow Drift Games <cit.>). 

In this work, motivated by examples of complex systems involving competition and sharing of physical space, we considered a different type of agent preference - local crowd avoidance -  in a spatial setting. In the model, competing agents strive for advantageous locations by avoiding the local crowd, trying to keep the local population density in their neighborhood below their tolerance threshold by relocation. We study the resultant emergent behavior on a 1D lattice system. The crowd avoidance model we define thus lies in the broad class of spatial models that investigate how a group of agents respond and form collective action and emergence based on spatial or neighborhood preferences. On a preliminary level, there is some similarity with Schelling's model of segregation in that both models trace patterns of emergence based on the requirement of having (or not having) a minimum fraction of a kind of agent in the neighborhood. However, the present model stands apart in that it focuses on understanding spatial crowd-avoidance behavior. The micro-mechanisms used in defining the rules and settings of the problem reflect this macro-agenda. For example, there is only one kind of agent, and empty sites that have neither preferences nor actions are actually the resource for which they compete. Furthermore, on the macroscale, unlike the Schelling's model which focuses on the emergence of segregation, we investigate the competitive allocation of spatial resources.

We analytically find the densest possible arrangement of agents for a given value of neighborhood size and tolerance threshold and arrive at the expressions for the highest density (termed the peak sustainable density) at which the system can be packed, ensuring that all the agents are winners. We simulated the model and analyzed the emergent behavior. The dependence of its macroscopic features - inefficiency, inequality, and the fraction of frozen agents in the system - on the population density and the information accessibility of the agents is studied. 

We find two crucial density values at which the macroscopic behavior of the system suddenly changes. The first one is the critical density (ρ_c). A dynamic system manifests the emergence of order and maintains zero inefficiency up to the critical density. If the density is below ρ_c, the system evolves into one of its absorbing states. The second is the peak sustainable density (ρ_p), above which there are guaranteed losers in the system. Between ρ_c and ρ_p, inefficiency is non-zero but remains lower than that of a random arrangement signaling emergent coordination between the agents. When the system's density is above ρ_p, the global inefficiency of the system starts to decline and approaches that of a randomly arranged configuration of agents. 
 
 Interestingly, the global inefficiency shows a non-monotonic dependence on the information accessibility of agents and displays two opposite characters above and below ρ_p. Below ρ_p, highly efficient system organization and agent adaptation are observed for smaller standardized information radius revealing how smaller information inputs may favor better system efficiency. There also exists an optimal information radius at which the inefficiency is minimum. Above the peak sustainable density, the system reverses its behavior in response to variations in the information radius.  In this regime, higher inefficiency occurs at lower information radii. The varying impact of information inputs on the global system behavior suggests that the prudent choice of information cannot be based on either the quantity or the quality of the information but depends on the parameters of the system and the resultant complex interaction. 

In contrast, the variation of the Gini coefficient, which measures the inequality between agents, with respect to information radius displays a consistent behavior for most density ranges. High inequality is observed at lower information radii, and the inequality falls off rapidly with an increase in the information radius for most of the density values except for a narrow density range near ρ_c. Therefore, although the role of information in reducing inequality is contingent on other conditions, its role in bringing down inequality is significant. 

Our study opens multiple avenues for further investigations into the unexplored realms of spatial crowd avoidance behavior and the resulting dynamics. An automatic extension of the problem is to look into the features of higher-dimensional systems and spaces with different structures. Attempts can be made to solve the system analytically, with the relocation of agents restricted to only the neighboring sites of an agent which could be more amenable to analytical treatments. The current model employs agents obeying a static and homogeneous rule with a sequential update scheme. This can be generalized by introducing adaptive and heterogeneous behaviors and updating rules. An interesting area where the heterogeneity of agents can bring useful insights is in addressing questions regarding the impact of information asymmetry on outcomes at both individual and global levels when some agents have access to more information. This will help us to understand the role of information in complex systems in a more comprehensive manner. The quantifiable nature of individual-level and collective information in our model provides ample scope for investigating the emergence of collective intelligence in the context of spatial resource sharing in future studies. 




§ GENERAL EXPRESSIONS FOR THE PEAK SUSTAINABLE DENSITY
 
We formulate general expressions for the peak sustainable density in terms of the neighborhood size z and tolerance threshold τ of the agents. In formulating these expressions, we limit our search to configurations with clusters of equal size.  Following the method described in Section <ref>, we compose configurations of winning agents with cluster sizes varying from the maximum possible value of τ z +1 to the minimum possible value of 1. The highest density among these configurations is identified as the peak sustainable density. 

We have seen that the neighborhoods of the agents in a cluster begin to expand into neighboring clusters and cluster gaps when the cluster size is gradually decreased from τ z +1. As stated in section <ref>, if removing one agent from a cluster results in all the other agents gaining additional vacant sites in their neighborhood then the cluster gap size too can be reduced accordingly. On the other hand, the cluster gap size cannot be reduced if removing one agent results in the neighborhood of any of the remaining agents in the cluster extends into an occupied site as a result. Using these facts, in the following, we formulate the general expressions for the peak sustainable density. We treat the cases τ < 1/2 and τ≥1/2 separately because they differ in the access of the agents in the largest possible cluster to the nearest cluster gaps. 



 §.§ For tolerance threshold tau < 1/2


In this case, the largest possible cluster size is (τ z +1) with a cluster gap of size z/2. The subsequent lower-sized clusters of winning agents with maximum density can be created by successively removing the agents, keeping the cluster gap size at an essential minimum while doing so. It can be seen that initially, for the removal of a few agents from the largest possible cluster, the cluster gap size can be reduced along with the cluster size. Fig. <ref>(a) shows the variation of density of agents with respect to variation in cluster size for such a sequence of configurations for a particular value of z and τ.

Let i denotes the number of agents removed from the biggest possible winning cluster described in the manner mentioned above. The expression for the density of agents for the series of configurations generated for increasing values of i can be written as,


    ρ= τ z+1-i/(τ z+1-i)+ (z/2 -i) ; where   i=0,1,2 … i_max







Here, i_max is the maximum number of agents that can be removed from the biggest possible winning cluster, such that the cluster gap can also be reduced concurrently. Thus, i_max corresponds to the case when the neighborhoods of all the agents cover the two nearest cluster gaps on either side and any further removal of agents results in the neighbourhood of at least one of the agents in the cluster extending into the next nearest cluster (i.e., to an occupied site). At this point, the cluster gap size will be (⌈z-τ z/2⌉), and the cluster will be of size (τ z +1-i_max).  Therefore, i_max can be expressed as,


    i_max=z/2- ⌈z-τ z/2⌉


It is easy to see that the highest value for ρ from Eq. <ref> is obtained for i=0. This is because, for τ < 1/2, peak sustainable density (ρ_p) is less than or equal to 1/2, and subtracting one from the numerator and two from the denominator of a fraction that is less than 1/2 always yields a smaller fraction. 

When the cluster size is reduced even further, the cluster gap size can not be reduced correspondingly since agents require at least z - τ z vacancies in their neighborhood to win. This vacancy requirement should be met by the two cluster gaps of fixed size ⌈z-τ z/2⌉ on both sides. Hence, packing density decreases at a faster rate when the cluster size is reduced beyond i_max (see Fig. <ref>(a)). (If z-τ z is an odd number, then configurations with cluster gaps of sizes  ⌈z-τ z/2⌉ and ⌊z-τ z/2⌋ asymmetrically distributed on both sides are possible. However, it should be noted that such configurations have a lower packing density compared to configurations with larger cluster sizes). 

This decreasing trend of the packing density will continue until the neighborhood of all the agents in the cluster starts to have access to the cluster gaps which are further away (third, fourth, etc.) at which the packing density will show momentary spiking as visible in Fig. <ref>(a).



 §.§ For tolerance threshold tau >= 1/2

 
 Fig. <ref>(b) shows the variation in the density of configurations where all agents win when the cluster size is decremented from the maximum to the minimum possible value for a sample system with τ≥1/2. In this case, the largest cluster size possible with all agents winning is (τ z +1), but with a cluster gap of size (z-τ z). Here, unlike in the earlier case, the agents at the edge of the biggest possible winning cluster have access to only one cluster gap. Therefore, the cluster gap size cannot be reduced initially, when decrementing the cluster size from the largest possible value, as all agents must remain winning. 
 
 Let i denote the number of agents removed from the largest possible cluster without reducing the cluster gap size as described above. Then, the densities of the configurations realized by decrementing the cluster size can be written as follows:


    ρ    = τ z+1-i/(τ z+1-i)+ (z -τ z)
        = τ z+1-i/z+1-i ; where   i=0,1,2 … i_max


 Here, i_max represents the maximum number of agents that can be removed from the largest possible winning cluster while maintaining a fixed minimum cluster gap size of z-τ z. This also pertains to the case when the cluster is of the minimum size, with the agent at the edge accessing only one nearest cluster. Consequently, at this juncture, the cluster size (τ z +1-i_max) is equal to (z/2+1). Therefore, i_max can be represented as


    i_max=(τ z +1) -(z/2+1)


It is easily seen that the highest value among the set of possible values for different i given by Eq. <ref> is for i=0, given by,


    ρ  =  τ z+1/z+1


 When the cluster size is further reduced from (z/2 + 1), the neighborhoods of all agents in the cluster begin to expand to the second nearest cluster gap. In that case, the cluster gap size can also be reduced along with the reduction in the cluster size. Hence, the expression for the density of these configurations when the agents have access to the two nearest cluster gaps can be written as, 


    ρ = 
        z/2-j/(z/2-j)+(z-τ z- (j+1)) ;
     where   j=0,1,2 … j_max


The first increasing trend of packing density when the cluster size is reduced, as seen in Fig. <ref>(b), happens when the neighbourhood of all the agents in the cluster gain access to the second nearest cluster gap and thus follows Eq. <ref>. This increasing trend in packing density will continue only until the neighborhoods of all the agents in the cluster entirely cover the two nearest cluster gaps, i.e. when b= ⌈z-τ z/2⌉. Thereafter, the packing density again starts to decrease. Hence, in Eq. <ref>, the upper limit of j can be determined using the condition,


    (z-τ z-(j+1)) ≥z-τ z/2


or,

    j ≤z-τ z/2-1


which implies,

    j_max = ⌊z-τ z/2⌋ -1


The highest value for Eq. <ref> is obtained when j=j_max, since for τ≥1/2, peak sustainable density (ρ_p) is greater than 1/2, which is given by, 


    ρ = 
        z/2-j_max/(z/2-j_max)+ ⌈z-τ z/2⌉


It can be seen that the configurations generated with cluster sizes smaller than (z/2- j_max) have a packing density less than or equal to the case when the neighborhoods of all the agents in the cluster span the two nearest cluster gaps. This constraint is due to the requirement of having (z- τ z) vacancies in the neighborhood of each agent. The local maxima in the packing density following the initial two peaks, as seen in Fig. <ref>(b), are attained when all agents in the cluster access the subsequent cluster gaps entirely.


 Thus, the peak sustainable density (ρ_p) for τ≥1/2 is the largest value among the set of fractions obtained from equations <ref> and <ref>. i.e., the densest packing is either for the one with the largest possible cluster size with all agents winning or for the one with a maximal cluster size with the neighborhood of all the agents in the cluster spanning the two nearest cluster gaps entirely. Note that numerically there is not much difference between the values given by Eqs. <ref> and <ref> in the parameter range of τ we are interested in. 



 Paper Title: Topological Anderson insulator phases in one dimensional quasi-periodic mechanical SSH chains.





Tata Institute of Fundamental Research, Hyderabad 500046, India

§ ABSTRACT 
In this paper, we investigate the transition between topological phases in a Su-Schrieffer-Heeger (SSH) model composed of springs and masses in which the intracellular Aubry-André disorder modulates the spring constants. We observe the presence of a topological Anderson insulating (TAI) phase within a specific range of quasi-periodic modulation strength and calculate the phase transition boundary analytically. We examine the localization properties of normal modes using their inverse participation ratio (IPR) of eigenstates of the dynamical matrix, and the corresponding fractal dimension associated with quasiperiodic modulation. We also examine the stability of the TAI phase across a range of modulation strengths and comments on the presence of mobility edge that separate localized modes from non-localized ones. We demonstrate the fact that special analytical techniques are needed to compute an exact expression for mobility edges in such scenarios.


