# Reconciling nonlinear dissipation with the bilinear model of two Brownian
particles

Elisa I. Goettems<EMAIL_ADDRESS>Instituto de Física de São Carlos,
Universidade de São Paulo, CP 369, 13560-970, São Carlos, SP, Brazil  Ricardo
J. S. Afonso<EMAIL_ADDRESS>Instituto de Física de São Carlos,
Universidade de São Paulo, CP 369, 13560-970, São Carlos, SP, Brazil  Diogo O.
Soares-Pinto  Instituto de Física de São Carlos, Universidade de São Paulo, CP
369, 13560-970, São Carlos, SP, Brazil  Daniel Valente  Instituto de Física,
Universidade Federal de Mato Grosso, 78060-900 Cuiabá MT, Brazil

(August 28, 2024)

###### Abstract

The Brownian motion of a single particle is a paradigmatic model of the
nonequilibrium dynamics of dissipative systems. In the system-plus-reservoir
approach, one can derive the particle’s equations of motion from the
reversible dynamics of the system coupled to a bath of oscillators
representing its thermal environment. However, extending the system-plus-
reservoir approach to multiple particles in a collective environment is not
straightforward, and conflicting models have been proposed to that end. Here,
we set out to reconcile some aspects of the nonlinear and the bilinear models
of two Brownian particles. We show how the nonlinear dissipation originally
derived from exponential system-reservoir couplings can alternatively be
obtained from the bilinear Lagrangian, with a modified spectral function that
explicitly depends on the distance between the particles. As applications, we
discuss how to avoid the anomalous diffusion from the standard nonlinear
model, as well as how to phenomenologically model a hydrodynamic interaction
between a pair of Brownian particles in a viscous fluid.

brownian motion, spectral function

##  I Introduction

The Brownian motion is a representative example of nonequilibrium dissipative
dynamics. Theoretically, the irreversible dynamics of a subsystem can emerge
from a reversible dynamics of the global system. To achieve that, one way is
to phenomenologically model the particle’s environment as a set of independent
oscillators, each linearly coupled to the system of interest. This is the so
called system-plus-reservoir approach. By tuning the spectral function, which
determines the weight of each frequency mode on the particle’s dissipation
rate, one can recover experimental observations. This framework has proven
useful across classical and quantum domains [1, 2, 3].

Let us consider two particles immersed in the same bath. This can be relevant,
for instance, when environment-induced effects on multiple degrees of freedom
are being investigated, as in the cases of biologically-inspired problems [4,
5], of non-Markovianity [6], of synchronization [7] and of quantum
entanglement [8, 9, 10, 11], to name a few. One way to address this problem is
to employ the so called bilinear model, which assumes that each particle is
linearly coupled to the same set of oscillators [9, 12, 10, 4, 7, 6]. However,
such a bilinear coupling may lead to unphysical results, namely, the free-
particle motion of the relative coordinate, and the absence of mutual effects
between proximal particles, as pointed out by Duarte and Caldeira [13]. To
solve these issues, a nonlinear model for the system-environment couplings has
been devised, which not only recovers the well-known single-particle case, but
also predicts dissipation rates that are nonlinear functions of the distance
between the pair of Brownian particles [13].

Despite the successes of the nonlinear model for two Brownian particles, the
bilinear approach has its merits. Experimentally, it can yield correct
results, as in the study of heat flow between Brownian particles [14], and in
a recent demonstration of environment-induced entanglement in the optical
domain [15]. Theoretically, linearity allows the model to be exactly solvable
in the quantum and the classical regimes, making it a desirable tool.
Additionally, the original spirit of the system-plus-reservoir approach as a
search for a simplified description of an otherwise untractable interaction
with a complex environment sounds more in line with the approach of the
bilinear model.

Here, we address the following question. Is there an alternative way to avoid
the shortcomings of the standard bilinear model, without recurring to
nonlinear system-environment couplings? To that end, we introduce a physically
motivated spectral function that explicitly depends on the relative distance
between a pair of Brownian particles. By doing so, we obtain, from the
bilinear model, the nonlinear dissipation that had been originally derived
from exponential system-environment couplings [13]. Our method thereby reveals
a simpler and more versatile way to phenomenologically model nonlinear
environment-induced forces. In particular, we discuss how to avoid an
anomalous diffusion found in the nonlinear model, and also how our theory can
describe hydrodynamic correlations of multiple Brownian particles [16].

In Sec.II, we revisit the bilinear and nonlinear models. In Sec.III, we show
our main result, namely, how nonlinear dissipation can be obtained from the
bilinear Lagrangian. For that, we define a spectral function that depends on
the relative distance between the particles (Sec.III.1). We show how our
modified spectral function enables us to bridge the dissipation rates from the
standard nonlinear and the standard bilinear models (Sec.III.2). We discuss
how to avoid the anomalous diffusion of the original nonlinear model
(Sec.III.3), and how our approach can be applied to model a pair of Brownian
particles sharing a hydrodynamic environment (Sec.III.4). Finally, in Sec.IV
we present our conclusions and further considerations. Detailed derivations
are presented in the AppendixA.

