# Super-diffusion and crossover from diffusive to anomalous transport in a
one-dimensional system

Anupam Kundu  International Centre for Theoretical Sciences, Tata Institute of
Fundamental Research, Bengaluru 560089, India

###### Abstract

We study transport in a one-dimensional lattice system with two conserved
quantities – ‘volume’ and energy. Considering a slowly evolving local
equilibrium that is slightly deviated from an underlying global equilibrium,
we estimate the correction to the local equilibrium distribution. This
correction arises mainly through the space-time correlations of the local
currents. In the continuum limit, we show that the local equilibrium
distribution along with the correction yields drift-diffusion equation for the
‘volume’ and super-diffusion equation for the energy in the linear response
regime as macroscopic hydrodynamics. We find explicit expression of the super-
diffusion equation. Further, we find diffusive correction to the super-
diffusive evolution. Such a correction allows us to study a crossover from
diffusive to anomalous transport. We demonstrate this crossover numerically
through the system size scaling of the stationary current in non-equilibrium
steady state prepared by two reservoirs of different temperatures.

##  1 Introduction

In low dimensional systems, the transport of energy on macroscopic scale is
often anomalous as manifested by emergence of super-diffusion [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14]. According to the Green-Kubo formula the
average current in a non-equilibrium system is related to the time integral of
the equilibrium total current-current correlation in the linear response
regime. Several numerical studies as well as theoretical arguments reveal that
super-diffusion of energy is associated to the power-law tail of the current-
current correlation at long time. The non-linear fluctuating hydrodynamic
(NFHD) theory provides a general framework (applicable to a wide class of
systems both Hamiltonian as well as stochastic) to understand this super-
diffusion [4, 5, 15]. This theory describes the evolution of conserved fields
on a mesoscopic scale in terms of hydrodynamic (HD) equations in which
corresponding currents are expanded to non-linear order in the deviation from
their values in a underlying global equilibrium (GE). For this, one assumes a
slowly varying and slowly evolving local-equilibrium picture that is slightly
deviated from a global equilibrium state of the system. Further the
dissipation and the noise terms, obeying fluctuation-dissipation relation, are
added phenomenologically to the currents in order to describe fluctuations. By
decomposing the hydrodynamic evolution of the conserved fields into evolution
of sound and heat modes (also known as normal modes), this theory reveals the
connection between the super-diffusion (or anomalous transport) in
translationally invariant Hamiltonian systems with short range interaction
with the Kardar-Parisi-Zhang (KPZ) universality class [4, 5, 15, 16]. This
connection brought out by identifying the structural similarity between the
stochastic HD equation of the sound mode fields with (coupled) noisy Burgers
equations. However for the heat mode, which is non-propagating, one requires
to study the sub-leading correction which is achieved through a mode-coupling
approximation. The NFHD theory successfully applies to a wider class of
Hamiltonian systems with short range interactions and the predictions of this
theory classifies Hamiltonians into different universality classes depending
on their transport behaviours.

In this paper, we study anomalous transport in a simple model defined on a one
dimensional lattice of size $N$. Each lattice site contains a ‘volume’
variable $\eta_{i},$ that evolves according to

$\displaystyle\begin{split}\dot{\eta}_{i}=&~{}V^{\prime}(\eta_{i+1})-V^{\prime}(\eta_{i-1})\\\
&+\text{stochastic~{}exchange~{}$\eta$~{}between~{}neighbouring
sites~{}at~{}rate~{}}\gamma\end{split}$ (1)

for $i=1,2,...,N$, where $V(\eta)=\frac{k_{o}}{2}\eta^{2}$ with $k_{o}>0$. We
consider periodic as well as open boundary conditions in different situations,
details of which will be provided in the particular sections. The stochastic
exchange terms represent exchanging the variables across a bond at random with
rate $\gamma$. This model was first introduced in [17] and was called
“harmonic chain with volume exchange” (HCVE) system in [18]. The stochastic
exchange terms are added to the dynamics to make the system posses good
ergodic properties such that the system reach a thermal equilibrium state by
itself. It is easy to see that this model has two globally conserved
quantities, namely the total ‘volume’ $\sum_{i}\eta_{i}$ and total energy
$\sum_{i}V(\eta_{i})$ which yields two conservation equations for the
corresponding locally conserved fields, again namely ‘volume’ density field
and energy density field. Previously this model was shown to exhibit anomalous
energy transport in the non-equilibrium steady state (NESS) [17, 15, 18] and
super-diffusion of space-time correlation in close system set-up (i.e. not
connected to reservoirs)[15, 10].

Following different microscopic approaches it was shown in [17] and [18], that
the energy field (with the convective part subtracted) follows a super-
diffusive evolution equation on the macroscopic scale. In the first part of
the paper we provide a simpler alternative derivation of the macroscopic
hydrodynamic equations corresponding to the conserved fields of the system in
a close system set-up. We show that the non-convective part of the energy
field evolves according to a super-diffusion equation and the ‘volume’ density
field evolves diffusively in the linear response (LR) regime. We find explicit
expressions of these equations. Additionally we find the diffusive correction
to the super-diffusion equation explicitly. Our derivation is based on
estimating the correction to the local-equilibrium distribution. Such a
correction includes the contribution of current-current correlation to the
computation of the average current which would appear in the macroscopic
hydrodynamics. To compute these correlations, we invoke fluctuating
hydrodynamics (FHD) on a mesoscopic scale. Our derivation is intuitive and
reveals the importance and significance of the various approximations that
goes into deriving the hydrodynamics on a macroscopic scale.