Topological Anderson insulator phases in one dimensional quasi-periodic mechanical SSH chains
    Sayan Sircar
    July 29, 2024
=============================================================================================





§ INTRODUCTION

The study of topological phases of matter <cit.> has gardened significant research interest in the last decades. Among this, topological insulators <cit.> is a fascinating phase characterized by bulk states with a gap at the Fermi level, while hosting conducting edge modes in the bulk gap, <cit.> characterized by a non-trivial winding number, as per bulk-edge correspondence <cit.>. The Su–Schrieffer–Heeger (SSH) model <cit.> is a fundamental system consisting of two bands and showing non-trivial band topology , has been introduced to examine conductivity and transport properties of polyacetylene chain, particularly focusing on fractional charge soliton excitations, <cit.> and non-trivial conducting edge modes <cit.> in open boundary condition (OBC). Conducting edge states are maintained by internal symmetries and are resilient to specific types of disorder. Disorder in translational invariant systems disrupts the translational invariance of the system, leading to the assumption that it could undermine the topological properties.  The emergence of topological Anderson insulators (TAIs) indicates that disorder can trigger the transition from topologically trivial to non-trivial phases, <cit.>.  Anderson localization was proposed in a model of disordered
 system <cit.> first  in 1958, which since then has been performed in various experimental settings <cit.>.
 Typically, real materials exhibit Anderson-type disorder <cit.>, characterized by random potentials generated by limited concentration of contaminants dispersed through out the material. Depending on spatial dimensions, potential intensity δ has a critical value for example in three dimensions <cit.>. In contrast, in one dimension, for an infinitely large system, most of the states remain localized for arbitrary finite value of δ. The emergence of twisted bilayer-graphene <cit.> introduces a novel form of disorder known as quasi-periodic disorder. The best known model to track quasi-periodicity in condensed matter systems is Aubry-Andre-Harper (AAH) model. The 1D Aubry-Andre-Harper (AAH) model <cit.> illustrates a metal-insulator transition governed by self-duality point <cit.>. This transition occurs when the quasi-periodic modulation strength surpasses a critical threshold. Aubry and Andre introduced a one dimensional hopping model in tight-binding approximation, in which electrons experience a sinusoidal electrostatic potential that is incommensurate with the lattice. They demonstrated that the transition occurs at δ^AA=2J, where J represents the nearest-neighbour electron hopping amplitude. In this scenario, all eigenmodes become exponentially localized for δ^AA>2J or extend as plane waves for δ^AA<2J. These results differ from Anderson localization phenomenon that occurs in one dimensional systems. These conclusions have also been extended to non-Hermitian systems <cit.>. The presence of mobility edges in disordered systems, particularly in 3D Anderson insulators, has been demonstrated even in the 1D generalized AAH model. Compact analytic results for mobility edges under broken self-duality have been obtained <cit.> and for slowly varying modulation incommensurate with the lattice <cit.>.These works focus on calculating mobility edges that differentiate localized from non-localized modes. Recent studies have expanded this concept to anomalous mobility edges, which separate localized states from critical ones in one-dimensional disordered systems, differing from conventional mobility edges, <cit.>. Disorder-induced topological Anderson insulators (TAI) have been experimentally observed in 2D photonics <cit.>, 1D wires with controllable random disorder <cit.> protected by chiral symmetry, superconductors with spin-orbit coupling <cit.>. The TAI phase also emerges in this model (SSH) modulated with quasi-periodic intercell hopping, where most of the eigenstates are localized <cit.>. Recently there are interest in exploring similar phases in several acoustic systems and elastic <cit.>. Hence its a natural urge to search for a similar phase in various mechanical settings <cit.> as well. 
In this article, starting from the model as shown in Ref. <cit.>, we apply quasi-periodic modulation in on-site intra-cellular spring stiffness in a mechanical SSH chain, and calculate the topological phase transition boundary, localization properties with modulation δ^AAcos(2πβ j^u+ϕ), where δ^AA is the quasi-periodic modulation strength and u=1.




§ MECHANICAL SPRING-MASS MODEL

This section provides introduction to the SSH mechanical model used, and examines its topological properties in real space, akin to Ref. <cit.>.




 §.§ Dynamical matrix of the spring-mass chain

We are studying a one-dimensional finite chain whose ends are held fixed (FBC), and is made up of N unit cells <cit.>. 

One can identify two masses (having same mass) in each unit cell by a sub-lattice labelling α=A,B along with a unit cell labelling j ∈ [1,N]. There are two types of spring stiffness constants: K_a for intra-cellular and K_e for inter-cellular interactions, analogous to the hopping parameters in the original SSH chain. By tuning those parameters in this mechanical setting, different topological phases, akin to the original quantum SSH model, can be realized. The equation of motion for each mass can be formulated using Newton's second law or, equivalently, the Euler-Lagrange equations, which are commonly employed in normal modes analysis. The equations can be compactly expressed in matrix notation, assuming each mass undergoes harmonic displacement over time,

    u_j(t)=u_je^-iω_jt.

The second-order equations of motion can be converted into a linear matrix equation. 

This matrix, known as the dynamical matrix, contains the eigenvectors and eigenvalues that represent the normal modes and their frequencies,

    DU=ω^2U,

where D is the dynamical matrix of the system having dimensions 2N×2N, and U is displacement vector associated with the displacement of each mass U=[u_1^A,u_1^B,u_2^A,u_2^B,···,u_N^A,u_N^B]^T with u_j^α is the displacement of mass with unit cell index j and sub-lattice index α=A/B. The explicit form for the matrix is 

    D= [ K_a+K_b    -K_a       …       0;    -K_a K_a+K_b    -K_b       …;       ⋮       ⋮       ⋱    -K_a;       …       …    -K_a K_a+K_b;         ] .


The matrix has a tri-diagonal structure. Unlike the initial SSH Hamiltonian, changes in inter-cellular hopping only affect the off-diagonal elements, while in our mechanical scenario, adjustments to the inter-cellular stiffness constant also impact the diagonal elements. However, since all diagonal elements are identical, the system maintains chiral symmetry; these diagonal elements merely shift the eigenvalue spectrum to ω_0^2=K_a+K_b. We measure all other eigenvalues relative to this value. We can define the matrix D^' = D - ω_0^2I, that anti-commutes with the chiral operator Γ, centering the eigenvalue spectrum around ω_0^2=0, as shown below:

    Γ(D-ω_0^2I)+(D-ω_0^2I)Γ=0,

where I is the identity matrix having dimension 2N×2N, and Γ is the chiral operator, whose matrix representation is:

    Γ=  [  1  0  …  0;  0 -1  0  …;  ⋮  ⋮  ⋱  0;  …  …  0  1;    ] .





















 §.§ Characterizing the topology in real space

In quantum tight-binding hopping models, we compute the winding number in momentum space (where translational invariance is preserved) for topological characterization <cit.>. In large systems, translational symmetry holds in the bulk but is broken near the boundary. In the clean limit (without disorder), our system maintains this symmetry in the bulk, but disorder disrupts it. We will conduct the analysis in real space <cit.> and characterize the topology using local topological marker (LTM) <cit.>. An LTM is defined for each unit cell, which converges to the winding number computed using translational symmetry in the periodic system, when averaged away from the boundaries. The primary advantage of the LTM is its ability to handle disordered systems <cit.>. For computing LTMs for a chain composed of N unit cells, we need to create two N× N matrices of eigenvectors from the dynamical matrix: U_- = [U_1, U_2, …, U_N] for eigenvectors below the Fermi level, and U_+ = [U_N+1, U_N+2, …, U_2N] for those above it. Using these matrices, we can build the projectors for the bands as P_- = U_- U_-^T and P_+ = U_+ U_+^T, allowing us to construct a flat-band Hamiltonian <cit.>. The flat-band Hamiltonian, topologically equivalent to D, is expressed as Q = P_+ - P_-. This can be spectral decomposed into sub-lattices A and B as Q = Q_AB + Q_BA = Γ_AQΓ_B + Γ_BQΓ_A, with 

    Γ_A/B= [ 1/0   0   0   0   …;   0 0/1   0   0   …;   0   0 1/0   0   …;   …   …   …   …   …;     ]   ,

are the spectral decomposition operators and Γ=Γ_A-Γ_B is the chiral operator. The mathematical definition of LTM <cit.> is


    ν(k)=1/2∑_α=A,B(Q_BA[X,Q_AB])_kα,kα+(Q_AB[Q_BA,X])_kα,kα ,


X is defined as the position operator, whose matrix representation has dimensions 2N × 2N. The specific form of X is given by X = [-N, -N, -(N-1), -(N-1), …, (N-1), (N-1)], all arranged in as diagonal elements with off-diagonal components being set to 0. The local topological marker (LTM) converges to the winding number and can have only two values: ν = 0, characterizes topologically trivial phase, while ν=1 characterizes topologically non-trivial phase, respectively. Each phase is tuned to a specific ratio between the parameters K_a and K_b, as shown in Fig. <ref>. The ratio K_b/K_a > 1 corresponds to the non-trivial phase, which is characterised by the system possessing conducting edge states, while K_b/K_a < 1 corresponds to the trivial phase, that does not host conducting edge states.






§ MECHANICAL MODEL MODULATED WITH INTRACELL AUBRY-ANDRE SPRING CONSTANTS



 §.§ Dynamical matrix of the modulated spring-mass system

Our model is a nearest-neighbour mechanical model <cit.> that is influenced by AA modulation. The AA modulation is defined by a sinusoidal term whose periodic characterisation is in-commensurate with the underlying lattice structure, because of the irrational number β, defined in our case as the inverse of golden ratio, g^-1=2/1+√(5). 

The equations of motion of our model for type A mass in jth unit cell will be 

    ü^A_j   = -[K_a +  K_b + δ^AAcos(2πβ j + ϕ)] u^A_j
          + [K_a+δ^AAcos(2πβ j + ϕ)] u^B_j + [K_b] u^B_j-1.

Similarly for type B mass in jth unit cell, 

    ü^B_j   = -[K_a + K_b + δ^AAcos(2πβ (j) + ϕ)] u^B_j
          + [K_a+δ^AAcos(2πβ j + ϕ)] u^A_j + [K_b] u^A_j+1.

where δ^AA is the AA amplitude (with j∈[1,N] as unit cell index), the spring constants are defined as 

    k_j=k_j+δ_j^AA=K_a+δ^AAcos(2πβ j+ϕ),
 for j even.
Different values of ϕ are possible; however, in this article, we will fix ϕ at 0 when discussing incommensurate modulation. One can also use an ensemble of different ϕ values and then compute a disorder average of all relevant quantities. In this article, as we have already fixed the value of ϕ, hence no disorder averaging is needed. Perturbing the stiffness of the springs introduces new diagonal terms, which do not appear in the original SSH model. This perturbation alters both diagonal and off-diagonal terms, hence disrupting the chirality of the dynamical matrix. To maintain the chiral symmetry, we introduce local springs to each mass.
The equations of motion of our model for type A mass in jth unit cell will be 

    ü^A_j   = -[K_a  + K_b + δ^AAcos(2πβ j + ϕ)+K^0,A_j] u^A_j
          + [K_a + δ^AAcos(2πβ j + ϕ)] u^B_j + [K_b] u^B_j-1.




Similarly for type B mass in jth unit cell will be 

    ü^B_j   = -[K_a + K_b + δ^AAcos(2πβ (j) + ϕ)+K^0,B_j] u^B_j
          + [K_a + δ^AAcos(2πβ j + ϕ)] u^A_j + [K_b] u^A_j+1,

where K^0,A_j=δ^AA[1-cos(2πβ j+ϕ)]=K^0,B_j .
This selection of on-site spring stiffness preserves the chirality of the matrix and guarantees that all eigenvalues are positive. Assuming a normal mode solution, the two equations can be expressed as follows:

    ω^2u^A_j   = (K_a + K_b + δ^AA)u^A_j
          -(K_a +δ^AAcos(2πβ (j) + ϕ) )u^B_j
          -(K_b)u^B_j-1 ,


   similarly for B type mass one has 

    ω^2u^B_j   = (K_a + K_b + δ^AA )u^B_j
          -(K_a + δ^AAcos(2πβ (j) + ϕ))u^A_j
          -(K_b)u^A_j+1.