##  II Preliminary Remarks

###  II.1 Standard bilinear model

In the bilinear model [9], the Lagrangian of two classical Brownian particles
immersed in a collective environment reads

$\displaystyle\begin{split}L=&\frac{m}{2}(\dot{x}_{1}^{2}+\dot{x}_{2}^{2})-\frac{1}{2}\sum_{k}R_{k}\left(C_{k}^{(1)}x_{1}+C_{k}^{(2)}x_{2}\right)\\\
&-\frac{1}{2}\sum_{k}\frac{1}{m_{k}\omega_{k}^{2}}\left(C_{k}^{(1)}x_{1}+C_{k}^{(2)}x_{2}\right)^{2}+\sum_{k}\frac{m_{k}}{2}(\dot{R}_{k}^{2}-\omega_{k}^{2}R_{k}^{2}).\end{split}$
(1)

Here, $\dot{x}_{1,2}$ and $x_{1,2}$ are the velocity and the positions of the
particles with mass $m$. $R_{k}$ is the position of the $k$-th bath
oscillator, with frequency $\omega_{k}$ and mass $m_{k}$. The coupling between
the system and each oscillator is assumed to be linear in their positions,
with distinct coupling strengths $C_{k}^{(i)}$. The counterterm is added so as
to offset the environment-induced modification on the external potential.

One can derive the Euler-Lagrange equations for both the system and the bath.
From now on, we define the center of mass and relative coordinates as

$\displaystyle\begin{split}q=\frac{x_{1}+x_{2}}{2},\\\ \end{split}$ (2)

and

$\displaystyle\begin{split}u=x_{1}-x_{2},\end{split}$ (3)

which yield the following equations of motion,

$\displaystyle\begin{split}m\ddot{q}(t)+(\eta+\eta_{12})\dot{q}(t)=f_{q}(t),\\\
m\ddot{u}(t)+(\eta-\eta_{12})\dot{u}(t)=f_{u}(t).\end{split}$ (4)

One can interpret $f_{q}(t)$ and $f_{u}(t)$ as the fluctuating forces for the
center of mass and the relative coordinate,

$\displaystyle\begin{split}f_{q}(t)&=-\sum_{k}\frac{(C_{k}^{(1)}+C_{k}^{(2)})}{2}\\\
&\times\left(\dot{R}_{k}(0)\frac{\sin{\omega_{k}t}}{\omega_{k}}+\tilde{R}_{k}(0)\cos{\omega_{k}t}\right),\end{split}$
(5) $\displaystyle\begin{split}f_{u}(t)&=\sum_{k}(C_{k}^{(1)}-C_{k}^{(2)})\\\
&\times\left(\dot{R}_{k}(0)\frac{\sin{\omega_{k}t}}{\omega_{k}}+\tilde{R}_{k}(0)\cos{\omega_{k}t}\right).\end{split}$
(6)

where
$\tilde{R}_{k}(0)=R_{k}(0)+[C_{k}^{(i)}x_{i}(0)+C_{k}^{(j)}x_{j}(0)]/m_{k}\omega_{k}^{2}$.
The statistical properties of these forces stem from the initial state of the
total system.

For the dissipation term, we assume an Ohmic bath [3], so the usual spectral
functions read

$\displaystyle
J_{i}(\omega)=\frac{\pi}{2}\sum_{k}\frac{C_{k}^{(i)2}}{m_{k}\omega_{k}}\delta(\omega-\omega_{k})\equiv\eta\omega\Theta(\Omega-\omega),$
(7)

with the high frequency cutoff $\Omega$ [17, 18], and with $\Theta$ being the
Heaviside step function. Similarly, a mixed spectral function appears,

$\displaystyle
J_{12}(\omega)=\frac{\pi}{2}\sum_{k}\frac{C_{k}^{(1)}C_{k}^{(2)}}{m_{k}\omega_{k}}\delta(\omega-\omega_{k})\equiv\eta_{12}\omega\Theta(\Omega-\omega),$
(8)

as an indication of bath-mediated interactions between the particles. This
introduces the dissipation rates $\eta$ and $\eta_{12}$.