In the second part of the paper we demonstrate a crossover from diffusive
transport to anomalous transport as one goes from mesoscopic length scale
$\Lambda$ (s.t. $1\ll\Lambda\ll N$) to macroscopic length scale. In NFHD, one
can expect such a crossover from the structure of the the HD currents which
have two parts: the Euler part that comes from the deviation from the global
equilibrium and the second part constituting of dissipation (in the form of
diffusion) and noise. The later part generically originates while coarse
graining the conserved quantities, both in space and time. In the coarse
graining procedure one usually replaces the values of the locally conserved
fields by values averaged with respect to a local (canonical) thermal
equilibrium distribution, which is different from the actual distribution of
the microscopic degrees of freedom. The later distribution involves
correlations of the locally conserved fields among themselves as well as
across space and time. The NFHD theory, in fact, computes contribution of
these correlation to the steady state currents which could be anomalous in the
leading order (of system size).

Although the actual values of the diffusion and noise terms in NFHD do not
affect the leading anomalous behaviour of the energy current and similarly the
super-diffusion, their presence is crucial for deriving the anomalous
transport or super-diffusive behaviour. For a given Hamiltonian, deriving the
fluctuating HD equations, especially the dissipation and the noise terms, is a
difficult problem and until recently only a few attempts have been made. Using
the projection operator technique and Markovian approximation, the study in
[19] derives the diffusion and noise terms for a class of Hamiltonian systems
defined on a one-dimensional lattice, although the expressions are not
explicit. Another study that derives these terms involves one dimensional
ideal gas of identical point particles undergoing stochastic collisions (both
momentum and energy conserving) among three consecutive particles in addition
to their Newtonian dynamics [20]. In this paper, we also provide a heuristic
derivation of the dissipation and noise terms for the ‘volume’ field of the
HCVE model in the LR regime. The presence of the stochastic exchange terms in
Eq. (1) makes it possible to derive the noise terms explicitly.

While the diffusion terms do not affect the leading anomalous system size
scaling of the steady state current in the LR regime, they can provide the
sub-leading correction which has normal scaling as one finds in the Fourier’s
Law i.e. inversely proportional to the system size [20]. This suggests a
crossover from diffusive to anomalous transport as one increases the system
size from a mesoscopic scale to macroscopic scale in comparison to the
microscopic scale (e.g. lattice spacing, interaction core or mean free path).
In this paper we demonstrate such a crossover, both analytically and
numerically through the system size scaling of the stationary current of the
system in the non-equilibrium steady state (NESS) in the open system set-up.

The paper is organized as follows. In sec. 2, we describe the system and
define the conserved quantities and, write the corresponding continuity
equations. In the next section 3, we provide the derivation of the linearized
hydrodynamic equations. This section starts with the Fokker-Planck equation
and the solution of it. This solution is used to derive the hydrodynamic
equations in two stages which are presented in sections 3.1 and 3.2. To
complete the derivation of the linearized hydrodynamics as will be shown one
requires to compute space-time correlation of the local currents, which is
done in sec. 3.2.1. This section is followed by a demonstration of crossover
from diffusive to anomalous transport in open system set-up in sec. 4. Finally
in section 5, we provide a summary of our results along with possible future
directions of study.

Figure 1: A schematic diagram of the HCVE model. Each site contains some
variable $\eta_{i}$ which are called ‘’volume’. These variables are subjected
to harmonic potential $V(\eta)=\frac{k_{0}\eta^{2}}{2}$ at each lattice site.
In addition to the deterministic evolution, ’volume’ variables on successive
sites are exchanged with rate $\gamma$.

##  2 Conservation Laws and the equilibrium state

We first consider the HCVE model on a circular lattice of size $N$ with
periodic boundary condition $\eta_{i+N}=\eta_{i}$ with $i=1,2,...,N$. A
schematic of the system is given in fig. 1. It is easy to see that the
dynamics in Eq. (1) keeps the total volume and the total energy invariant i.e.
$\frac{d}{dt}\sum_{i}\eta_{i}=0$ and $\frac{d}{dt}\sum_{i}V(\eta_{i})=0$. The
sum structure of these global conservations suggests the following two local
conserved quantities, namely

$\displaystyle\begin{split}\text{local~{}`volume':}&~{}~{}\hat{h}_{i}(\vec{\eta})=\eta_{i},\\\
\text{local~{}`energy':}&~{}~{}\hat{e}_{i}(\vec{\eta})=V(\eta_{i})=\frac{k_{o}}{2}\eta_{i}^{2},\end{split}$
(2)

where $\vec{\eta}=(\eta_{1},\eta_{2},...,\eta_{N})$. One can write the
following conservation laws for these local quantities

$\displaystyle\partial_{t}\hat{h}_{i}$
$\displaystyle=\hat{j}^{(h)}_{i-1,i}-\hat{j}^{(h)}_{i,i+1},$ (3)
$\displaystyle\partial_{t}\hat{e}_{i}$
$\displaystyle=\hat{j}^{(e)}_{i-1,i}-\hat{j}^{(e)}_{i,i+1},$ (4)

where $\hat{j}^{(h)}_{i,i+1}$ and $\hat{j}^{(e)}_{i+1,i}$ are the local
currents corresponding to the ‘volume’ field and the energy field,
respectively. From the explicit form the dynamics given in Eq. (1) one finds
that the local currents have the following form

$\displaystyle\begin{split}\hat{j}^{(h)}_{i,i+1}(\vec{\eta},t)&=-k_{o}(\hat{h}_{i}+\hat{h}_{i+1})+(\hat{h}_{i}-\hat{h}_{i+1})(\gamma-\xi_{i+1/2}(t))\\\
\hat{j}^{(e)}_{i,i+1}(\vec{\eta},t)&=-k_{o}^{2}(\hat{h}_{i}\hat{h}_{i+1})+(\hat{e}_{i}-\hat{e}_{i+1})(\gamma-\xi_{i+1/2}(t)),\end{split}$
(5)

with $\xi_{i+1/2}(t)=\frac{dN_{i+1/2}(t)}{dt}-\gamma$ where $N_{i+1/2}(t)$
represents the Poisson process describing the exchanges happening at the bond
$(i,i+1)$ [represented by the subscript $(i+1/2)$] with rate $\gamma$. It was
shown in [17] that, for large $N$, the evolution given in Eq. (1) makes the
system ergodic. Hence, using ensemble equivalence, one can describe the system
by the following canonical ensemble distribution