The two equations can be compactly expressed in matrix form using a dynamical matrix, similar to Eq. (<ref>) as


    D=[             K_a+K_b+δ^AA      -K_a-δ^AAcos(2πβ+ϕ)                        …                     -K_b;      -K_a-δ^AAcos(2πβ+ϕ)             K_a+K_b+δ^AA                     -K_b                        …;                        ⋮                        ⋮                        ⋱ -K_a-δ^AAcos(2πβ(N-1)+ϕ);                     -K_b                        … -K_a-δ^AAcos(2πβ(N-1)+ϕ)             K_a+K_b+δ^AA;                          ]  .





In order to achieve genuine quasi-periodicity, it is important to work with an infinite system. To simulate this, we apply periodic boundary conditions within the lattice and find a suitable rational approximation β̃=F_n-1/F_n, where the numerator and denominator are successive terms in the Fibonacci sequence.


 §.§ Topological characterisation 

We characterize the topology by computing the LTM, as indicated in Eq. (<ref>), for each K_a and δ^AA while fixing K_b=1. This approach leads to convergence with the winding number. Specifically, we set ϕ=0, use the rational approximation β̃=377/610, include N=610 unit cells, and vary K_a∈[0,5] and δ^AA∈[0,6]. The topological phase transition in clean systems i.e. with δ^AA=0 occurs when K_a=K_b <cit.>. Secondly, we will examine the behavior of localization length, which is known to diverge at the Fermi level during topological phase transitions in electronic systems <cit.>. This divergence aligns with the closing of the band gap. To validate this, we analytically calculate the localization length Λ and verify its divergence in the transition region. We measure the entire eigenvalue spectrum relative to the middle frequency ω_0^2=K_a+K_b+δ^AA, so that the Fermi level can be centered at zero energy level i.e. ω^2-ω_0^2=0. Hence the equation of motions for states at the Fermi level are:

    (K_a+δ^AAcos(2πβ j+ϕ))u^B_j+(K_b)u^B_j-1=0,


    
    (K_a+δ^AAcos(2πβ j+ϕ))u^A_j+(K_b + δ^AAcos(2πβ (j) + ϕ))u^A_j+1=0.



Simplifying the equation for A sub-lattice we have

    u^A_j+1=(-(K_a+δ^AAcos(2πβ j+ϕ))/K_b)u^A_j.

The above equation leads to recursive relation as 

    u^A_N=(-1)^N-1∏_j=1^j=N-1(-(K_a+δ^AAcos(2πβ j+ϕ))/K_b)u^A_1.

We can derive the Lyapunov exponent γ, which is the inverse of the localization length <cit.>,

    γ=-lim_N →∞1/Nln|u^A_N/u^A_1|,


    γ=-lim_N →∞1/N∑_j=1^N-1ln|K_b/K_a+δ^AAcos(2πβ j+ϕ)|.

Due to the AA modulation being incommensurate with the underlying lattice, the summation can be expressed as a Riemann integral. Utilizing the properties of irrational rotation <cit.>, integrating it over angular variable θ∈ [0, 2π], one has

    γ=-1/2π∫_0^2πln|K_b/K_a+δ^AAcos(θ)|dθ .

The Eq. (<ref>) can be evaluated to 

    γ=ln(K_a+√(K_a^2-(δ^AA)^2)/2K_b) ,


for K_a>δ^AA.
And for K_a<δ^AA the expression evaluate to 

    γ=ln(δ^AA/2K_b) .

Setting the Lyapunov exponent to 0 computes the boundary of the topological phase transition, indicating a divergent localization length.
Equations for the critical diverging line are 

    K_a=(δ^AA)^2/4K_b+K_b,
 

for K_a>δ^AA 
and 

    δ^AA=2K_b
 for K_a<δ^AA.
The initial phase ϕ in the modulation does not need to vary, as its information is lost during integration with incommensurate modulation. The localization length at the central frequency can be numerically computed using the transfer matrix method <cit.> as a function of relevant parameters.



 §.§  Calculation of inverse participation ratio (IPR)

To gain insight into the localization properties of the model, we can analyze the inverse-participation ratios (IPR). The IPR for a specific eigenstate is defined as 


    IPR(ψ⟩)=∑_j=1^2N|u_j|^4,

where u_j are the components of the normalized eigenstate |ψ⟩ in a space of dimension 2N.
For extended eigenmodes, the inverse participation ratio (IPR) scales as 1/N, where N is the system size. In contrast, fully localized eigenmodes exhibit an IPR that scales as 1. Eigenmodes that do not fit these patterns define the critical phase of the phase transition <cit.>. The IPR of eigenstates can be plotted against energy eigenvalues and Aubry-André strength δ^AA. These plots facilitate the visualization of transitions among non-localized, localized, and critical phases in both real and momentum spaces.
When studying localization with AA modulation, a crucial parameter to calculate is fractal dimension D_f, which does not depend on the size of the system, and influences the behavior of IPR (or average of all state IPR) as size N varies. The scaling relation is IPR≈ (2N)^-D_f. To derive D_f from finite size systems, we use the size-dependent quantity d_f(N), which approaches the fractal dimension D_f as N increases. The expression for d_f(N) is as follows:

    d_f(N)=-ln(IPR)/ln(2N).

In the context where 2N denotes the system size, D_f equals 1 in the non-localized phase and 0 in the localized phase <cit.>. Numerical computations as shown in 

Fig. <ref>, shows that as N increases, the value of d_f(N) approaches D_f. The value of d_f(N) converges to 1 in case of the non-localized phase and to 0 in case of the localized phase, where a finite portion of states remains localized. In the critical region, d_f(N) stabilizes at value between 0 and 1.



§ CONCLUSION

In this study, we expanded the exploration of topological phase transitions in a mechanical SSH model, as described in <cit.>, by applying Aubry-André modulation only to the intra-cellular spring constants, rather than the intercellular ones. This approach resulted in distinct localization properties and the emergence of mobility edges, which could not be analytically derived from Avila's global theory <cit.>. We preserved the chiral symmetry of the dynamical matrix throughout our calculations. Consequently, we utilized a topological invariant from class-AIII or DBI, along with its real-space covariant form, to analyze and characterize the topological phases. We developed an analytical expression for the Lyapunov exponent, enabling us to predict the boundary of the topological phase transition based on system parameters (K_a, K_b, δ^AA). We examined the localization properties of the eigenmodes by calculating the inverse participation ratio (IPR) and fractal dimensions. We have used fractal dimension <cit.> to quantitatively estimate the localized and non-localized phases as well as to detect the presence of critical phase. The article emphasized that the global theory <cit.> did not enable the exact calculation of mobility edges in such a scenario. We planned to develop novel analytical techniques for calculating the exact expression of mobility edge in our future work.



apsrev4-2




 Paper Title: Simulation of the continuous-time random walk using subordination schemes.












 








Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
 








             
§ ABSTRACT 
The continuous time random walk model has been widely applied in various fields, including physics, biology, chemistry, finance, social phenomena, etc.  In this work, we present an algorithm that utilizes a subordinate formula to generate data of the continuous time random walk in the long time limit.
The algorithm has been validated using commonly employed observables, such as typical fluctuations of the positional distribution, rare fluctuations, the mean and the variance of the position, and breakthrough curves with time-dependent bias, demonstrating a perfect match.

















                             

                              
Simulation of the continuous-time random walk using subordination schemes
    Wanli Wang
    August 5, 2024
=========================================================================



§ INTRODUCTION

The continuous-time random walk (CTRW) <cit.> is a stochastic process for a random walk that jumps from one position to another one.
As an extension of the discrete random walk process, it was originally discussed by
Montroll and Weiss <cit.>, where a random walk is subordinated to a renewal process.  Waiting times and displacements of the random particles are independent and identically distributed (IID) <cit.>. For the waiting times, it may be drawn from a heavy or narrow-tailed distribution. Different combinations of waiting times and displacements distributions yield different types of diffusion statistics, ranging from sub-diffusion, normal diffusion, and super diffusion. The CTRW model has attracted great attention from researchers due to its wide applications.  For example, it was used in many physical and chemical phenomena, including the transport of amorphous materials <cit.>, contamination in disorder systems <cit.>, the U.S. dollar in Deutsche mark future exchange <cit.>, diffusion of polymers in attractive nano-particle polymer mixtures <cit.> and so on.


The key issue is to discuss and simulate the positional distribution of the CTRW model. Two alternative approaches are the  fractional diffusion equation <cit.> and Langevin equations <cit.>. For example, Fogedby used decouple Langevin equations concerning power law step and waiting time distributions <cit.> to illustrate the CTRW model. One advantage is that the drift force field of the system is natural to be added. This method was further extended in Ref. <cit.>. See also discussions of coupled cases <cit.>. Here we use the statistics of the number of renewals in the long time limit, eliminating the need to generate waiting times statistics.



This manuscript focuses on the case when waiting time has a finite mean and an infinite variance, and the displacement follows a Gaussian distribution with a non-zero mean. However, the methods we provided are much more general. Due to the asymmetric displacement distribution, sometimes it was called the biased CTRW model. The biased CTRW has garnered significant attention and a vast number of specialized literatures <cit.>.
When the bias of the system or the mean of the displacement is zero, the process shows normal diffusion. While, when the bias is added, CTRW shows the enhanced or supper diffusion <cit.>, i.e.,

    ⟨ x^2⟩-⟨ x⟩^2  ∼ t^3-α

with 1<α<2.
In the long time limit, the positional distribution follows the asymmetric Lévy stable law <cit.> according to the Montroll-Weiss equation <cit.>. The fat tail is a key characteristic of the asymmetric Lévy distribution.   Generally, when seeking data on CTRW, we obtain a random count representing the number of renewals occurring within the time interval from zero to t, then use subordinated processes to get the  particle's position. A drawback is that it takes a long time to find the number of renewals when the observation time is long.  Thus, it would be interesting to find a new way to generate the number of renewals and the position of particles.


Recall that in the long time limit, the number of renewals follows the asymmetric Lévy stable law <cit.>. By instinct, one possible way is to generate the number of renewals based on the mentioned Lévy stable law. The Lévy stable law is perfectly correct, but it does have some drawbacks. For example, the far tail of Lévy distribution tends to infinity, which indicates that the number of the renewals tends toward negative infinity.  In turn, the Lévy stable law gives an infinite mean square displacement <cit.> (MSD) of the number of renewals. It is certainly not true, as the far tail should have a cutoff <cit.> for a finite observation time t.  In addition, as suggested in <cit.>, the MSD of the number is governed by an infinite density. On the other hand, the positional distribution in the context of simulations converges to the Lévy stable law rather slowly, especially when the system's bias is weak.
In some sense, the convergence problem was solved in <cit.> using the fractional advection-diffusion-asymmetry equation. To be more exact, the solution of the equation is the convolution of the Lévy stable law and a Gaussian distribution. Unfortunately, the solution of the fractional advection-diffusion-asymmetry equation yields a divergent MSD <cit.>.


Our goal in this manuscript is to construct a PDF to describe the number of renewals, which is not only valid for the typical fluctuations of the number of renewals but also effective for rare events. The main idea is that we use the Lévy stable law  and the non-normalized density to illustrate the central part and the far tail of the number of renewals, respectively. Below, we will demonstrate that these two laws exhibit distinct scaling limits on the number of renewals and then utilize commonly employed observables to illustrate the effectiveness of the constructed PDF.




The organization of the manuscript is as follows.  In Sec. <ref>, we give the definition of the CTRW model and the algorithm to generate the number of renewals in the long time limit. Specifically, the modified Lévy stable law is introduced. We further check the theoretical prediction using the widely used observables in Sec. <ref>, including  typical fluctuations of the position, large deviations of the position describing the far tail, the mean and the variance of the position, and breakthrough curves with the time-dependent bias. Finally, we conclude the manuscript with a discussion in Sec. <ref>.



















§ MODEL AND ALGORITHM

Below, we present the definition of the CTRW model and the algorithm for simulating data on the number of renewals and positions.



 §.§ Model


In this subsection, we introduce the CTRW model, which is defined as follows. A random walk starts from its initial position. The particle waits at its initial position for time τ_1 drawn from ϕ(τ), and then makes a jump to x_1 with x_1 following f(χ). The particle traps at x_1 for time τ_2 and then jumps to the position x_1+x_2 with x_2 generated from f(χ). Then, the process is renewed.  All along the manuscript, the waiting times τ_i are IID random variables, and the same holds for χ_i. The number of renewals, denoted as N_t, obeys the relation ∑_i=1^N_tτ_i+B_t=t, where B_t is the backward recurrence <cit.> and t is the total observation time.