Two properties of this model call our attention. First, the case of identical
couplings, $C_{k}^{(1)}=C_{k}^{(2)}$, (a reasonable hypothesis, as far as two
proximal particles in the same environment are concerned) implies that
$\eta=\eta_{12}$ and $f_{u}(t)=0$, leading to vanishing dissipation and
fluctuating forces (see Eqs.(4) and (6)). It means that a free-particle motion
is found, namely,

$\displaystyle\ddot{u}(t)=0.$ (9)

Second, even for nonidentical couplings one finds that $\eta_{12}$ is
independent of the distance between the particles. Both considerations suggest
instantaneous effects between spatially separate entities. As discussed by
Duarte and Caldeira [13], these are undesirable features, arising from the
lack of an appropriate length scale for environment-mediated phenomena.

###  II.2 Standard nonlinear model

In the nonlinear model as introduced by Duarte and Caldeira in Ref. [13], the
Lagrangian reads

$\displaystyle\begin{split}&L=\frac{m}{2}(\dot{x}_{1}^{2}+\dot{x}_{2}^{2})+\frac{1}{2}\sum_{k}m_{k}\left(\dot{R}_{k}\dot{R}_{-k}-\omega_{k}^{2}R_{k}R_{-k}\right)\\\
&-\frac{1}{2}\sum_{k}\left[(C_{-k}(x_{1})+C_{-k}(x_{2}))R_{k}+(C_{k}(x_{1})+C_{k}(x_{2}))R_{-k}\right],\end{split}$
(10)

where

$\displaystyle C_{k}(x)=\kappa_{k}e^{ikx}.$ (11)

The exponential couplings between the bath and the particles guarantee
homogeneity and translational invariance. Note that the index $k$ now has
dimensions of $[L]^{-1}$, explaining why it introduces the required length
scale.

The equations of motion are

$\displaystyle\begin{split}m\ddot{x}_{i}(t)&+\int_{0}^{t}\mathbf{K}(x_{i}(t)-x_{i}(t^{\prime}),t-t^{\prime})\dot{x}_{i}(t^{\prime})dt^{\prime}\\\
&+\int_{0}^{t}\mathbf{K}(x_{i}(t)-x_{j}(t^{\prime}),t-t^{\prime})\dot{x}_{j}(t^{\prime})dt^{\prime}\\\
&+\dfrac{\partial}{\partial x_{i}}V(x_{i}(t)-x_{j}(t))=F_{i}(t)\end{split}$
(12)

with $i,j=1,2$, and once again $F_{i}(t)$ can be interpreted as the
fluctuating force. The dissipation kernels are

$\displaystyle\mathbf{K}(r,\tau)$
$\displaystyle=\sum_{k}\int_{0}^{\infty}d\omega 2k^{2}\kappa_{k}\kappa_{-k}$
$\displaystyle\times\frac{\mathrm{Im}\chi_{k}^{(i)}(\omega)}{\pi\omega}\cos{\omega_{k}\tau}\cos{kr},$
(13)

written in terms of the imaginary part of the dynamical response of the
environment oscillators, namely,

$\displaystyle\mathrm{Im}\chi_{k}^{(i)}(\omega)\equiv\frac{\pi}{2m_{k}\omega_{k}}\delta(\omega-\omega_{k}).$
(14)

In the present model, $\mathrm{Im}\chi_{k}^{(i)}(\omega)$ is equivalent to the
spectral function in that it allows for the transformation of a discrete set
of oscillators into a continuum. The delta function is thus replaced by a
Lorentzian peaked around $\omega_{k}$. The next step is to focus on the low-
frequency limit of that Lorentzian so as to recover the Ohmic regime (linear
in $\omega$). This justifies the approximation

$\displaystyle\mathrm{Im}\chi_{k}^{(i)}(\omega)\approx\tilde{f}(k)\omega\Theta(\Omega-\omega).$
(15)

Nonlinear equations of motion are obtained, such that

$\displaystyle
m\ddot{q}(t)+\left(\eta+\eta_{e}[u(t)]\right)\dot{q}(t)=f_{q}(t),$ (16)

and

$\displaystyle
m\ddot{u}(t)+\left(\eta-\eta_{e}[u(t)]\right)\dot{u}(t)+V_{e}^{\prime}(u(t))=f_{u}(t).$
(17)

Here,

$\displaystyle V_{e}(u)\equiv-\frac{2\Omega\eta}{\pi
k_{0}^{2}(k_{0}^{2}u^{2}+1)}$ (18)

represents an environment-induced potential, which depends on the relative
distance. Similarly,

$\displaystyle\eta_{e}[u]\equiv\eta\frac{(1-3k_{0}^{2}u^{2})}{(k_{0}^{2}u^{2}+1)^{3}}$
(19)

describes a distance-dependent bath-mediated dissipation rate. The constant
$k_{0}$ is a characteristic inverse length introduced when the summation over
$k$ is transformed into an integral, $\sum_{k}\rightarrow(L/2\pi)\int dk$, and
a density of spatial modes is postulated,

$\displaystyle\eta g(k)\equiv\frac{L}{2\pi}\kappa_{k}\kappa_{-k}\tilde{f}(k),$
(20)

such that $\int_{0}^{\infty}dkg(k)k^{2}=1$. Finally, the choice for

$\displaystyle g(k)=\left(\frac{1}{2k_{0}^{3}}\right)e^{-k/k_{0}}$ (21)

explains how $k_{0}$ is defined in Ref. [13].