$\displaystyle P_{GE}(\\{\eta_{i}\\})=\prod_{i=1}^{N}\sqrt{\frac{k_{o}}{2\pi
T_{0}}}e^{-\frac{k_{o}}{2T_{0}}\left(\eta_{i}+\frac{\tau_{0}}{k_{o}}\right)^{2}},$
(6)

where $T_{0}$ is the temperature and $\tau_{0}$ is the ‘pressure’ of the
system. We have set the Boltzmann constant $k_{B}=1$ throughout the paper. The
quantities $T_{0}$ and $\tau_{0}$ are determined from the micro-canonical
constraints for the total energy and total volume

$\displaystyle\sum_{j=1}^{N}\langle\hat{h}_{i}(\vec{\eta})\rangle_{P_{GE}}=Nh_{0},~{}~{}\sum_{j=1}^{N}\langle\hat{e}_{i}(\vec{\eta})\rangle_{P_{GE}}=Ne_{0},$
(7)

with $h_{0}$ and $e_{0}$ being the ‘volume’ and the energy per particle,
respectively which are constants both over space and time. In this paper
$\langle....\rangle_{P}$ denotes average with respect to a joint distribution
$P$. In particular one finds the following equations for the equilibrium state

$\displaystyle h_{0}$
$\displaystyle=-\frac{\tau_{0}}{k_{o}},~{}~{}~{}~{}~{}e_{0}=\frac{T_{0}}{2}+\frac{\tau_{0}^{2}}{2k_{o}}.$
(8)

##  3 Derivation of the hydrodynamic equations

Since the dynamics of the system is ergodic (in our case evolves to a
homogeneous equilibrium state given by Eq. (6) at $t\to\infty$) and the
microscopic currents in Eqs. (5) depend only on local variables, one may
expect hydrodynamic evolutions to emerge for the locally conserved quantities,
namely the ‘volume’ density and the energy density over coarse grained length
and time scales. We assume a slowly evolving local equilibrium state for the
system which is characterized by slowly varying conserved density fields at
each time.

We start with the Fokker-Planck (FP) equation corresponding to the dynamics
given in Eq. (1) which describes the evolution of the joint probability
density $P(\vec{\eta},t)$ of $\vec{\eta}=(\eta_{1},\eta_{2},...,\eta_{N})$ at
time $t$. The FP equation is given by

$\displaystyle\partial_{t}P(\vec{\eta})=\mathcal{L}P(\vec{\eta}),~{}~{}\text{with}~{}~{}\mathcal{L}=\mathcal{L}_{\ell}+\mathcal{L}_{ex}.$
(9)

Here $\mathcal{L}_{\ell}$ is the Liouvillian part and $\mathcal{L}_{ex}$
represents the contribution from the stochastic exchange events. Explicit
expressions of these operators are given by

$\displaystyle\mathcal{L}_{\ell}P(\vec{\eta},t)$
$\displaystyle=\sum_{i=1}^{N}\left[V^{\prime}(\eta_{i-1})-V^{\prime}(\eta_{i+1})\right]\partial_{\eta_{i}}P(\vec{\eta},t),$
(10) $\displaystyle\mathcal{L}_{ex}P(\vec{\eta},t)$
$\displaystyle=\sum_{i=1}^{N}\left[P(\vec{\eta}_{i,i+1},t)-P(\vec{\eta},t)\right],$
(11)

where $\vec{\eta}_{i,i+1}$ represents the configuration after exchanging the
$\eta$ variables at sites $i$ and $i+1$ and we impose periodic boundary
condition. Note, for other boundary conditions the expressions of the FP
operators will change. For these cases we will provide the expressions of the
corresponding FP operators in the relevant sections later.

To solve the FP equation we follow a method similar to the one described in
[21]. Starting from a LE state that is slightly deviated from the GE state,
the solution of the FP equation (9) at a later time $t$ can be formally
written as sum of the local equilibrium distribution $P_{LE}(\vec{\eta})$ plus
a deviation $P_{d}(\vec{\eta},t)$ from it

$\displaystyle P(\vec{\eta},t)=P_{LE}(\vec{\eta},t)+P_{d}(\vec{\eta},t).$ (12)

The LE distribution $P_{LE}(\vec{\eta},t)$, characterised by the local
temperature field $T_{i}(t)$ and the ‘pressure’ field $\tau_{i}(t)$, is given
by

$\displaystyle P_{LE}(\vec{\eta},t)=\prod_{i=1}^{N}\sqrt{\frac{k_{o}}{2\pi
T_{i}(t)}}e^{-\frac{k_{o}}{2T_{i}(t)}\left(\eta_{i}+\frac{\tau_{i}(t)}{k_{o}}\right)^{2}},$
(13)

which implies the following local equations of state

$\displaystyle\begin{split}h_{i}&=\langle\hat{h}_{i}\rangle_{P_{LE}}=\langle\eta_{i}\rangle_{P_{LE}}=-\frac{\tau_{i}}{k_{o}},\\\
e_{i}&=\langle\hat{e}_{i}\rangle_{P_{LE}}=\langle
V(\eta_{i})\rangle_{P_{LE}}=\frac{T_{i}}{2}+\frac{\tau_{i}^{2}}{2k_{o}}.\end{split}$
(14)

The deviation $P_{d}(\vec{\eta},t)$ from the LE distribution satisfies

$\displaystyle\partial_{t}P_{d}(\vec{\eta},t)-\mathcal{L}P_{d}(\vec{\eta},t)=\mathcal{L}P_{LE}(\vec{\eta},t)-\partial_{t}P_{LE}(\vec{\eta},t).$
(15)