All along the manuscript, waiting times follow a power law distribution <cit.>

    ϕ(τ)={[            0,        τ<τ_0;; ατ_0^α/τ^1+α,         τ≥τ_0 ].

with 1<α<2. As mentioned, waiting times  τ have a finite mean and an infinite variance. Performing the Laplace transform with respect to ϕ(τ), we have ϕ(s)=∫_0^∞exp(-st)ϕ(τ)dτ∼ 1-⟨τ⟩+(τ_0)^α|Γ(1-α)|s^α with ⟨τ⟩=ατ_0/(α-1) being the mean of waiting times.

For displacements, we focus on Gaussian distribution with a finite mean a and a finite variance σ^2

    f(x)=1/√(2σ^2π)exp[-(x-a)^2/2σ^2].




Let Q_t(N) be the PDF of the number of renewals and f(x|N) be the conditional probability of finding the particle on the position x conditioned it made N jumps. The PDF of the position follows

    P(x,t)   =∑_N=0^∞Q_t(N)f(x|N)
       →∫_0^∞Q_t(N)f(x|N) dN.

Eq. (<ref>) describes the particle at position x after N steps on conditional that  exactly N steps were
made up to time t. Eq. (<ref>) can be interpreted as the so-called integral formula
of subordination <cit.>, i.e., a random walk is subordinated to a renewal process.
The subordination scheme has been employed in mathematics and physics within various contexts, including fractional kinetic equations <cit.>, Jacobi stochastic volatility models <cit.>, and financial models <cit.>. Below, we will use Eq. (<ref>) to generate the data of positions.









 Recall that  displacements are IID random variables, thus f(x|N) obeys

    f(x|N)=1/√(2σ^2π N)exp[-(x-aN)^2/2σ^2N].

As described in <cit.>, in the long time limit, the number of renewals follows the asymmetric Lévy distribution

    Q_t(ξ)∼ℒ_α(ξ)

with ξ=(N-t/⟨τ⟩)/(t/t̅)^1/α and t̅=⟨τ⟩^1+α/(τ_0^α|Γ(1-α)|). Eq. (<ref>) is plotted by the red solid line in Fig. <ref>.
Here the symbol ℒ_α(ξ) is the Lévy stable distribution, whose Fourier transform is exp((ik)^α). Typical fluctuations Eq. (<ref>) describes the case when N-t/⟨τ⟩ is of the order of t^1/α, i.e., the central part of the distribution. Using the asymptotic behavior of the Lévy distribution, the far  left tail of N follows Q_t(ξ)∼ (-ξ)^-1-α/Γ(-α). It indicates that Eq. (<ref>) does not give the information of the MSD <cit.>.





 §.§ Algorithm



According to  Eqs. (<ref>, <ref>, <ref>), in the long time limit, the positional distribution follows

    P(x,t)∼   1/(t/t)^1/α∫_0^∞ℒ_α(N-t/⟨τ⟩/(t/t)^1/α)
       exp(-(x-aN)^2/2σ^2 N)/√(2πσ^2 N) dN,

Changing the random variables, the above equation leads to

    P(x,t)∼   ∫_-t/⟨τ⟩(t/t)^1/α^∞ℒ_α(ξ)
       ×exp(-(x-at/⟨τ⟩-aξ(t/t)^1/α)^2/2σ^2(t/⟨τ⟩+ξ(t/t)^1/α))/√(2σ^2π(t/⟨τ⟩+ξ(t/t)^1/α)) dξ,

Eq. (<ref>) is the main result of this section used to generate the random variables. Note that ℒ_α(ξ) in Eq. (<ref>) should be modified, denoted as ℒ_α^*(ξ).
Otherwise, it will not accurately account for the MSD and the far tail of the position distribution.
For that, we generate the statistics of ℒ_α^*(ξ), stemming from typical fluctuations and the infinite density <cit.> and then use it to get the data of the position. Afterwards, positions of particles are drawn from a Gaussian distribution with the mean at/⟨τ⟩+aξ(t/t̅)^1/α and the variance δ^2(t/⟨τ⟩+ξ(t/t̅)^1/α)^2.





Rare fluctuations of N were investigated, describing the scaling when N-t/⟨τ⟩ is of the order of t. According to the result given in <cit.>, large deviations of N obey

    Q_t(ϵ)∼α(τ_0)^α[t(-ϵ⟨τ⟩)^-α-1+1-α/α(-ϵ⟨τ⟩)^-α]

with ϵ=N-t/⟨τ⟩.  Consider the random variable, ξ=ϵ/(t/t̅)^1/α, we have

    Q_t(ξ)∼   α(τ_0)^α(t/t̅)^1/α
       ×[t/(-ξ(t/t̅)^1/α⟨τ⟩)^α+1+(1-α)/α/(-ξ(t/t̅)^1/α⟨τ⟩)^α].

See the black dashed line in Fig. <ref>.
Recall that Eq. (<ref>) is valid for small ξ corresponding to small N.
As shown in Fig. <ref>, Eq. (<ref>) is valid for the central part of the number of renewals together with the right tail, and the non-normalized density Eq. (<ref>) works for the left far tail. By instinct, we can utilize the effective part of Eqs. (<ref>) and (<ref>) to generate data of the number of renewals.



In order to generate the random variable based on Eq. (<ref>), we construct a PDF by adding a condition z on Eq. (<ref>),

    Q_t^I(ξ)=α(τ_0)^α[t̅/(-ξ⟨τ⟩)^α+1+(1-α)/α/(-ξ⟨τ⟩)^α(t/t̅)^1-1/α]

with b̅<ξ≤ z<0. Here, z is an undetermined constant.
As N≥ 0, the bottom limit of ξ should be great than -t^1-1/αt̅^1/α/⟨τ⟩, i.e.,

    b̅=-(α/Γ (2-α))^1/α(t/τ_0)^1-1/α.

When 1<α<2, in the long time limit, b̅→ -∞. While for a finite observation time t, ξ can not tend to infinity. Thus, the modified Lévy stable law ℒ_α^*(ξ) should have a cutoff at the tail.


We further introduce the cumulative distribution function, describing the probability of ξ when ξ is less than y

    h(y)=∫_b̅^y Q_t^I(ξ) dξ.

Performing the integral from b̅ to z, Eq. (<ref>) leads to

    τ_0 z ((α-1) t)^1/α | Γ (1-α)| ^1/α+t (ατ_0)^1/α/t (ατ_0)^1/α (-z)^α | Γ (1-α)| =1.

Eq. (<ref>) can be used to determine the upper limit
z defined in Eq. (<ref>). When α=3/2, using a cubic equation, the formula
for the roots of Eq. (<ref>) is easy to find. While, if α=5/4, Eq. (<ref>) reduces to a 5-th degree polynomial. Thus, according to the Abel-Ruffini theorem, the equation concerning z shown by Eq. (<ref>) is generally unsolvable. For that, to have a universal program valid for all kinds of α, we have to abandon some methods that are z- dependent.






As shown in Fig. <ref>, the modified Lévy stable has three parts, labeled as A, B, and C, respectively. Part A represents the  probability for the normalization, describing the fade probability of random variables equaling to b̅. For part B, we utilize the information of the modified infinite density Eq. (<ref>).
In part C, random variables ξ are  drawn from the Lévy stable law Eq. (<ref>).
The dot D is the intersection of the part B and the part C.




Below, we calculate the probability of random variables located in regions B and C, respectively.  For the probability of rare fluctuations in region B, we have

    P_rare(ξ^*)   =h(ξ^*)
       =τ_0 z ((α-1) t| Γ (1-α)|)^1/α+t (ατ_0)^1/α/t (ατ_0)^1/α (-ξ^*)^α | Γ (1-α)| ,

as mentioned before, ξ^* is the transition point of the region B and the region C. Recall that part C is illustrated by the Lévy stable law, the probability in the region C follows

    P_typical(ξ^*)=∫_ξ^*^∞ℒ(ξ) dξ.

When t→∞, P_rare(ξ^*)+P_typical(ξ^*) tends to unity. While for a finite t, P_rare(ξ^*)+P_typical(ξ^*)<1. Therefore, we further introduce the modified or fake survival probability describing the non-traveling particles

    P_survival(ξ^*)=1-P_rare(ξ^*)-P_typical(ξ^*).

In other words, it is related to scenarios where the number of renewals is zero, which leads to the non-moving particles in the context of the CTRW model.


Now we discuss the details of generating random variables ξ drawn from ℒ_α^*(ξ). Firstly, we generate a random variable drawn from a uniform distribution, denoted as η. Based on the value of
η, we determine which part it belongs to.
If η > P_survival(ξ^*) + P_rare(ξ^*), we generate a random variable ξ from ℒ(ξ) under the condition that ξ≥ξ^*. Instructions for generating a Lévy stable distribution can be found on <cit.>
or the MATLAB Central File Exchange, as detailed in the work by Veillette <cit.>.
When P_survival(ξ^*) < η < P_survival(ξ^*) + P_rare(ξ^*),  ξ are drawn from the modified infinity density Eq. (<ref>) using the accept/reject algorithm. If η < P_survival(ξ), the random variable is fixed, i.e., ξ = b̅. Note that the MSD is sensitive to lagging particles far behind the mean <cit.>. Therefore, when simulating the position for the MSD, the random variables η are larger than P_survival(ξ^*) using the theory given in Eq. (<ref>). Throughout the manuscript, the theories marked in  legends of figures refer to the algorithm discussed in this subsection. Specifically, we generate ξ drawn from the modified Lévy law ℒ_α^*(ξ) and then use it to determine the position statistics given by the expression

    J(x)=exp(-(x-at/⟨τ⟩-aξ(t/t)^1/α)^2/2σ^2(t/⟨τ⟩+ξ(t/t)^1/α))/√(2σ^2π(t/⟨τ⟩+ξ(t/t)^1/α)).

Here, J(x) represents the distribution of the position
x at the time t within the context of the CTRW model, where  ξ are drawn from ℒ_α^*(ξ).































§ THEORETICAL PREDICTIONS COMPARED WITH ASYMPTOTIC BEHAVIORS AND SIMULATIONS


In this section, we generate the data of the position obtained from the method outlined in Sec. <ref> to obtain  typical fluctuations of the position, rare fluctuations, the mean of position, the variance of the position, and breakthrough curves. Additionally, we generate the positional distribution data using simulations and outline the well-known asymptotic behaviors for comparative analysis.







 §.§ Typical fluctuations

Several authors have studied typical fluctuations of the position for the biased CTRW; see Refs. <cit.>.  In the long-time approximation, the scaled positional distribution follows the asymmetric Lévy stable distribution <cit.>

    P(ξ)∼ℒ_α(ξ)

with ξ=(x - at/⟨τ⟩)/l(t) ⟩ and l(t) = a(t/t)^1/α. In Eq. (<ref>), it is crucial to emphasize that a must be nonzero; otherwise, the positional distribution would yield a Gaussian distribution. In the general scenario, P(x,t) is described by the convolution of a Lévy stable distribution and a Gaussian distribution <cit.>

    P(x,t)∼∫_-∞^∞ℒ_α(ξ)exp(-(x-at/⟨τ⟩-aξ(t/t)^1/α)^2/2σ^2t/⟨τ⟩)/√(2πσ^2t/⟨τ⟩) dξ,

describing typical fluctuations when x-at/⟨τ⟩∼ t^1/α, i.e., the central part of the positional distribution. As shown in Fig. <ref>, Eq. (<ref>) fails for the left tail of the positional distribution. While the theoretical prediction given in the manuscript is valid for both tails. It indicates that if we use the modified Lévy stable law in Eq. (<ref>), Eq. (<ref>) does work perfectly. In addition, we plot the positional distribution at a short time t, see Figs. <ref> and <ref>. For this case, the positional distribution follows a nearly Gaussian distribution, affirming the validity of our findings. All along the manuscript, the theoretical predictions are based on the algorithm described in Sec. <ref>, which uses the modified Lévy stable law to generate statistics of the number of renewals.