Equations (16) to (19) describe rich environment-induced behaviors for
proximal particles, while recovering the independent Brownian movement for
arbitrarily large distances. The free-particle anomaly found in the bilinear
model is no longer present, since both the dissipative and the fluctuating
forces are finite for any finite $u(t)$. The statistical properties of the
fluctuating forces are such that (see Appendix A)

$\displaystyle\langle{f_{q}(t)}\rangle$
$\displaystyle=\langle{f_{u}(t)}\rangle=0,$
$\displaystyle\langle{f_{q}(t)f_{q}(t^{\prime})}\rangle$
$\displaystyle=k_{B}T(\eta+\eta_{e}[u(t)])\delta(t-t^{\prime}),$
$\displaystyle\langle{f_{u}(t)f_{u}(t^{\prime})}\rangle$
$\displaystyle=4k_{B}T(\eta-\eta_{e}[u(t)])\delta(t-t^{\prime}),$

in agreement with the fluctuation-dissipation theorem.

##  III Results

###  III.1 Nonlinear dissipation from a bilinear Lagrangian: modified spectral
function

In this section we show how the nonlinear dissipation term $\eta_{e}[u(t)]$
from Eqs.(16) and (17) can be obtained from a bilinear model. Following Ref.
[9], we assume a Lagrangian that breaks translational invariance both in the
system degrees of freedom and in the couplings to the bath,

$\displaystyle\begin{split}L&=\frac{m}{2}(\dot{x}_{1}^{2}+\dot{x}_{2}^{2})-V(x_{1})-V(x_{2})-m\
c_{12}x_{1}x_{2}\\\
&+\sum_{k=1}^{N}\left(\frac{m_{k}\dot{R}_{k}^{2}}{2}-\frac{m_{k}}{2}\omega_{k}^{2}R_{k}^{2}\right)-\sum_{k=1}^{N}(C_{k}^{(1)}x_{1}+C_{k}^{(2)}x_{2})R_{k}.\end{split}$
(22)

We have maintained the direct coupling between the particles, proportional to
$c_{12}$, as done in Ref.[9]. Note that, in order to avoid the anomalous free-
particle motion for the relative coordinate ($\ddot{u}=0$), we have to assume
that each particle has a distinct coupling parameter ($C_{k}^{(1)}\neq
C_{k}^{(2)}$), otherwise the bath decouples from $u=x_{1}-x_{2}$. Using
similar techniques as in the previous sections, one gets to the following
equations of motion for each particle

$\displaystyle\begin{split}m\ddot{x}_{i}+&\dfrac{dV(x_{i})}{dx_{i}}+mc_{12}x_{j}\\\
&+\sum_{k}\frac{C_{k}^{(i)2}}{m_{k}\omega_{k}^{2}}\int_{0}^{t}\cos{\omega_{k}(t-t^{\prime})}\dot{x}_{i}(t^{\prime})dt^{\prime}\\\
&+\sum_{k}\frac{C_{k}^{(i)}C_{k}^{(j)}}{m_{k}\omega_{k}^{2}}\int_{0}^{t}\cos{\omega_{k}(t-t^{\prime})}\dot{x}_{j}(t^{\prime})dt^{\prime}=f_{i}(t),\end{split}$
(23)

where $i\neq j=1,2$, and $f_{i}(t)$ term can be interpreted as the fluctuating
force

$\displaystyle\begin{split}f_{i}(t)=-\sum_{k}C_{k}^{(i)}\left[\dot{R}_{k}(0)\frac{\sin{\omega_{k}t}}{\omega_{k}}+\tilde{R}_{k}(0)\cos{\omega_{k}t}\right],\end{split}$
(24)

where
$\tilde{R}_{k}(0)=R_{k}(0)+(C_{k}^{(i)}x_{i}(0)+C_{k}^{(j)}x_{j}(0))/(m_{k}\omega_{k}^{2})$.