A formal solution of this equation is given by

$\displaystyle P_{d}(\vec{\eta},t)$
$\displaystyle=\int_{t_{0}}^{t}dt^{\prime}e^{\mathcal{L}(t-t^{\prime})}[\Phi(\vec{\eta},t^{\prime})-\Phi_{LE}(\vec{\eta},t^{\prime})]P_{LE}(\vec{\eta},t^{\prime}),$
(16)

with $P_{d}(\vec{\eta},t_{0})=0$ and

$\displaystyle\begin{split}\Phi(\vec{\eta},t)&=P_{LE}(\vec{\eta},t)^{-1}\mathcal{L}P_{LE}(\vec{\eta},t)=\sum_{i=1}^{N}\left[-\beta_{0}^{2}\nabla_{i}T_{i}\hat{\mathscr{Y}}^{(e)}_{i,i+1}+\nabla_{i}\left(\frac{\tau_{i}}{T_{i}}\right)\hat{\mathscr{Y}}^{(h)}_{i,i+1}\right],\\\
\Phi_{LE}(\vec{\eta},t)&=P_{LE}(\vec{\eta},t)^{-1}\partial_{t}P_{LE}(\vec{\eta},t)=\sum_{i=1}^{N}\left[-\beta_{0}^{2}(\partial_{t}T_{i})_{LE}~{}\hat{\mathscr{Z}}_{i}^{(e)}-\beta_{0}(\partial_{t}\tau_{i})_{LE}~{}\hat{\mathscr{Z}}_{i}^{(h)}\right],\end{split}$
(17)

where $\nabla_{i}f_{i}=f_{i+1}-f_{i}$ represents the discrete forward
difference and $\beta_{0}=1/T_{0}$ . The expressions of
$\mathscr{Y}_{i,i+1}^{(h)}$, $\mathscr{Y}_{i,i+1}^{(e)}$,
$\mathscr{Z}_{i}^{(h)}$ and $\mathscr{Z}_{i}^{(e)}$ are given explicitly as

$\displaystyle\hat{\mathscr{Y}}_{i,i+1}^{(h)}(\vec{\eta})$
$\displaystyle=-k_{o}(\hat{h}_{i}+\hat{h}_{i+1})-\gamma(\hat{h}_{i}-\hat{h}_{i+1}),$
(18) $\displaystyle\hat{\mathscr{Y}}_{i,i+1}^{(e)}(\vec{\eta})$
$\displaystyle=-k_{o}^{2}(\hat{h}_{i}\hat{h}_{i+1})-\gamma(\hat{e}_{i}-\hat{e}_{i+1}),$
(19) $\displaystyle\hat{\mathscr{Z}}_{i}^{(h)}(\vec{\eta})$
$\displaystyle=\left(\hat{h}_{i}+\frac{\tau_{i}}{k_{o}}\right),$ (20)
$\displaystyle\hat{\mathscr{Z}}_{i}^{(e)}(\vec{\eta})$
$\displaystyle=\frac{T_{0}}{2}-\frac{k_{o}}{2}\left(\hat{h}_{i}+\frac{\tau_{i}}{k_{o}}\right)^{2},$
(21)

where the functions $\hat{h}_{i}(\vec{\eta})$ and $\hat{e}_{i}(\vec{\eta})$
are provided in Eq. (2). To arrive at the expressions in Eq. (17) to Eq. (21),
we have used the explicit form of $P_{LE}(\vec{\eta},t)$ given in Eq. (13).
Moreover, while writing the explicit form of $\Phi_{LE}(\vec{\eta},t)$ we have
assumed the local deviation of the temperature and ‘pressure’ from their
global equilibrium values $T_{0}$ and $\tau_{0}$ [see Eq. (8)] given,
respectively, by $|T_{i}(t)-T_{0}|\ll T_{0}$ and
$|\tau_{i}(t)-\tau_{0}|\ll|\tau_{0}|$ are small. The subscript $(...)_{LE}$ in
the expression of $\Phi_{LE}$ indicates the rate of change of the $T_{i}(t)$
and $\tau_{i}(t)$ in local equilibrium state [shown later in Eq. (28)].

Note, the currents $\hat{\mathscr{Y}}_{i,i+1}^{(h)}$ and
$\hat{\mathscr{Y}}_{i,i+1}^{(e)}$ are generated due to spatial inhomogeneity
of the local temperature and ‘pressure’ fields in the LE state, whereas the
the quantities $\hat{\mathscr{Z}}_{i}^{(h)}$ and $\hat{\mathscr{Z}}_{i}^{(e)}$
appearing due the time variations of these local fields. We will later see
that the deviation from the LE, characterized by $P_{d}(\vec{\eta},t)$ would
incorporate the contributions from the space-time correlations of the local
currents in the system.

The ansatz for the form of the joint distribution $P(\vec{\eta},t)$ in Eq.
(12) is sensible as a solution of the FP equation (9), when the deviations
from the global equilibrium characterized by $\tilde{T}_{i}(t)=T_{i}(t)-T_{0}$
and $\tilde{\tau}_{i}(t)=\tau_{i}(t)-\tau_{0}$ are small i.e. the LE state is
slightly deviated from the GE state. In this ansatz, we have introduced the
space-time varying local temperature field $T_{i}(t)$ and the local ‘pressure’
field $\tau_{i}(t)$. Question is how should these fields evolve so that the
ansatz for the $P(\vec{\eta},t)$ in Eq. (12) holds to be a valid solution of
the FP equation (9). The space-time evolution of these fields are determined
by evaluating the continuity equations in (3) and (4) for the average ‘volume’
and energy fields. Performing average over the state $\vec{\eta}$ at time $t$
with respect to the joint distribution $P(\vec{\eta},t)$ and average over the
noises at the bonds $(i-1,i)$ and $(i,i+1)$ appearing from the exchange
events, one finds