 §.§ Rare fluctuations

To proceed, we discuss the limit of the Montroll-Weiss equation (<ref>)  but when |s|/|k| is fixed. Specifically, s and k are small and in the same order.  As a  complementary to the Lévy law Eq. (<ref>), the rare fluctuations illustrate the far tail of the positional distribution.
Following the result given in <cit.>, rare fluctuations are

    P(x,t)∼(τ_0)^α/at^α(α(1-x/a/t/⟨τ⟩)^-1-α-(α-1)(1-x/a/t/⟨τ⟩)^-α),

describing the scaling when x is of the order of the time t.
Eq. (<ref>) gives a prediction for  particles that are near the initial position. In other words, the mentioned slowing-moving particles stem from these longest single waiting times drawn from Eq. (<ref>). As shown in Fig. <ref>, our theory is effective for both the positional distribution's central part and far tails, whereas the Lévy stable law Eq. (<ref>) tends to infinity.










 §.§ Moments of the position

We investigate the mean of the position denoted by ⟨ x(t) ⟩, which refers to the sum of all positions divided by the number of particles. It is a measure that reflects the tendency of the position.  Let us start from the Montroll-Weiss equation, which in Fourier-Laplace space is as follows <cit.>

    P(k,s)=1-ϕ(s)/s1/1-f(k)ϕ(s).

Using the relation

    ⟨x̂^q(s)⟩=(-i)^q∂^q P(k,s) /∂ k^q|_k=0,

in the long time limit Eq. (<ref>) yields

    ⟨ x(t)⟩∼ at/⟨τ⟩.

Recall that the waiting time has a finite mean ⟨τ⟩, then for a fixed t, the mean of the number of renewals is ⟨ N(t)⟩∼ t/⟨τ⟩ <cit.>. Thus ⟨ x(t)⟩∼⟨ x⟩⟨ N(t)⟩∼ at/⟨τ⟩ is obtained again, where ⟨ x⟩ is the mean of the displacements. As shown in Figs. <ref> and <ref>, the theoretical prediction grows linearly with the observation time t, showing a perfect match with simulations and Eq. (<ref>).




The MSD is the most commonly employed quantifier for diffusion processes. Due to the effect of the bias, the biased CTRW shows superdiffusion <cit.>. Fascinatingly, the MSD is controlled by these sluggish particles that significantly lag far behind the average position, as opposed to the swift particles near the average. In that sense, the rare events, governed by the infinite density, are responsible for the enhanced diffusion <cit.>.  Mathematically, the variance of the position grows as

    Var(x)   =⟨ x^2⟩-⟨ x⟩^2
       ∼2a^2τ_0^α/(2-α)(3-α)⟨τ⟩^3t^3-α+σ^2t/⟨τ⟩,

measuring the time evolution of its spatial extension. In the long time limit, the leading term of Eq. (<ref>) is t^3-α, and when the observation time is short, the correction term is of importance. As shown in Fig. <ref>, theoretical results are consistent with simulations. In particular, when the bias is weak, for example, a=0.5, our theory gives a perfect prediction if compared with Eq. (<ref>).
In linear-linear scale, small deviations exist between  theoretical predictions and simulation results; see Fig. <ref>. The reason is as follows. When the bias is strong, particles are pushed by the bias and move quickly. In this scenario, the asymptotic Lévy stable law given by Eq. (<ref>) applies. Consequently, the modified infinite density becomes insignificant, as its probability is very low compared to the weak bias case (see Fig. (<ref>)). Therefore, the algorithm provided in Sec. <ref> essentially utilizes Eq. (<ref>) with a cutoff at the tail to compute the MSD. Besides, fluctuations of the far tail of the positional distribution are large when the bias are strong. It leads to deviations, but it does not like the results given by Eq. (<ref>), which predicts an infinite MSD.





















 §.§ Breakthrough curves

Now we further discuss the positional distribution, but from aspects of breakthrough curves, which are essential tools in measuring the contamination in porous materials <cit.>. We assume that all the particles are injected immediately when we start to observe the process, wherein the positional concentration is measured at a fixed position x_b. To be more exact, the breakthrough curves, the probability of particles being at x_b at a specific observation time t, are measured in terms of the distribution.

Based on the discussions in <cit.>, we delve into the time-dependent bias to validate the program outlined in Sec. <ref>. The process entails four states. In state (i), particles are promptly injected; thereafter, displacements follow a Gaussian distribution with a mean of a and a variance of σ^2 from time zero until t_a. All along the process, the variance of displacements remains the same. Subsequently, in state (ii), the bias of the system is enhanced from time t_a to t_b, i.e., the displacements are drawn from Gaussian distribution with the mean 4aη/(1+η), where η>2/3. State (iii) witnesses a bias of the system represented by a/(2/3+η) over the time interval (t_b,t_c), constituting the weakest bias among the four states. Ultimately, the bias reverts to a after t_c until the end of the observation time. The corresponding theoretical prediction was considered using the fractional advection-diffusion-asymmetry equation in Ref. <cit.>, where the main strategy is that the final position of each state is regarded as the initial position of the next process. In the long time limit, the positional distribution follows <cit.>


    P(x,t)∼∫_-∞^∞1/√(4π c_m1)exp[-(x-y-c_m2)^2/4c_m1]1/(c_m3)^1/βℒ_β[y/(c_m3)^1/β] dy

with

    c_m1={[ tσ^2/2⟨τ⟩ ,        m=1;;  tσ^2/2⟨τ⟩,        m=2;;  tσ^2/2⟨τ⟩,        m=3;; tσ^2/2⟨τ⟩ ,        m=4, ].



    c_m2={[                                            a_1t/⟨τ⟩,                                                 m=1;;                            [a_1t_1+a_2(t-t_1)]1/⟨τ⟩,                                                 m=2;;               [a_1t_1+a_2(t_2-t_1)+a_3(t-t_2)]1/⟨τ⟩,                                                 m=3;; [a_1t_1+a_2(t_2-t_1)+a_3(t_3-t_2)+a_4(t-t_3)]1/⟨τ⟩ ,                                                 m=4, ].

and

    c_m3={[                                                 a_1^α t1/t,                                                        m=1;;                                [a_1^α t_1+a_2^α(t-t_1)]1/t,                                                        m=2;;                  [a_1^α t+a_2^α(t_2-t_1)+a_3^α(t-t_2)] 1/t,                                                        m=3;; [a_1^α t_1+a_2^α(t_2-t_1)+a_3^α(t_3-t_2)+a_4^α(t-t_3)] 1/t,                                                        m=4. ].


with m=1,2,3,4 being the number of the state. This is plotted in  Fig.  <ref> for different times t. Note that when 0<t<t_1, the solution Eq. (<ref>) reduces to  Eq. (7) in the main text.








§ CONCLUSION

In this manuscript, we introduced a way to generate statistics of the positional distribution that is consistent with the CTRW model for a long observation time t. The main idea is that we use the modified Lévy stable law  ℒ^*_α(ξ) to simulate the number of renewals, which yields a perfect prediction for the positional distribution. In some sense, the method avoids the generating of IID random waiting times, which saves a significant amount of time.
Mathematically, we can use the following formula to predict the positional distribution

    P(x,t)∼   ∫_-t/⟨τ⟩(t/t)^1/α^∞ℒ^*_α(ξ)
       ×exp(-(x-at/⟨τ⟩-aξ(t/t)^1/α)^2/2σ^2(t/⟨τ⟩+ξ(t/t)^1/α))/√(2σ^2π(t/⟨τ⟩+ξ(t/t)^1/α)) dξ,

with

    ℒ^*_α(ξ)={[ ℒ^*_α(ξ),    ξ≥ξ^*;; Q_t^I(ξ),  b̅<ξ<ξ^*;       b̅,     else. ].

The method is not limited to examples discussed in the manuscript. On the one hand, it is valid for the case of waiting times having an infinite mean, where the number of renewals follows the one-sided Lévy stable distribution <cit.>. On the other hand, the displacements can be extended to situations when steps have a finite mean and a finite variance. When discussing the MSD, deviations exist between simulations and theoretical predictions. However, we did not find such deviations for the positional distribution and the mean of the position. This indicates that, for the biased CTRW model, the MSD is sensitive to the details of the positional distribution. Improving the presented work is a task left for the future.          


Recently, researchers have been increasingly drawn to the nearly exponential decay observed in stochastic processes <cit.>. Unlike the approach investigated here, the exponential decay is easy to detect at a short time scale, such as, the total observation time is half. Consequently, Eq. (<ref>) fails. How to extend the present approach is left for future work.



§ ACKNOWLEDGMENTS

WW expresses gratitude to Eli Barkai for the discussion that inspired this work. The work is supported by the National Natural Science Foundation of China under Grant No.
12105243 and the Zhejiang Province Natural Science Foundation LQ22A050002.


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 Paper Title: Experimentally probing Landauer's principle in the quantum many-body regime.






These authors contributed equally to this work: 
mailto:stefan.aimet@gmail.comstefan.aimet@gmail.com, mailto:amintajik.physics@gmail.comamintajik.physics@gmail.com
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

These authors contributed equally to this work: 
mailto:stefan.aimet@gmail.comstefan.aimet@gmail.com, mailto:amintajik.physics@gmail.comamintajik.physics@gmail.com
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
Department of Physics and Astronomy, and Stewart Blusson Quantum Matter Institute,
University of British Columbia, V6T1Z1 Vancouver, Canada

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Institute of Theoretical and Computational Physics, University of Crete, 71003 Heraklion, Greece
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Dipartimento di Fisica,  Università di Pavia, 27100 Pavia, Italy
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

§ ABSTRACT 
Landauer’s principle bridges information theory and thermodynamics by linking the entropy change of a system during a process to the average energy dissipated to its environment. Although typically discussed in the context of erasing a single bit of information, Landauer’s principle can be generalised to characterise irreversibility in out-of-equilibrium processes, such as those involving complex quantum many-body systems. 
Specifically, the relationship between the entropy change of the system and the energy dissipated to its environment can be 
decomposed into changes in quantum mutual information and a difference in relative entropies of the environment. 
Here we experimentally probe Landauer’s principle in the quantum many-body regime using a quantum field simulator of ultracold Bose gases. Employing a dynamical tomographic reconstruction scheme, we track the temporal evolution of the quantum field following a global mass quench from a Klein-Gordon to a Tomonaga-Luttinger liquid model and analyse the information-theoretic contributions to Landauer’s principle for various system-environment partitions of the composite system. Our results agree with theoretical predictions, interpreted using a semi-classical quasiparticle picture. Our work demonstrates the potential of ultracold atom-based quantum field simulators to experimentally investigate quantum thermodynamics.



Experimentally probing Landauer's principle in the quantum many-body regime
    Jens Eisert
    Received ..... .., ....; accepted ......, ....
===========================================================================

Information theory and thermodynamics constitute foundational pillars of modern technology, underpinning our understanding of modern computers and heat engines, respectively. While these disciplines may seem distinct, they are intricately connected. This connection is encapsulated in Landauer's principle, first articulated in 1961 <cit.>. Landauer understood that erasing a bit of information in a computer is not for free but is instead accompanied by a minimal energy cost dissipated to the environment. Subsequently, various experimental studies <cit.> have confirmed this lower bound on energy dissipation for near-reversible bit erasure. Logical operations thus imply irreversibility, posing a fundamental theoretical limitation to the design of any small-scale energy-efficient information-processing technologies <cit.>. However, extending beyond the case of bit erasure, recent influential work <cit.> has generalised the link between information theory and thermodynamics. Using a quantum statistical mechanics framework, their work reinterprets Landauer's principle to relate the entropy change of a system to the energy dissipated to its environment in very general out-of-equilibrium processes. This relationship can be quantified by a measure of process irreversibility <cit.>. Such an extension is thus particularly relevant for quantum many-body systems, where contributions to irreversibility remain an area of active research, notably in phenomena such as equilibration and thermalisation <cit.>. 

In this work, we experimentally probe Landauer's principle for an out-of-equilibrium process in the quantum many-body regime by measuring the time evolution of quantum information-theoretic quantities that characterise irreversibility. We present a crisp information-inspired interpretation of correlations present in states of quantum many-body systems in terms of entropic expressions. 

Concretely, we consider a system-environment composite initially in the state ϱ_ SE(0), evolving under a global unitary time evolution U to the state ϱ_ SE(t)=Uϱ_ SE(0)U^†. The reduced state of the system S (environment E) is ϱ_ S(E)(t)=_E(S)[ϱ_ SE(t)]. 