To evidence the center of mass and the relative coordinate, we rewrite the
equations of motion as

$\displaystyle\begin{split}&m\ddot{q}(t)+\frac{1}{2}\left(\dfrac{dV(x_{1})}{dx_{1}}+\dfrac{dV(x_{2})}{dx_{2}}\right)+mc_{12}q(t)\\\
&+\frac{1}{2}\int_{0}^{t}\left[K_{1}(t-t^{\prime})\dot{x}_{1}(t^{\prime})+K_{2}(t-t^{\prime})\dot{x}_{2}(t^{\prime})\right]dt^{\prime}\\\
&+\int_{0}^{t}K_{ij}(t-t^{\prime})\dot{q}(t^{\prime})dt^{\prime}=f_{q}(t),\end{split}$
(25)

$\displaystyle\begin{split}&m\ddot{u}(t)+\left(\dfrac{dV(x_{1})}{dx_{1}}-\dfrac{dV(x_{2})}{dx_{2}}\right)-mc_{12}u(t)\\\
+&\int_{0}^{t}\left[K_{1}(t-t^{\prime})\dot{x}_{1}(t^{\prime})-K_{2}(t-t^{\prime})\dot{x}_{2}(t^{\prime})\right]dt^{\prime}\\\
&-\int_{0}^{t}K_{ij}(t-t^{\prime})\dot{u}(t^{\prime})dt^{\prime}=f_{u}(t).\end{split}$
(26)

The dissipation kernels in Eqs. (25) and (26) are different from the ones in
Ref. [13] in the lack of a spatial dependence of the environment-induced
effects, as we show below,

$\displaystyle\mathbf{K}_{i}(t-t^{\prime})=\sum_{k}2C_{k}^{(i)2}\int\displaylimits_{0}^{\infty}d\omega\frac{\imaginary\chi_{k}^{(i)}(\omega)}{\pi\omega}\cos{\omega(t-t^{\prime})},$
(27)
$\displaystyle\mathbf{K}_{ij}(t-t^{\prime})=\sum_{k}2C_{k}^{(i)}C_{k}^{(j)}\int\displaylimits_{0}^{\infty}d\omega\frac{\imaginary\chi_{k}^{(ij)}(\omega)}{\pi\omega}\cos{\omega(t-t^{\prime})}.$
(28)

We have also used Eq.(14) to write the kernels in terms of the imaginary part
of the bath susceptibility functions.

As far as the single-particle dissipation rates are concerned, we define

$\displaystyle\eta\equiv\sum_{k}C_{k}^{(i)2}\tilde{f}(k),$ (29)

where $\tilde{f}(k)$ comes from the Ohmic approximation in Eq.(27), i.e.,
$\imaginary\chi_{k}^{(i)}(\omega)\approx\tilde{f}(k)\omega\Theta(\Omega-\omega)$
(similarly to Eq.(15)). To obtain the continuum limit, we replace
$\sum_{k}C_{k}^{(i)2}\tilde{f}(k)$ by $\eta\int dk\ k^{2}g(k)$, where
$g(k)=\exp(-k/k_{0})/(2k_{0}^{3})$ (as in Sec.II.2).

The key step in our derivation concerns the two-particle susceptibility
$\chi_{k}^{(ij)}$. We consider that, when two Brownian particles are
sufficiently close, the environment acting on each particle is composite: It
is jointly formed by the free environment dynamics plus the perturbation of
the other particle dynamics on that environment. Put differently, each
particle “sees” an effective bath “dressed” by the state of the other one,
when these are close enough. As a consequence, our insight translates into a
response function that should depend on the distance between the particles or,
more generally, on the relative coordinate $u$. We thus postulate that, in our
model,

$\displaystyle\imaginary\chi_{k}^{(ij)}(\omega)\equiv\imaginary\chi_{k}^{(ij)}(\omega,u).$
(30)

We obtain the Ohmic regime by choosing a linear function in $\omega$, namely,

$\displaystyle\imaginary\chi_{k}^{(ij)}(\omega,u)$ $\displaystyle\approx
h(k,u)\omega\Theta(\Omega-\omega),$ (31)

where $h(k,u)$ is to be defined. The explicit choice for $h(k,u)$ will allow
us to define the bath-related length scale. Because our main goal is to
reobtain the nonlinear dissipation from the nonlinear model, we write $h(k,u)$
in the form

$\displaystyle h(k,u)=\tilde{F}(k)G(k,u).$ (32)

Here, $\tilde{F}(k)$ is analogous to $\tilde{f}(k)$ in that it allows us to
define

$\displaystyle\eta_{\mathrm{eff}}[u]\equiv\eta\int dk\
g_{\mathrm{eff}}(k)G(k,u)$ (33)

as the continuum limit of $\sum_{k}C_{k}^{(i)}C_{k}^{(j)}\tilde{F}(k)G(k,u)$.
This summation is obtained from applying Eqs. (31) and (32) to (28). We have
also defined

$\displaystyle g_{\mathrm{eff}}(k)\equiv k^{2}g(k),$ (34)

motivated by the fact that we recover the single-particle dissipation rate by
choosing $G(k,u)=1$.