$\displaystyle\begin{split}\partial_{t}h_{i}(t)&=j_{i-1,i}^{(h)}(t)-j_{i,i+1}^{(h)}(t),\\\
\partial_{t}e_{i}(t)&=j_{i-1,i}^{(e)}(t)-j_{i,i+1}^{(e)}(t),\end{split}$ (22)

where the average currents are

$\displaystyle\begin{split}j^{(h)}_{i,i+1}(t)&=\langle\hat{j}^{(h)}_{i,i+1}(\vec{\eta},t)\rangle_{P}\\\
j^{(e)}_{i,i+1}(t)&=\langle\hat{j}^{(e)}_{i,i+1}(\vec{\eta},t)\rangle_{P}.\end{split}$
(23)

Our aim is to express these average currents in terms of the average fields
$h_{i}(t)=\langle\hat{h}_{i}(\vec{\eta})\rangle_{P}$,
$e_{i}(t)=\langle\hat{e}_{i}(\vec{\eta})\rangle_{P}$. Assuming the
distribution $P$ in $\langle...\rangle_{P}$ is given by Eq. (12) along with
Eqs. (13) and (16), one can get evolution equations for the average conserved
fields $h_{i}(t)$ and $e_{i}(t)$,— hence for the fields $\beta_{i}(t)$ and
$\tau_{i}(t)$ through the equations of state in Eq. (14). We emphasize that we
are interested to obtain linearized hydrodynamic evolutions of these fields
valid at macroscopic scale to linear order in deviations from the GE i..e. in
the LR regime. Our strategy consists of two steps. In the first step, we find
the equations for the fields averaged only over the LE distribution in the LR
regime. In this computation we will get a linearized HD equation which are
diffusive in the form because one ignores the space-time correlations of the
currents in this step (since the LE distribution in Eq. (13) is product in
structure). Such correlations can provide extra contributions to the average
currents at linear order in deviations from the GE state. To incorporate the
effects of such correlations, we, in the second step, incorporate the
contribution from the deviation $P_{d}(\vec{\eta},t)$ from the LE
distribution.

In order to get the proper space-time continuous hydrodynamic equations one
uses the fact that the conserved fields vary slowly over space, i.e. their
values change appreciably only over large number of lattice sites and the
system size is very large i.e. $N\to\infty$. Equivalently, one can say that
the temperature and ‘pressure’ fields vary slowly over the lattice as

$\displaystyle T_{i}(t)=T(\epsilon
i,\bar{\epsilon}t)~{}\text{and}~{}\tau_{i}(t)=\tau(\epsilon
i,\bar{\epsilon}t),$ (24)

where $\epsilon^{-1}$ and $\bar{\epsilon}^{-1}$ are macroscopic space and time
scales measured in lattice (microscopic) units. In such situations one can
formally replace the fields $e_{i}(t)$ and $h_{i}(t)$ by density functions
$e(x,s)$ and $h(x,s)$, respectively where $x=i\epsilon$ and
$s=t\bar{\epsilon}$. Similarly, the differences would get replaced by
derivatives such as $\nabla_{i}f_{i}=f_{i+1}-f_{i}=\epsilon\partial_{x}f(x)$,
$\Delta_{i}f_{i}=f_{i+1}-2f_{i}+f_{i-1}=\epsilon^{2}\partial_{x}^{2}f(x)$ and
the sums by integrals such as $\sum_{i}\to\int dx$. Also time derivative gets
changed as $\partial_{t}f=\bar{\epsilon}\partial_{s}f$. We will use these
continuum limits. However, we continue with the discrete notation for now and
take the the continuum limits at appropriate stages.

Next we compute the average fields
$h_{i}(t)=\langle\hat{h}_{i}(\vec{\eta})\rangle_{P}$,
$e_{i}(t)=\langle\hat{e}_{i}(\vec{\eta})\rangle_{P}$ using the ansatz for
$P(\vec{\eta},t)$ in Eq. (12) and also evaluate the average currents
$j^{(h)}_{i,i+1}(t)$ and $j^{(e)}_{i,i+1}(t)$ following the two steps
mentioned above.

###  3.1 Contribution to the Linearized hydrodynamics from LE distribution

In order to compute the average fields
$h_{i}(t)=\langle\hat{h}_{i}(\vec{\eta})\rangle_{P}$,
$e_{i}(t)=\langle\hat{e}_{i}(\vec{\eta})\rangle_{P}$, we first make the
approximation $P(\vec{\eta},t)\approx P_{LE}(\vec{\eta},t)$ assuming the
deviations from the global equilibrium characterized by
$\tilde{T}_{i}(t)=T_{i}(t)-T_{0}$ and
$\tilde{\tau}_{i}(t)=\tau_{i}(t)-\tau_{0}$ are small. Keeping terms up to
linear order in deviations, we get [from Eq. (14)]

$\displaystyle\begin{split}h_{i}(t)&\approx
h_{0}+\tilde{h}_{i}(t),~{}~{}~{}e_{i}(t)\approx
e_{0}+\tilde{e}_{i}(t),~{}~{}~{}\text{with}\\\
\tilde{h}_{i}(t)&=-\frac{\tilde{\tau}_{i}(t)}{k_{o}},~{}~{}\text{and}~{}~{}\tilde{e}_{i}(t)=\frac{\tilde{T}_{i}(t)}{2}+\frac{\tau_{o}\tilde{\tau}_{i}(t)}{k_{o}},\end{split}$
(25)

for the average values of the conserved fields and for the corresponding
average currents, we get

$\displaystyle\begin{split}j^{(h)}_{i,i+1}(t)\big{|}_{LE}&\approx
2\tau_{0}-k_{o}(\tilde{h}_{i}+\tilde{h}_{i+1})+\gamma(\tilde{h}_{i}-\tilde{h}_{i+1})\\\
j^{(e)}_{i,i+1}(t)\big{|}_{LE}&\approx-\tau_{0}^{2}-k_{o}^{2}h_{0}(\tilde{h}_{i}+\tilde{h}_{i+1})+\gamma(\tilde{e}_{i}-\tilde{e}_{i+1}).\end{split}$
(26)

where $h_{0}$ and $\tau_{0}$ are given in Eq. (8). Inserting these equations,
on both sides of Eq. (22), and simplifying we get