First, let us assume that there are no initial correlations between the system and environment, and the environment is in a thermal state γ_ E^β_ E=e^-β_ EH_ E/[e^-β_ EH_ E] at inverse temperature β_ E with respect to its Hamiltonian H_ E. In this case, the entropy production Σ(t), which is a measure of irreversibility <cit.>, can be decomposed as

    Σ(t):=I_ SE(t)+D(ϱ_ E(t)||γ_ E^β_ E)  .

Here, the quantum relative entropy

    D(ϱ_ E(t)||γ_ E^β_ E)=[ϱ_ E(t)logϱ_ E(t)]-[ϱ_ E(t)logγ_ E^β_ E]   ,

quantifies the deviation of the environment from its initial thermal state. The quantum mutual information 

    I_ SE(t)=S(ϱ_S(t))+S(ϱ_E(t))-S(ϱ_SE(t))

measures the system-environment correlations, where 

    S(ϱ)=-[ϱlogϱ]

denotes the von Neumann entropy of a state ϱ.



Now we move on to a more general scenario where the system and the environment are initially correlated, and the state of the environment deviates from thermal equilibrium. We thereby introduce the generalised entropy production ΔΣ :=Σ(t)-Σ(0) where the initial correlations and deviations from thermal equilibrium are accounted for by a resource cost Σ(0)=I_ SE(0)+D(ϱ_ E(0)||γ_ E^β_ E) <cit.>, with β_ E being an effective inverse temperature <cit.>. Landauer's principle can now be formulated <cit.> as an equality as 

    ΔΣ=β_ EΔ E_ E+Δ S=Δ I+Δ D   ,

relating the system's entropy 
change Δ S:=S(ϱ_ S(t))-S(ϱ_ S(0)) to the energy dissipated to the environment Δ E_ E:=[(ϱ_ E(t)-ϱ_ E(0))H_ E].
Alternatively, ΔΣ also decomposes into the change of quantum mutual information, Δ I=I_ SE(t)-I_ SE(0), expressing how much the system-environment correlations change along the out-of-equilibrium process, as well as the term Δ D=D(ϱ_E(t)||γ_E^β_E)-D(ϱ_E(0)||γ_E^β_E) that quantifies how much both the initial and final state of the environment are different from the reference equilibrium thermal state. We provide a more thorough motivation for the generalised entropy production ΔΣ in the Methods section. 

Expressed in this form, Landauer's principle serves as a means of tracking changes in information-theoretic quantities, which compose generalised entropy production ΔΣ in two different ways via a single equation. Herein lies the power at the heart of thermodynamics: just as knowing the precise microscopic degrees of freedom, such as the position and momentum of each particle in a classical gas, is not useful for drawing meaningful conclusions about its dynamics, knowing the density matrix of a quantum many-body system is similarly not descriptive. In traditional thermodynamics, macroscopic variables like changes in pressure or volume effectively characterise the dynamics. Similarly, analysing the information-theoretic quantities in Landauer's principle allows for a simple characterisation of the irreversibility in the out-of-equilibrium dynamics of a quantum many-body system.

So far, there has been no systematic experimental pursuit of probing Landauer's principle in the quantum many-body regime. Here, we address this gap by measuring the time evolution of the different terms in Eq. (<ref>) in a (1+1)D quantum field simulator of tunnelling-coupled ultracold Bose gases. This experimental platform has proven to be a suitable testbed for simulating one-dimensional quantum field theories. Previous studies using this setup have examined the area law of quantum mutual information in thermal equilibrium <cit.> 
and the propagation of second- <cit.> or higher-order correlations <cit.>. 

In this experiment, two parallel ultracold clouds of  atoms (illustrated in Fig. <ref>) are confined in highly anisotropic magnetic traps produced by an atom chip <cit.>. In the axial direction (z), the clouds are confined in a parabolic trap superposed with an optical dipole potential to produce hard walls, while in the radial directions (x,  y), by a double-well trap with an adjustable barrier in between, created by RF dreessing <cit.>. The single-particle tunnelling rate can be modified by the amplitude of a radio-frequency field created by two parallel wires on the atom chip.

The bosonic quantum field operator for each condensate can be written using the phase-density representation as ψ_n(z) = √(ρ_n(z))e^iθ_n(z), where θ_n and ρ_n denote the phase and density of the respective condensate, indexed by n=1,2. We prepare the Bose-Einstein condensates in a global thermal state of a massive Klein-Gordon Hamiltonian 

    H_ KG   = ∫_0^L dz [ g_ 1Dδρ^2(z) + ħ^2 n_ 1D/4m (∂_zφ(z))^2 
        + ħ J n_ 1Dφ^2(z) ]

which models low-energy phononic excitations. Here, ħ is the reduced Planck constant, m is the atomic mass, g_ 1D is the effective one-dimensional atomic interaction strength, L=49 is the axial length of the condensates, n_ 1D≈ 70^-1 is the average linear density and J≈ 2π× 0.8 Hz is the tunnelling rate introduced before. The operators φ(z) = θ_1(z) - θ_2(z) and δρ(z) = [ρ_1(z) - ρ_2(z)]/2 represent the relative phase and relative density, respectively, and satisfy the commutation relation [φ(z), δρ(z')] = iδ(z-z'). 

To initiate out-of-equilibrium dynamics in the experimental system and measure the evolution of information-theoretic quantities, we rapidly quench J to zero by ramping up the barrier between the condensates within approximately 2. This change corresponds to a global mass quench of the Klein-Gordon Hamiltonian. The condensates then evolve independently under the post-quench Tomonaga-Luttinger liquid Hamiltonian <cit.> (H_KG with J=0) for times t, up to 65.



 

For every time step t, we turn off all the traps and let the atoms fall freely for 15.6. The clouds will expand and interfere, resulting in absorption pictures similar to the one shown in Fig. [fig:setupandtomography]<ref>fig:setupandtomographyb, allowing us to measure the spatially resolved relative phase φ(z) between them. Since the detection process is destructive, the measurements are repeated to gather statistics. In our current experimental setup, relative density fluctuations δρ are not directly measurable, which prompted the development of a dynamical tomographic reconstruction technique <cit.> to access all of the elements of the covariance matrix 

    Γ(t)=[ Γ_φφ(t) Γ_φρ(t); Γ_ρφ(t) Γ_ρρ(t);         ]  .

Here, the elements are defined as [Γ_φφ(t)]_m,n=⟨φ(z_m, t)φ(z_n, t)⟩, [Γ_ρρ(t)]_m,n=⟨δρ(z_m,t)δρ(z_n,t)⟩, and [Γ_φρ(t)]_m,n=[Γ_ρφ(t)]^T_m,n=⟨1/2{φ(z_m, t),δρ(z_n, t)}⟩, all on a discrete grid with N pixels (m,n∈{1,…,N}), which are determined by resolution constraints of our imaging system, thus introducing an ultraviolet (UV) cutoff.

The phase–phase correlations for different evolution times after the quench can be measured directly from the extracted relative phase profiles. Assuming that the short-time dynamics after the quench are governed by the Tomonaga–Luttinger liquid (TLL) Hamiltonian, as time progresses, the initial eigenmodes of the relative density transform into the phase quadrature, and the phase quadrature transforms into the relative density. This transformation allows us to extract information about these eigenmodes by fitting the initial second-order correlation functions of phase–density, and density–density with the observed evolution of the phase–phase correlations in momentum space. 

The schematics in Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc illustrate our dynamical tomographic reconstruction scheme. We follow the technique in Ref. <cit.> for various input intervals with varying starting points. By scanning the starting points of these input intervals throughout the trapping times, we can reconstruct the full covariance matrix at every time t, as depicted in Fig. <ref>. The interval length (32.5 close to L/c ≈ 27) is selected to be sufficiently long for the slowest eigenmode to acquire enough dynamical phase for a stable reconstruction, yet short enough to prevent mode interactions from affecting the reconstruction. A detailed overview of this reconstruction process is provided in the Methods section.

The quadratic form of the pre- and post-quench Hamiltonians allows us to work within the framework of Gaussian quantum information theory <cit.>. In this framework, the covariance matrix Γ captures all the accessible information about the state of the composite system during the dynamics, from which we can extract the information-theoretic quantities of interest in Eq. (<ref>). As has been demonstrated in various experiments <cit.>, the quadratic approximation of the Hamiltonian accurately captures the dynamics for the considered time scales. Even if the true dynamics deviate from the Gaussian regime and violate Tomonaga-Luttinger liquid theory, a Gaussian extremality argument presented in the Supplementary Material justifies the Gaussian tomography scheme and provides bounds on the information-theoretic quantities. Thus, based on our technique of dynamical tomographic reconstruction of covariance matrices, Landauer's principle can also be meaningfully experimentally probed for interacting models.

Having experimentally reconstructed the post-quench time evolution of the covariance matrices, we partition the one-dimensional field of length L into two distinct subregions and split the covariance matrix accordingly (Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc). Since the quantum field is isolated from its surroundings, one subregion serves as the system S with length L_S, while the other subregion functions as the environment E with length L_E = L - L_S. We probe Landauer's principle (Eq. (<ref>)) for various system-environment bipartitions by calculating β_ EΔ E_ E, Δ S, Δ I and Δ D. The results are presented in Fig. <ref> as a function of time for different subregion size ratios and in Fig. <ref> as a function of subregion size for different times. The time scales are shown in units of ct/L, where c=√(g_ 1Dn_ 1D/m)≈ 1.8  / is the speed of sound. Overall, a very good fit of the data (circles) is obtained compared to the theoretical calculations (shaded areas), which use the lowest N=7 modes and consider the imaging resolution of the experiment. The low-lying modes already capture the dynamics of the continuum theory very well. The error bars representing the 68% confidence intervals are obtained via bootstrapping <cit.> with 999 samples, considering the uncertainty in the tunnelling rate and the estimated initial global temperature. The effective inverse temperature of the environment β_ E with respect to the post-quench Tomonaga-Luttinger liquid Hamiltonian is computed using field-theoretic simulations by constructing a global thermal state of the Klein-Gordon Hamiltonian with the estimated initial global temperature from the experimental data. 

First, we examine the decomposition of the generalized entropy production ΔΣ into β_ EΔ E_ E and Δ S. The term associated with the energy dissipated to the environment, β_ EΔ E_ E, reveals that a small amount of energy flows into (or out of) the environment for small (or large) systems around ct/L_ S=1. However, this term contributes only minimally to the irreversible post-quench dynamics of the quantum field. In contrast, the entropy change of the system, Δ S, dominates the dynamics and shows a clear linear increase up to ct/L_ S<1, followed by more pronounced growth.

In comparison, the second decomposition involving Δ I and Δ D sheds a slightly different focus on the out-of-equilibrium dynamics. The largest contribution to ΔΣ, the change in quantum mutual information, Δ I, exhibits similar behaviour to Δ S but with a higher magnitude. This increased magnitude is accounted for by the additional entropy change in the environment. However, Δ I competes with the term proportional to the change of the environment, Δ D, which accounts for both the entropic and energetic changes to the environment. This makes Δ D a particularly interesting quantity to characterise the out-of-equilibrium dynamics of the environment experimentally. In our case, Δ D increases negatively, mirroring otherwise the behaviour of Δ S, due to the small energetic contribution. 

To interpret the results, we need to consider the effects of Neumann and Dirichlet boundary conditions on the quantum field, as shown with field-theoretic simulations in Fig. <ref>. These boundary conditions may be contrasted with the curved
backgrounds giving rise to effective boundary conditions discussed in Refs. <cit.>. The experimental system exhibits Neumann boundary conditions (∂_zφ(z)|_z=0, L=0) due to the vanishing particle current at the edges. Plots contrasting the scaling with subregion size are shown in the Supplementary Material. 

However, we first consider simulations for Dirichlet boundary conditions (φ(z)|_z=0, L=0), which are simpler to understand. Here, the well-established semi-classical quasiparticle picture <cit.> offers a clear and intuitive model for the post-quench dynamics of global mass quenches <cit.>, depicting the propagation of short-range initial correlations via ballistically moving quasiparticles. Following the homogeneous global quench, the initial Klein-Gordon thermal state becomes a non-equilibrium state with excess energy relative to the post-quench Tomonaga-Luttinger liquid Hamiltonian, which governs the time evolution. This initial state acts as a source of quasiparticle pairs emitted globally from every point across the length of the composite system. Each pair in the bulk consists of two quasiparticles correlated with each other and moving in opposite directions at the same speed. For our set of parameters, due to the finite initial correlation length, there exists a small localised region where quasiparticles are correlated.