Using Eqs.(29) and (33), and taking the limit $\Omega\rightarrow\infty$ in
Eqs. (27) and (28), the dissipation kernels become

$\displaystyle\mathbf{K}_{i}(t-t^{\prime})$
$\displaystyle=2\eta\delta(t-t^{\prime})$ (35)
$\displaystyle\mathbf{K}_{ij}(t-t^{\prime})$
$\displaystyle=2\eta_{\mathrm{eff}}[u]\delta(t-t^{\prime}).$ (36)

The length scale is now explicit, given by the dependence of the kernel on the
relative coordinate $u$. By tuning $G(k,u)$, one controls the nonlinear
dissipation force appearing in the equations of motion, be it in order to
recover a certain theoretical model or to explain a specific experiment.

Finally, our equations of motion in the case of free Brownian particles,
$V(x_{1})=V(x_{2})=0$, read

$\displaystyle\begin{split}&m\ddot{q}(t)+\left(\eta+\eta_{\mathrm{eff}}[u]\right)\dot{q}(t)=f_{q}(t)\\\
&m\ddot{u}(t)+\left(\eta-\eta_{\mathrm{eff}}[u]\right)\dot{u}(t)=f_{u}(t),\end{split}$
(37)

where center of mass and relative fluctuating forces are given by
$f_{q}(t)=(f_{1}+f_{2})/2$ and $f_{u}(t)=f_{1}-f_{2}$. Note that, although
nonlinear dissipation forces have been found, the effective bath-induced
potential $V_{e}[u]$ from Eq.(17) could not be recovered in our modified
bilinear model. This suggests that, if a given experiment reveals bath-
mediated conservative forces, the theoretical model should probably start from
nonlinear system-bath couplings.

###  III.2 Recovering the dissipation rates

In order to recover the nonlinear dissipation $\eta_{e}[u]$ from Eq.(19), we
choose

$\displaystyle G(k,u)=\cos(ku).$ (38)

We use this in Eq.(33), thus finding that

$\displaystyle\eta_{\mathrm{eff}}[u]=\eta\int_{0}^{\infty}dk\ k^{2}\
\frac{e^{-k/k_{0}}}{2k_{0}^{3}}\
\cos(ku)=\eta\frac{(1-3k_{0}^{2}u^{2})}{(k_{0}^{2}u^{2}+1)^{3}}.$ (39)

As expected, $\eta_{\mathrm{eff}}[u]=\eta_{e}[u]$.

We also recover the constant dissipation rate $\eta_{12}$ shown in Eq.(8), in
the context of the standard bilinear model. By choosing $G(k,u)=G_{0}$, we
find that

$\displaystyle\eta_{\mathrm{eff}}[u]=\eta G_{0}=\eta_{12}.$ (40)

This means that our result bridges the standard bilinear and the standard
nonlinear models, as far as dissipation is concerned.

###  III.3 Avoiding the anomalous diffusion

It is worth discussing the anomalous diffusion due to the specific form of
$\eta_{\mathrm{eff}}[u]$ in Eq.(39). This can be seen from the diffusion
coefficients appearing in the correlation functions of the Langevin forces,
namely,

$\displaystyle\langle{f_{1}(t)f_{2}(t^{\prime})}\rangle=2D_{12}(u)\delta(t-t^{\prime}),$
(41)

$\langle{f_{u}(t)f_{u}(t^{\prime})}\rangle=2D_{u}(u)\delta(t-t^{\prime})$, and
$\langle{f_{q}(t)f_{q}(t^{\prime})}\rangle=2D_{q}(u)\delta(t-t^{\prime})$. The
connection comes from the fact that

$\displaystyle D_{12}(u)=\eta_{\mathrm{eff}}[u]k_{B}T.$ (42)

Similarly, we also find

$\displaystyle D_{u}(u)=(\eta-\eta_{\mathrm{eff}}[u])2k_{B}T,$ (43)

and

$\displaystyle D_{q}(u)=(\eta+\eta_{\mathrm{eff}}[u])\frac{k_{B}T}{2}.$ (44)

The anomalous diffusion arises from

$\displaystyle\eta_{\mathrm{eff}}[|u|>k_{0}^{-1}/\sqrt{3}]<0,$ (45)

implying a reduction in the diffusion coefficient $D_{q}$ at intermediate
separations, as compared to arbitrarily far apart Brownian particles, as well
as an anticorrelation ($D_{12}<0$) between the Langevin forces acting on the
particles.