$\displaystyle\begin{split}\partial_{t}\tilde{h}_{i}(t)&=k_{o}\nabla_{i}(\tilde{h}_{i}+\tilde{h}_{i-1})+\gamma\Delta_{i}\tilde{h}_{i},\\\
\partial_{t}\tilde{e}_{i}(t)&=k_{o}^{2}h_{0}\nabla_{i}(\tilde{h}_{i}+\tilde{h}_{i-1})+\gamma\Delta_{i}\tilde{e}_{i},\end{split}$
(27)

where $\Delta_{i}$ represents discrete Laplacian defined as
$\Delta_{i}f_{i}=f_{i+1}-2f_{i}+f_{i-1}$. Using the equations (25), one can
rewrite these equations in terms of $\tilde{T}_{i}(t)$ and
$\tilde{\tau}_{i}(t)$ as

$\displaystyle\begin{split}\partial_{t}\tilde{\tau}_{i}&=k_{o}\nabla_{i}(\tilde{\tau}_{i}+\tilde{\tau}_{i-1})+\gamma\Delta_{i}\tilde{\tau}_{i}\\\
\partial_{t}\tilde{T}_{i}&=\gamma\Delta_{i}\tilde{T}_{i}\end{split}$ (28)

The Eqs. (27) represent the linearized hydrodynamic equations when the space-
time correlations among the currents are ignored. These equations can be
improved by incorporating such correlations. In systems exhibiting normal
transport, such correlations decay fast enough both in space and time that
they effectively lead to (on macroscospic scale) diffusion of the locally
conserved quantities. However, such correlations in our problem, as we will
see, decay as power law (in time) in the leading order of (macroscopic) coarse
graining scale leading to anomalous transport. In the next section we will see
how these correlations appear to modify the equations (27).

###  3.2 Adding contribution to the linearized HD from the correction $P_{d}$
to $P_{LE}$

Recall that the equations (27) were obtained by computing averages of the
locally conserved quantities with respect to $P_{LE}$. In this section we
include the contribution from $P_{d}$ also to compute the average currents
appearing in Eq. (22). The average currents for the conserved fields
$u=(h,e)$, are computed as follows

$\displaystyle j^{(u)}_{i,i+1}(t)$
$\displaystyle=\langle\hat{j}^{(u)}_{i,i+1}(t)\rangle_{P=P_{LE}+P_{d}}$
$\displaystyle=\langle\hat{j}^{(u)}_{i,i+1}(t)\rangle_{P_{LE}}+\int_{t_{0}}^{t}dt^{\prime}\hat{j}^{(u)}_{i,i+1}(\eta)\left\\{e^{\mathcal{L}(t-t^{\prime})}[\Phi(\vec{\eta},t^{\prime})-\Phi_{LE}(\vec{\eta},t^{\prime})]P_{LE}(\vec{\eta},t^{\prime})\right\\},$
$\displaystyle=\langle\hat{j}^{(u)}_{i,i+1}(t)\rangle_{P_{LE}}+\int_{t_{0}}^{t}dt^{\prime}\left\langle\hat{j}^{(u)}_{i,i+1}(t)[\Phi(t^{\prime})-\Phi_{LE}(t^{\prime})]\right\rangle_{P_{LE}},$
$\displaystyle=j^{(u)}_{i,i+1}(t)\big{|}_{LE}+\int_{t_{0}}^{t}dt^{\prime}\left\langle\hat{j}^{(u)}_{i,i+1}(t)[\Phi(t^{\prime})-\Phi_{LE}(t^{\prime})]\right\rangle_{P_{GE}}+O(\tilde{T}^{2},\tilde{\tau}^{2},\tilde{T}\tilde{\tau}).$
(29)

While going from the third to fourth line we have changed the average
$\langle...\rangle_{P_{LE}}$ inside the integral to
$\langle...\rangle_{P_{GE}}$ because $[\Phi-\Phi_{LE}]$ is already in the
linear order of the deviations from the GE characterised by $\tilde{T}_{i}$
and $\tilde{\tau}_{i}$. To see this more clearly, let us write
$[\Phi-\Phi_{LE}]$ explicitly. First we recall
$T_{i}(t)=T_{0}+\tilde{T}_{i}(t)$ and
$\tau_{i}(t)=\tau_{0}+\tilde{\tau}_{i}(t)$. Using the forms of
$(\partial_{t}\tilde{T}_{i})_{LE}$ and $(\partial_{t}\tilde{\tau}_{i})_{LE}$
from Eq. (28) in Eq. (17) and performing some manipulations we get

$\displaystyle\begin{split}\Phi(\vec{\eta},t)-\Phi_{LE}(\vec{\eta},t)=&\sum_{i=1}^{N}\left[-\beta_{0}^{2}\nabla_{i}T_{i}(t)\left\\{\hat{\mathscr{Y}}^{(e)}_{i,i+1}+\tau_{0}\hat{\mathscr{Y}}^{(h)}_{i,i+1}\right\\}\right.\\\
&\left.+\beta_{0}\left\\{\nabla_{i}\tau_{i}(t)\hat{\mathscr{Y}}^{(h)}_{i,i+1}+k_{o}\nabla_{i}(\tau_{i}+\tau_{i-1})\hat{\mathscr{Z}}_{i}^{(h)}\right\\}\right]\\\
&~{}+\beta_{0}\gamma\sum_{i=1}^{N}\left[\beta_{0}\Delta_{i}T_{i}~{}\hat{\mathscr{Z}}_{i}^{(e)}+\Delta_{i}\tau_{i}~{}\hat{\mathscr{Z}}_{i}^{(h)}\right].\end{split}$
(30)

Now using the definitions of the currents $\hat{\mathscr{Y}}^{(u)}$ and the
quantities $\hat{\mathscr{Z}}^{(u)}$ from Eqs. (18) - (21) for $u=(h,e)$ we
get