In this quasiparticle picture, the change of quantum mutual information Δ I between the system and environment is proportional to the number of pairs of quasiparticles that are shared between the two different subregions. The spread of correlations of short-range interacting models is described by a linear effective light cone originating at the system-environment boundary. The increase in correlations is thus proportional to the section of the light cone, or in other words, it is proportional to the distance between the quasiparticles of the pair emitted at the system-environment boundary (Fig. [fig:DBC_main]<ref>fig:DBC_mainc). If the system is smaller than the environment, once the first quasiparticle of this boundary pair reaches the edge of the system at ct/L_ S=1 and undergoes reflection, the section of the system-environment boundary pair light cone remains constant. The behaviour of Δ I transitions to a plateau value. From a system perspective, the composite system appears to have locally equilibrated to a generalised Gibbs ensemble steady state <cit.>. For finite-size composite systems, the plateau eventually ends again at ct/L_ E=1 when the other quasiparticle of the boundary pair reaches the edge. The distance between the two quasiparticles decreases until they meet again at ct/L=1, explaining the phenomenon of recurrences <cit.> (with respect to the correlation functions). 



On the other hand, Δ I behaves differently for Neumann boundary conditions. Neumann boundary conditions introduce a zero mode, whose variance does not evolve harmonically like the other momentum modes, but quadratically (phase diffusion <cit.>). The global fluctuations of the zero mode contribute to entropic quantities <cit.> and as they increase with time after a quench they eventually dominate over the contribution of other modes. Since the quasiparticle picture cannot capture this zero-mode feature, its predictions for ct/L_ S>1, when a plateau value should be reached, need to be modified under the present Neumann boundary conditions. The above features of the dynamics also apply to the change of system entropy Δ S. In addition, the zero mode restricts the validity of the tomography scheme to ct/L<1. 

The energetic contribution, β_ EΔ E_ E, can also be explained using the quasiparticle picture. Initially, β_ EΔ E_ E remains constant because quasiparticles traveling between the system and the environment carry equal energy, resulting in zero net energy flux. However, a finite-size energy flow becomes apparent when quasiparticles from the edge of the system cross the system-environment boundary. The translational invariance of the homogeneous quench is broken due to the higher energy density at the edges of the composite system for Neumann boundary conditions (the reverse is true for Dirichlet boundary conditions, where the energy density is lower at the edges). This edge region has a size of the order of the initial correlation length. The resulting behaviour of the change of the environment Δ D reflects both the entropic and energetic fluxes. 

In this study, we have experimentally probed Landauer's principle in the quantum many-body regime following a global mass quench in an ultracold atom-based quantum field simulator. By reconstructing the dynamics of the state of the composite system, we examined the information-theoretic quantities related by Landauer's principle, which we interpret using a semi-classical quasiparticle picture. Our approach underscores the general utility of Landauer's principle for characterising the irreversibility of out-of-equilibrium dynamics in quantum many-body systems. The way we express correlations in terms of entropic quantities with an 
information-theoretic meaning can be viewed as a vehicle to capture many-body correlations in 
non-equilibrium. The Gaussian extremality argument may offer a pathway to extend our methodology to capture features of out-of-equilibrium processes in interacting models, despite the challenges posed by non-Gaussian effects. Looking forward, progress has already been made towards investigating a local quench involving two Bose-Einstein condensates of different temperatures being joined together <cit.>. This protocol represents a crucial primitive towards developing a quantum field thermal machine <cit.>, potentially allowing for Landauer erasure as a mechanism to reduce the entropy of a subregion in the quantum many-body regime, hence functioning as an effective cooling mechanism. Our current work demonstrates the potential of ultracold one-dimensional gases as testbeds for quantum thermodynamics in the many-body regime, where complexity, quantum effects, and finite-size effects play a crucial role.


We thank Bernhard Rauer for conducting the experimental measurements. In addition, we are grateful to Marcus Huber, Patrick Emonts, Ivan Kukuljan, Mauro Paternostro, Philippe Faist, Florent Goulette, Marek Gluza, and Igor Mazets for helpful discussions and comments. This work has been supported by the DFG Research Unit FOR 2724 on `Thermal machines in the quantum world', the FQXi, the Quantum Flagship (`Millenion' and `PasQuans2'), the Einstein Research Unit, the BMBF, Berlin Quantum, and the ERC-AdGs `Emergence in Quantum Physics' and 
`Delineating the boundary
between the computational
power of quantum and
classical devices'.

S.S. acknowledges support by
the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No.  101030988. 



§ METHODS




 §.§ Landauer's principle

Consider a system S and an environment E with a bipartite Hilbert space ℋ=ℋ_S⊗ℋ_E. For an initial product state ϱ_SE=ϱ_S⊗γ_E^β_E, where the environment is in a thermal state γ_E^β_ E=e^-β_ E H_ E/[e^-β_E H_E], the entropy production under global unitary dynamics is given by <cit.>

    Σ(t)=I_ SE(t)+D(ϱ_E(t)||γ_E)=D(ϱ_ SE(t)||ϱ_S(t)⊗γ_E^β_ E)  .

Since it is expressed as a quantum relative entropy, Σ is non-negative due to Klein's inequality <cit.>, which is a manifestation of the second law of thermodynamics. More generally, as in our work, we start with a resourceful initial state that includes both initial correlations I_ SE(0) and athermality of the environment D(ϱ_E(0)||γ_E^β_ E). While defining temperature for a non-equilibrium state is ambiguous, we determine the effective inverse temperature of the environment β_ E by demanding that the environment's energy is equal to that 
of a Gibbs state, such that [ϱ_ EH_ E]≡[γ_E^β_ EH_ E] <cit.>. This choice also minimises the athermality term. Thus, we define the resource cost of the initial state <cit.>

    Σ(0)=I_ SE(0)+D(ϱ_E(0)||γ_E^β_ E)=D(ϱ_ SE(0)||ϱ_S(0)⊗γ_ E^β_ E).


Although there is no unique consensus on entropy production for initial non-equilibrium environment states <cit.> and initial correlations <cit.>, we define Landauer's principle Eq. (<ref>) in terms of generalised entropy production ΔΣ:=Σ(t)-Σ(0) as

    ΔΣ :=β_EΔ E_ E+Δ S=Δ I+Δ D  .

This can be derived similarly to Ref. <cit.>. One caveat is that ΔΣ may not necessarily be non-negative, which is a necessary criterion for a strictly valid definition of entropy production. However, this makes sense as ΔΣ may be considered a finite-difference version of the entropy production rate ΔΣ=Σ(t)-Σ(0)=∫_0^t dt' dΣ(t')/dt', which can also be negative <cit.>. Thus, ΔΣ intuitively measures the "extra" irreversibility associated with the protocol, excluding the initial resources.



 §.§ Experimental implementation

The experimental setup consists of a pair of tunnelling-coupled one-dimensional  quasicondensates, following a protocol implemented previously <cit.>. The magnetic fields required to trap the gas in a double-well potential are generated by an atom chip <cit.>, achieving transverse trapping frequencies of ω_⊥/2π = 1.4 kHz. After preparing a thermal equilibrium state at a temperature of 49 nK, all traps are turned off, and absorption imaging of the atoms is performed during a free-fall time of 15.6 ms to measure the two-dimensional atomic density distributions. This procedure enables us to detect the relative phase between the condensates across the length of the quasicondensate using matter interferometry (Fig.  [fig:setupandtomography]<ref>fig:setupandtomographyb). Due to the destructive nature of the detection process, many repetitions of the experiment are necessary to accumulate numerous phase correlation images throughout the time evolution, thereby also ensuring statistically accurate expectation values.



 §.§ Quantum field simulation

We have the Klein-Gordon Hamiltonian as a model for our quantum field simulator

    H_ KG=∫_0^Ldz [g_ 1Dδρ(z)^2 +ħ Jn_ 1Dφ(z)^2+ħ^2n_ 1D/4m(∂_zφ(z))^2] ,

where the length of the composite system is set to L =49 with a mean density of n_ 1D = 70^-1. The coupling strength between the particles is chosen as g_ 1D = 8.594× 10^-39kg·m^3·s^-2, and we have an atomic mass of m = 1.433× 10^-25kg.
The initial single-particle 
tunnelling rate is J= 2π× 0.76Hz. The post-quench evolution is described by the Tomonaga-Luttinger liquid Hamiltonian

    H_ TLL=∫_0^L dz[g_ 1D(δρ(z))^2+ħ^2 n_ 1D/4m(∂_zφ(z))^2
     ]  .




 §.§ Dynamical tomographic reconstruction of covariance matrices

The tomography scheme for reconstructing the initial thermal covariance matrix is detailed in Refs. <cit.>. Here, we extend this method to dynamically reconstruct the state over time. We provide a brief overview below.
Given Neumann boundary conditions ∂_zφ(z)|_z=0,L=0, the relative phase φ(z) and relative density fluctuations δρ(z) can be expanded in Fourier space in terms of cosine eigenfunctions

    f_k^φ(z) = 
    2(1/kħπ√(mg_ 1D/n_ 1D))^1/2cos(kπ/Lz)   if  k > 0 ,
    
    1    if  k = 0,
    
    
    f_k^ρ(z) = 1/L(kħπ√(n_ 1D/mg_ 1D))^1/2cos(kπ/Lz)   if  k > 0 ,
    1/L   if  k = 0.


The Hamiltonian of the post-quench Tomonaga-Luttinger liquid Eq. (<ref>) is then diagonalised as 
(indices omitted for brevity from now on)

    H=ħ u/2δρ_0^2+∑_k=1^∞ħω_k/2[δρ_k^2+φ_k^2]

with u:=2g_ 1D/ħ L and 

    ω_k:=√(g_ 1Dn_ 1D/m)kπ/L  .

The interaction strength g_ 1D is given by 

    g_ 1D:=ħω_⊥a_s2+3a_s n_ 1D/(1+2a_s n_ 1D)  ,

where a_s=5.2 nm is the three-dimensional scattering length.

Consequently, the time evolution of modes (k>0) in Fourier space is

    φ_k(t)=φ_k(0)cos(ω_k t)+δρ_k(0)sin(ω_k t),
    δρ_k(t)=δρ_k(0)cos(ω_k t)-φ_k(0)sin(ω_k t),

with non-harmonic time evolution of the zero mode

    φ_0(t)= utδρ_0(t)+φ_0(0),
    δρ_0(t)=δρ_0(0)= const   .

As mentioned in the main text and shown in Fig. <ref>, the covariance matrices can be dynamically tomographically reconstructed by measuring the relative phase profiles using matter interferometry at various times. Subsequently, the relative phase correlations are computed as
Φ^2_m,n(t)=⟨(φ(z_m,t)-φ(z_0, t))(φ(z_n,t)-φ(z_0, t))⟩ with z_0 as a reference position. 
We then expand in Fourier space, see Eq. (<ref>), and expand the time evolved operators to express the measured phase correlation function in terms of the covariance matrix elements prior to the time evolution

    Φ^2_m,n(t)=⟨(φ(z_m,t)-φ(z_0, t))(φ(z_n,t)-φ(z_0, t))⟩
    
    =∑_k,l=1^N-1f_k,l^m,n⟨φ_k(t)φ_l(t)⟩

where

    f_k,l^m,n=(f_k^φ(z_m)-f_k^φ(z_0))(f_l^φ(z_n)-f_l^φ(z_0))

and so we see that the covariance matrix Γ(t) can be obtained by fitting the coefficients f_k,l^m,n using the phase-space correlations

    Φ^2_m,n(t+Δ t)
    		=∑_k,l=1^N-1f_k,l^m,ncos(ω_k Δ t)cos(ω_lΔ t)Γ_φφ; k,l(t)
    +∑_k,l=1^N-1f_k,l^m,nsin(ω_k Δ t)sin(ω_l Δ t)Γ_ρρ; k,l(t)
    +∑_k,l=1^N-1(f_k,l^m,n+f_l,k^m,n)cos(ω_k Δ t)sin(ω_l Δ t)Γ_φρ; k,l(t)  .