To avoid the anomalous diffusion, we tune $G(k,u)$. This allows us to derive a
different behavior that could still fulfill the fluctuation-dissipation
relation, while eliminating the anomalous anticorrelation effect. To be
concrete, let us take the example of the spectral functions used to model
localized excitons interacting with a bath of acoustic phonons [19, 20]. Their
typical Gaussian features motivate us to set

$\displaystyle G(k,u)=e^{-\frac{k}{k_{0}}(k_{0}u)^{2}}.$ (46)

This results in an effective dissipation rate given by

$\displaystyle\eta_{\mathrm{eff}}[u]=\frac{\eta}{(1+k_{0}^{2}u^{2})^{3}}.$
(47)

That is, Eq.(46) guarantees that $\eta_{\mathrm{eff}}[u]\geq 0$, and also
recovers the independent Brownian motions of two arbitrarily distant particles
(in the limit of $|u|\rightarrow\infty$). In fact, any positive and convergent
function (i.e., $G(k,u)\geq 0$ and $G(k,|u|\rightarrow\infty)=0$) is
sufficient to guarantee that $\eta_{\mathrm{eff}}[u]\geq 0$, and that
$\eta_{\mathrm{eff}}[|u|\rightarrow\infty]=0$.

###  III.4 Hydrodynamics-inspired model

We now consider two Brownian particles immersed in a viscous fluid. The fluid
can mediate interactions between the particles, so that the dissipative forces
may depend on the interparticle distance, as shown in Ref. [16]. Here, we
restrict our discussion to the limit where the radius of each Brownian
particle is vanishingly small as compared to their relative distance (i.e, the
regime of validity of the Oseen tensor). We also assume the one-dimensional
limit of the Brownian motion. In that case, an effective dissipation rate
mediated by the hydrodynamic environment, as derived with the help of the
fluctuation-dissipation relation, is given by [16]

$\displaystyle\eta_{\mathrm{eff}}^{\mathrm{hydro}}[u]\approx\gamma_{h}|u|,$
(48)

where $\gamma_{h}$ is a constant proportional to the solvent viscosity. It is
worth mentioning that, in Ref. [16], no effective conservative forces are
emerging from the hydrodynamic environment. This means that the hydrodynamic
scenario discussed in Ref. [16] behaves more similarly to our modified
bilinear model than to the standard nonlinear model (which gives rise to
$V_{e}[u]$, as we have seen in Eq.(18)).

To model Eq.(48), we can simply choose

$\displaystyle G(k,u)=k|u|,$ (49)

since

$\displaystyle\eta_{\mathrm{eff}}[u]=\eta\int_{0}^{\infty}dk\ k^{2}\
\frac{e^{-\frac{k}{k_{0}}}}{2k_{0}^{3}}\ k|u|=\gamma_{m}|u|,$ (50)

with $\gamma_{m}=3\eta k_{0}$.

We have thus found a simpler way to map an environment-mediated dissipation of
hydrodynamic nature into a fictitious bath consisting of a continuous set of
harmonic oscillators. By contrast, it is not clear how to derive Eqs.(50), and
neither (47), from the standard nonlinear model, since the appropriate choice
for the system-environment couplings as functions of the particles positions
in the Lagrangian is not evident to us.

##  IV Conclusions

In summary, we have revisited the standard bilinear and the standard nonlinear
models for the dynamics of two Brownian particles in a collective environment.
In particular, we have addressed a controversy between these approaches,
namely, whether distance-dependent nonlinear dissipation forces mediated by
the environment, and affecting both the center of mass and the relative
coordinate, should or not exist, and in which circumstances.

Our main result was the derivation of a nonlinear effective dissipation rate
$\eta_{\mathrm{eff}}[u]$ departing from a bilinear Lagrangian. Our method was
based on the introduction of a distance-dependent nonlinear spectral function
(response function) $\chi_{k}^{(ij)}(\omega,u)$, which settles a length scale
to the dynamics of the Brownian particles (as also achieved with the standard
nonlinear model, but not with the standard bilinear model). This allows us to
interpolate the dissipation forces as derived from the standard bilinear and
the standard nonlinear models. We have also shown that the nonlinear effective
potential $V_{e}[u]$ obtained from the standard nonlinear model did not arise
from our modified bilinear model.

As applications, we have discussed some consequences of tuning our distance-
dependent spectral function. For instance, a change from a trigonometric
function to a Gaussian in $G(k,u)$ made it possible to avoid the anomalous
diffusion presented in the original nonlinear model. Also, we described
hydrodynamic correlations between a pair of Brownian particles in a viscous
fluid by means of our phenomenological distance-dependent spectral function.
These examples illustrate how our results represent a simple and versatile way
to express diverse nonlinear dissipative forces in the dynamics of pairs of
Brownian particles.

As a perspective, we would like to generalize our distance-dependent spectral
function to a larger number of Brownian degrees of freedom in a common
environment ($N$ particles in a three-dimensional space). We believe it to be
a feasible goal, given the pairwise character that typically underlies
effective interactions. This could allow us to characterize relaxation
processes and entropy production in nonequilibrium dissipative many-body
systems, across classical and quantum regimes [21]. For instance, we could
think of generalizing a recent study concerning entropy production of a single
quantum Brownian particle [22].