$\displaystyle\begin{split}\Phi(\vec{\eta},t)-\Phi_{LE}(\vec{\eta}&,t)=-\sum_{i=1}^{N}\left[\beta_{0}^{2}\nabla_{i}T_{i}(t)~{}(k_{o}^{2}h_{0}^{2}+\hat{\mathscr{T}}^{(e)}_{i,i+1})+\beta_{0}\nabla_{i}\tau_{i}(t)(2k_{o}h_{0}+\hat{\mathscr{T}}^{(h)}_{i,i+1})\right]\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+O(\Delta_{i}T,\Delta_{i}\tau,\Delta_{i}\hat{h},\Delta_{i}\hat{e}),~{}~{}\text{where},\\\
\hat{\mathscr{T}}^{(e)}_{i,i+1}&=-k_{o}^{2}\hat{\tilde{h}}_{i}\hat{\tilde{h}}_{i+1},~{}~{}\hat{\mathscr{T}}^{(h)}_{i,i+1}=-k_{o}(\hat{\tilde{h}}_{i+1}-\hat{\tilde{h}}_{i}),~{}~{}\text{with}~{}~{}\hat{\tilde{h}}_{i}=\hat{h}_{i}-h_{0}.\end{split}$
(31)

Note $\hat{\mathscr{T}}^{(e)}_{i,i+1}$ depends non-linearly on the deviations
$\hat{\tilde{h}}_{i}$. The $\Phi_{LE}$ term cancels the non-gradient type
convective terms that depend linearly on the deviations. Hence, as will see
later [see Eq. (35)], the additional contribution to the average currents
arising from the deviation $P_{d}$ appears through the space-time correlations
of the non-linear (in deviations of the fields from GE) parts of the currents.

Further note that $\hat{\mathscr{T}}^{(h)}_{i,i+1}$ is of the form
$\nabla_{i}\hat{\tilde{h}}_{i}$ and is accompanied with $\nabla_{i}\tau_{i}$.
It will not contribute at the leading order to the average current in Eq.
(29). Similarly, other terms of the same form in Eq. (31) can also be
neglected. Moreover the, constant parts of the currents, like
$k_{o}^{2}h_{0}^{2}$ and $-2k_{o}h_{0}$ in Eq. (31) will also not survive
after averaging over the GE in Eq. (29). Hence the right hand side of the
expression for the average current $j^{(u)}_{i,i+1}(t)$ in Eq. (29) simplifies
a lot and we finally get

$\displaystyle\langle\hat{j}^{(u)}_{i,i+1}(t)\rangle_{P}$
$\displaystyle\approx\langle\hat{j}^{(u)}_{i,i+1}(t)\rangle_{P_{LE}}-\beta_{0}^{2}\int_{0}^{t-t_{0}}dt^{\prime}\sum_{\ell=1}^{N}\nabla_{\ell}T_{\ell}(t)~{}\langle\hat{j}^{(u)}_{i,i+1}(t^{\prime})\hat{\mathscr{T}}^{(e)}_{\ell,\ell+1}(0)\rangle_{P_{GE}},$
(32)

with $u=(h,e)$, where we have used time translational invariance in global
equilibrium. Since we are interested in the evolution at macroscopic time
scale, we assume $t-t_{0}$ is very large and approximate the time integral by
performing integration over $(0,\infty)$. As a consequence we get

$\displaystyle\langle\hat{j}^{(u)}_{i,i+1}(t)\rangle_{P}$
$\displaystyle\approx\langle\hat{j}^{(u)}_{i,i+1}(t)\rangle_{P_{LE}}-\beta_{0}^{2}\int_{0}^{\infty}dt^{\prime}\sum_{\ell=1}^{N}\nabla_{\ell}T_{\ell}(t)~{}\langle\hat{j}^{(u)}_{i,i+1}(t^{\prime})\hat{\mathscr{T}}^{(e)}_{\ell,\ell+1}(0)\rangle_{P_{GE}}.$
(33)

In the next section we will see that $\hat{\tilde{h}}_{i}(t)$ satisfies a
linear fluctuating equation with white Gaussian noise. Additionally, in global
equilibrium the fields $\hat{\tilde{h}}_{i}$ are independent and distributed
according to Gaussian with zero mean. Hence average over any odd power of
$\hat{\tilde{h}}_{i}$, even at different times are zero. Since
$\langle\hat{j}^{(h)}_{i,i+1}(t^{\prime})\hat{\mathscr{T}}^{(e)}_{\ell,\ell+1}(0)\rangle_{P_{GE}}$
involves odd powers of $\hat{\tilde{h}}_{i}$, as can be seen the from the
expressions of the currents in Eq. (5) and Eq. (31), it is zero. Once again
ignoring contributions involving higher order derivatives of the fields, we
get

$\displaystyle\langle\hat{j}^{(h)}_{i,i+1}(t)\rangle_{P}$
$\displaystyle=\langle\hat{j}^{(h)}_{i,i+1}(t)\rangle_{P_{LE}}+O(\tilde{u}^{2},\Delta_{i}\tilde{u})$
$\displaystyle\langle\hat{j}^{(e)}_{i,i+1}(t)\rangle_{P}$
$\displaystyle=\langle\hat{j}^{(e)}_{i,i+1}(t)\rangle_{P_{LE}}-\beta_{0}^{2}\sum_{\ell=1}^{N}\nabla_{\ell}T_{\ell}(t)~{}\mathcal{M}_{i,\ell}+O(\tilde{u}^{2},\Delta_{i}\tilde{u})$
(34)
$\displaystyle\begin{split}\text{where}~{}&~{}~{}~{}~{}~{}\mathcal{M}_{i,\ell}=\lim_{\tau\to\infty}\int_{0}^{\tau}dt^{\prime}\langle\hat{\mathscr{T}}^{(e)}_{i,i+1}(t^{\prime})\hat{\mathscr{T}}^{(e)}_{\ell,\ell+1}(0)\rangle_{P_{GE}},\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with}~{}\hat{\mathscr{T}}^{(e)}_{i,i+1}(t)=-k_{o}^{2}\hat{\tilde{h}}_{i}(t)\hat{\tilde{h}}_{i+1}(t).\end{split}$
(35)

Note $\mathcal{M}_{i,\ell}$ is similar to a transport coefficient. In order to
compute this coefficient one needs to solve for the stochastic field
$\hat{\tilde{h}}_{i}(\vec{\eta})=\hat{h}_{i}(\vec{\eta})-h_{0}$ which is
evolving according to Eq. (1). We proceed to do that in the next section.