By gathering statistics, we employ the same method as in Ref. <cit.>, but simply shift the window of observation.

We obtain the covariance matrix in momentum space for the first N modes that give physical (positive) occupation numbers. The imaging system is additionally accounted for by a Gaussian point spread function with standard deviation of 3μ m to estimate the initial state. In this way, we can also confirm the global initial thermal state and extract a pre-quench temperature. We use discrete cosine eigenfunctions to convert the momentum space correlators to real-space correlators, from which we compute the quantities of interest.






 §.§ Probing Landauer's principle from covariance matrices

In this section, we delineate how the quantities in Landauer's principle, as expressed in Eq. (<ref>), are computed using the full covariance matrix Γ in real space, defined on a grid of N pixels.
First, the von Neumann entropy S(Γ) is computed using the symplectic eigenvalues λ_n of the covariance matrix Γ <cit.>. These eigenvalues correspond to the eigenvalues of iΩΓ, where 

    Ω:=[    0  1_N; -1_N    0 ]

is the 
standard kernel of the symplectic form capturing the canonical commutation relations. The expression for 
the von Neumann entropy 
S(Γ) of a Gaussian state is given by

    S(Γ)=∑_n=1^N[(λ_n+1/2)ln(λ_n+1/2)-(λ_n-1/2)ln(λ_n-1/2)].


Given two full covariance matrices corresponding to different times and partitioning it into system and environment subspace covariance matrices (Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc), this allows us to compute the entropy change of the system Δ S and the mutual information Δ I.

Next, we illustrate how to calculate the energy of the environment with respect to the post-quench Tomonaga-Luttinger liquid Hamiltonian. To obtain the correct environment Hamiltonian, we need to proceed with care. The field corresponding to the environment is discretely defined with N_ E=N (L_ E/L) pixels. A naive approach would discretise the continuous Hamiltonian defined over the length of the environment, but this nearest-neighbour phase coupling Hamiltonian would yield a non-linear dispersion relation, inadequately describing the physics of the environment for the continuum model used in the experiment. Instead, we start from the environment Hamiltonian of the continuous theory, diagonalise it to obtain a momentum decomposition of infinitely many modes, and then truncate the Hamiltonian <cit.> of the environment to a form with a linear dispersion relation, retaining only N_ E modes

    H_ E=ħ u_ E/2(δρ^ E_0)^2+∑_k=1^N_ E-1[(δρ_k^ E)^2+(φ_k^ E)^2]

with u_ E:=2g_ 1D/ħ L_ E and ω_k^ E:=√(g_ 1Dn_ 1D/m)kπ/L_ E. Converting back to real-space operators, we derive an effective resolution scale-dependent Hamiltonian

    H_ E=g_ 1DΔ z∑_n=N_ S^N-1δρ(z_n)^2
    +ħ^2 n_ 1D/4mΔ z∑_n,m=N_ S^N-1φ(z_n)φ(z_m)2π^2/N_ E^3κ_n,m^ E  ,

where 

    κ_n,m^ E   :=   ∑_k=1^N_ E-1{ k^2cos(kπ/N_ E(n-N_ S+1/2))
       ×   cos(kπ/N_ E(m-N_ S+1/2))}

encodes the long-range coupling of the effective Hamiltonian. Here, Δ z:=L/N is the pixel size, N_ S(E) is the number of pixels of the system (environment).

The energy of the environment can then be calculated <cit.> as [ϱ_ E H_ E]=[Γ^ EH̃_ E], where H̃_ E is the matrix expression related to H_ E=X^TH̃_ EX, with X=(φ(z_N_S), …,  φ(z_N-1), δρ(z_N_S), …, δρ(z_N-1))^T. Thus, to calculate the energetic contribution β_ EΔ E_ E, we need to obtain the inverse temperature of the environment. As indicated in the previous section and detailed in Ref. <cit.>, we have obtained the inverse temperature of the composite system with respect to the pre-quench Klein-Gordon Hamiltonian experimentally. Subsequently, we construct the thermal state corresponding to this temperature in the continuum limit simulation. We obtain the inverse temperature of the environment β_ E for different subregion sizes, by demanding the environment's energy to match that of a Gibbs state. We then interpolate the profile for each pixel. We note that the post-quench temperature is slightly higher than the temperature measured with respect to the pre-quench Hamiltonian and β_ E>0 
is nearly homogeneous across different subregion sizes.

To calculate the energy of a Gibbs state for the environment in the Tomonaga-Luttinger liquid model, we begin with the partition function Z_ E(β_ E)=Z^ zm_ E(β_ E)Z^ ho_ E(β_ E) <cit.>.
Due to the compactified nature of the field, the zero mode partition function can be expressed in terms of the Jacobi-theta function. Given our experimental parameters, it approximates to

    Z^ zm_ E(β_ E)=∑_n=-∞^∞e^-β_ Eħ u_ E/2n^2≈√(2π/β_ Eħ u_ E)  .

The harmonic oscillator partition function is

    Z_ E^ ho(β_ E)=∏_k=1^N_ E-11/2sinh(βħω^ E_k/2)  .

The energy is then E_ E(β_ E)=-∂_β_ E[logZ_ E(β_ E)]=E_ E^ zm(β_ E)+E_ E^ ho(β_ E). The zero-mode contribution to the energy is

    E^ zm_ E = 1/Z_ 0^ E∑_n=-∞^∞e^-β_ Eħ u_ E/2 n^2(ħ u_ E/2n^2 ) ,

which approximates to 1/(2β_ E) for our experimental parameters. The harmonic oscillator contribution to the energy is

    E_ E^ ho=∑_k=1^N_ E-1ħω_k^ E/2(β_ Eħω_k^ E/2)  .

Consequently, we can express then quantum relative entropy of the environment state ϱ_ E with respect to a Gibbs state at inverse temperature β_ E as

    D(ϱ_ E||γ_ E^β_ E)
    		=-S(ϱ_ E)+β_ E[ϱ_ E H_ E]+logZ_ E(β_ E)

and we obtain Δ D accordingly.






Supplementary Material













 §.§ Continuum limit and role of boundary conditions

In this section we compare the effect of Neumann boundary conditions (∂_zφ(z)|_z=0, L=0) and Dirichlet boundary conditions (φ(z)|_z=0, L=0) on the information-theoretic quantities within Landauer's principle Eq. (<ref>) in the continuum limit. The quantum field is discretised on a  sufficiently fine grid to ensure convergence, achieved when higher momentum modes no longer affect the results significantly. The simulation parameters closely match experimental values, assuming a homogeneous density profile n_ 1D.



The plots showing the scaling with subregion size at various times are presented in Fig. <ref>. The scaling of β_ EΔ E_ E can be explained by the initial symmetric energy density profile, which is higher at the edges than in the bulk for Neumann boundary conditions and lower for Dirichlet boundary conditions (Fig. [fig:DBC_main]<ref>fig:DBC_maina-b). The dominant contribution for the considered quench, Δ I, shows scaling symmetrically about L_ S/L=0.5 by construction. For Dirichlet boundary conditions, small systems (or environments) do not increase in correlations with its environment for ct/L_ S>1 as correlations reach a plateau phase in time (Fig. [fig:DBC_main]<ref>fig:DBC_maine). Neumann boundary conditions, however, actually yield Δ I to be higher for the case of a small system than for the case of system and environment being of equal size. As discussed in the main text, the zero mode causes the results for Dirichlet boundary conditions, which are well described by the quasiparticle picture, to diverge from those for Neumann boundary conditions after ct/L_ S > 1.  



 §.§ Exploiting the extremality of Gaussian states

The validity of the Gaussian tomography scheme has been confirmed by measuring the normalised averaged connected fourth-order correlation functions, as demonstrated in previous works <cit.>. However, leveraging properties of Gaussian quantum information, we argue that the information-theoretic quantities in Landauer's principle can also be estimated using our second-moment tomography scheme, even for time evolution under a non-Gaussian model with significant higher-order correlation functions. This justifies the potential use of our experimental protocol under the lens of Landauer's principle for more complex protocols and interacting models.

To begin, we define G as the Gaussian state that shares the same second moments as the true state ϱ that may be non-Gaussian, reflecting a preparation involving interactions. Central to the argument are several extremality properties of Gaussian states. Firstly, Gaussian states G have the largest von Neumann entropy S(G)≥ S(ϱ) among all states ϱ with a given covariance matrix.
Secondly, we can make use of the property

    [ϱ(t)logG(0)]=[G(t)logG(0)]  ,

as any Gaussian state can be written as a matrix exponent of a quadratic polynomial, thereby causing the matrix logarithm of the relative entropy to cancel out. Thirdly, Ref. <cit.> has demonstrated that the quantum conditional entropy is also maximised for Gaussian states, and the  inequality 

    S(ϱ_ S(t))-S(ϱ_ SE(t))≥ S(G_ S(t))-S(G_ SE(t))

is true. This is less obvious, as the expression under consideration is 
a difference of von Neumann entropies. 
The generalised entropy production of the true dynamics ΔΣ:=β_ EΔ E_ E+Δ S= Δ I+Δ D can be written as

    ΔΣ:=Δ I+Δ D
    
    =S(ϱ_ S(t))+S(ϱ_ E(t))-S(ϱ_ SE(t))-S(ϱ_ S(0))-S(ϱ_ E(0))+S(ϱ_ SE(0))
    -S(ϱ_ E(t))-[ϱ_ E(t)logγ_ E^β_ E]+S(ϱ_ E(0))+[ϱ_ E(0)logγ_ E^β_ E]
    
    =S(ϱ_ S(t))-S(ϱ_ S(0))-[ϱ_ E(t)logγ_ E^β_ E]+[ϱ_ E(0)logγ_ E^β_ E]
    
    =Δ S-[ϱ_ E(t)logγ_ E^β_ E]+[ϱ_ E(0)logγ_ E^β_ E]  ,

where we have used the unitary invariance of the von Neumann entropy. Clearly, the generalised entropy production ΔΣ^G:=β_ EΔ E_ E^G+Δ S^G=Δ I^G+Δ D^G assuming an initially Gaussian state and assuming Gaussian dynamics is given by

    ΔΣ^G:=Δ I^G+Δ D^G
    =Δ S^G-[G_ E(t)logγ_ E^β_ E]+[G_ E(0)logγ_ E^β_ E]

where Δ E_ E^G=[(G_ E(t)-G_ E(0))H_ E], Δ S^G=S(G_ S(t))-S(G_ S(0)), Δ I^G=I^G_ SE(t)-I^G_ SE(0) and Δ D^G=D(G_ E(t)||γ_ E^β_ E)-D(G_ E(0)||γ_ E^β_ E). By utilising the property in Eq. (<ref>) and considering that γ_ E^β_ E is Gaussian we can write

    ΔΣ = Δ S-[G_ E(t)logγ_ E^β_ E]+[G_ E(0)logγ_ E^β_ E]=Δ S-Δ S^G+Σ^G  .

This also implies that

    Δ E_ E = Δ E_ E^G

demonstrating that covariance matrices are entirely sufficient for capturing the energetic dynamics, even when the true dynamics are not Gaussian. This is important, as in this way, one can meausure second moments experimentally and still get bounds for the true state that may show signatures of interactions.

To proceed further, we must assume that the initial state post-quench is also Gaussian, i.e., ϱ_SE,S,E=G_SE,S,E. At this step, we have not made the additional assumption of Gaussian dynamics to bound the quantities in Landauer's principle. By the maximality of entropy for Gaussian states, we have Δ S≤Δ S^G, we quickly deduce that the generalised entropy production is upper-bounded by the Gaussian value, i.e., ΔΣ≤ΔΣ^G. Similarly, we infer that the change in mutual information is upper-bounded by the Gaussian value <cit.>, i.e., Δ I≤Δ I^G. Utilising Eq. (<ref>), we find that the change in relative entropy is lower-bounded by the Gaussian value, i.e., 

    Δ D≥Δ D^G  .

However, in both cases, these inequalities alone do not provide quantitative information on the extent to which these quantities dominate each other in the true state. To gain a more precise understanding of the relative contributions of mutual information and relative entropy in this context, a more comprehensive theoretical model and advanced experimental methods are required for an experiment modelled by an arbitrary interacting theory.









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