## Acknowledgements

EIG and RJSA ackowledge support from Coordenação de Aperfeiçoamento de Pessoal
de Nível Superior - Brasil (CAPES) - Finance Code 001. DOSP ackowledge support
by Brazilian funding agencies CNPq (Grant 307028/2019-4) and FAPESP (Grant No
2017/03727-0). DV was supported by the Serrapilheira Institute (Grant No.
Serra-1912-32056). EIG, RJSA, DOSP and DV acknowledge support from the
Instituto Nacional de Ciência e Tecnologia de Informação Quântica, CNPq INCT-
IQ (465469/2014-0), Brazil.

*

##  Appendix A Two-time correlation functions and the fluctuation dissipation
theorem

From the condition of thermal equilibrium, we have the following identities

$\displaystyle\langle{\tilde{R}_{k}(0)}\rangle$ $\displaystyle=0,$
$\displaystyle\langle{\dot{R}_{k}(0)}\rangle=0,$ (51)
$\displaystyle\langle{\tilde{R}_{k}(0)\dot{R}_{k^{\prime}}(0)}\rangle$
$\displaystyle=0,$
$\displaystyle\langle{\dot{R}_{k}(0)\tilde{R}_{k^{\prime}}(0)}\rangle=0,$ (52)
$\displaystyle\langle{\dot{R}_{k}(0)\dot{R}_{k^{\prime}}(0)}\rangle$
$\displaystyle=\frac{k_{B}T}{m_{k}}\delta_{kk^{\prime}},$
$\displaystyle\langle{\tilde{R}_{k}(0)\tilde{R}_{k^{\prime}}(0)}\rangle=\frac{k_{B}T}{m_{k}\omega_{k}^{2}}\delta_{kk^{\prime}},$
(53)

where $k_{B}$ is the Boltzmann constant. As defined in the main text, the
formal expressions for the fluctuating forces are given by

$\displaystyle\begin{split}f_{i}(t)=-\sum_{k}C_{k}^{(i)}\left[\dot{R}_{k}(0)\frac{\sin{\omega_{k}t}}{\omega_{k}}+\tilde{R}_{k}(0)\cos{\omega_{k}t}\right],\end{split}$
(54)

where the displaced equilibrium positions of the oscillators (due to their
couplings with the particles) are
$\tilde{R}_{k}(0)=R_{k}(0)+(C_{k}^{(i)}x_{i}(0)+C_{k}^{(j)}x_{j}(0))/(m_{k}\omega_{k}^{2})$.
We have also defined $f_{q}=(f_{1}+f_{2})/2$, and $f_{u}=f_{1}-f_{2}$. With
the above expressions at hands, we obtain the general form for the two-time
correlation functions,

$\displaystyle\langle
f_{\alpha}(t)f_{\beta}(t^{\prime})\rangle=2D_{\alpha\beta}(u)\delta(t-t^{\prime}),$
(55)

where $D_{\alpha\beta}(u)$ is a type of diffusion coefficient having a
different form according to the choice of forces we are dealing with. To
explicitly compute them, we apply the continuum limit in the same way we did
in Sec.III.1. We thus find Eqs.(42), (43), and (44) as results.

For the sake of completeness, and also to highlight that Eq.(55) is indeed a
general form of the fluctuation-dissipation theorem, we recall the original
version of the theorem below. The Langevin equation for a free Brownian
particle reads

$\displaystyle m\dot{v}(t)=-\eta v(t)+F(t),$ (56)

where $\eta$ is a friction coefficient, and $F(t)$ is a fluctuating force
caused by collisions of the particle with the atoms of the surrounding fluid.
The fluctuating force fulfills

$\displaystyle\langle{F(t)}\rangle=0,$ $\displaystyle\mbox{and}\
\langle{F(t)F(t^{\prime})}\rangle=2D\delta(t-t^{\prime}),$ (57)

where $D$ can be seen as a measure of the strength of the fluctuating force.
The delta function in time indicates that there is no correlation between
impacts at any distinct time intervals. The solution for the linear, first-
order, and inhomogeneous differential equation reads [1]

$\displaystyle v(t)=e^{-\eta
t/m}v(0)+\int_{0}^{t}dt^{\prime}e^{-\eta(t-t^{\prime})/m}F(t^{\prime})/m.$
(58)

We can get the mean squared velocity and evaluate for long times, thus

$\displaystyle\langle{v^{2}(\infty)}\rangle=\frac{D}{\eta m}.$ (59)

At thermal equilibrium, $\langle{v^{2}}\rangle_{eq}=k_{B}T/m$ (equipartition
theorem), hence

$\displaystyle D=\eta k_{B}T.$ (60)

This relates the strength $D$ of the random noise, or fluctuating force, to
the magnitude $\eta$ of the friction, or dissipation rate, explaining why it
is known as the fluctuation-dissipation theorem. It expresses the balance
between friction and noise that is required to have thermal equilibrium state
at long times.

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