####  3.2.1 Fluctuating hydrodynamics of the ‘volume’ field:

We first note that the coefficient $\mathcal{M}_{i,\ell}$ should be
independent of $i,\ell$ for a finite size $N$ of the system. This quantity is
$i,\ell$ dependent only when $N\to\infty$ limit is taken before the
$\tau\to\infty$ limit is taken. Hence, it is more appropriate to rewrite this
coefficient as

$\displaystyle\begin{split}\mathcal{M}_{i,\ell}&=k_{o}^{4}\lim_{\tau\to\infty}\lim_{N\to\infty}\int_{0}^{\tau}dt^{\prime}\langle\hat{\mathscr{T}}^{(e)}_{i,i+1}(t^{\prime})\hat{\mathscr{T}}^{(e)}_{\ell,\ell+1}(0)\rangle_{P_{GE}},\\\
\text{where}~{}&~{}\text{recall,}~{}~{}\hat{\mathscr{T}}^{(e)}_{i,i+1}(t)=-k_{o}^{2}\hat{\tilde{h}}_{i}(t)\hat{\tilde{h}}_{i+1}(t).\end{split}$
(36)

The NFHD theory provides a general method to compute the space-time current-
current correlation [22]. For the HCVE model it was shown in [15] that there
is a sound mode and a heat mode corresponding to the two conserved quantities.
For the particular choice of the harmonic potential
$V(\eta)=\frac{k_{o}\eta^{2}}{2}$, one finds that the sound mode satisfies a
drift-diffusion equation whereas the heat mode depends nonlinearly on the
sound mode. It was argued that the traveling peak of the space-time
correlations of sound mode satisfies diffusive scaling whereas the same for
the heat mode is described by a $\frac{3}{2}$-Lévy scaling function. Such a
scaling suggests super-diffusive contribution to the evolution from the
current-current correlation
$\langle\hat{\mathscr{T}}^{(e)}_{i,i+1}(t^{\prime})\hat{\mathscr{T}}^{(e)}_{\ell,\ell+1}(0)\rangle_{P_{GE}}$
in Eq. (36). A brief discussion on the NFHD theory for the HCVE model is
provided in A. In the next we provide a detail computation of the
$\langle\hat{\mathscr{T}}^{(e)}_{i,i+1}(t^{\prime})\hat{\mathscr{T}}^{(e)}_{\ell,\ell+1}(0)\rangle_{P_{GE}}$.

To compute $\mathcal{M}_{i,\ell}$ in $N\to\infty$ limit, one convenient way is
to start with the dynamics in Eq. (1) on infinite line. Since we are
interested on the evolution of the local fields on coarse grained length and
time scales, one may consider the following effective dynamics for
$\hat{\tilde{h}}_{i}=\hat{h}_{i}-h_{0}$ as

$\displaystyle\begin{split}\partial_{t}\hat{\tilde{h}}_{i}=&~{}k_{o}\nabla_{i}(\hat{\tilde{h}}_{i}+\hat{\tilde{h}}_{i-1})+\gamma\Delta_{i}\hat{\tilde{h}}_{i}+\nabla_{i}[\sqrt{\mathscr{B}}\xi_{i+1/2}(t)]\\\
\end{split}$ (37)

for $i=...,-2,-1,0,1,2,...$ with the boundary conditions
$\hat{\tilde{h}}_{i}(t)\to 0$ for $i\to\pm\infty$ at any $t$. The noise
$\xi_{i+1/2}(t)$ appearing from the exchange events at the $(i,i+1)$ bond is a
white Gaussian noise of zero mean and unit variance. The strength of the noise
is given by $\mathscr{B}=2\gamma\left(\frac{T_{0}}{k_{o}}+h_{0}^{2}\right)$. A
hueristic derivation of the above stochastic equation is given in B.

The formal solution of Eq. (37) can be written as

$\displaystyle\hat{\tilde{h}}_{i}(t)=\sum_{j}G_{i,j}(t)\hat{h}_{j}(0)+\sqrt{\mathscr{B}}\int_{0}^{t}dt^{\prime}\sum_{j}G_{i,j}(t-t^{\prime})\nabla_{j}\xi_{j+1/2}(t^{\prime}),$
(38)

where $G_{i,j}(t)$ is the Green’s function. Inserting this solution in Eq.
(36) and using the fact
$\langle\hat{\tilde{h}}_{i}(0)\hat{\tilde{h}}_{j}(0)\rangle_{P_{GE}}=\frac{\delta_{i,j}}{\beta_{0}k_{o}}$
[proved using Eq. (6) along with Eq. (2)], we get

$\displaystyle\mathcal{M}_{i,\ell}=\frac{k_{o}^{2}}{\beta_{0}^{2}}\int_{0}^{\infty}dt^{\prime}[G_{i,\ell}(t^{\prime})G_{i+1,\ell+1}(t^{\prime})+G_{i,\ell+1}(t^{\prime})G_{i+1,\ell}(t^{\prime})],$
(39)

where $\delta_{i,j}$ is the Kronecker delta. The Green’s function $G_{i,j}(t)$
satisfies the following equation

$\displaystyle\partial_{t}G_{i,j}=k_{o}\nabla_{i}(G_{i,j}+G_{i-1,j})+\gamma\Delta_{i}G_{i,j}+\delta_{i,j}\delta(t).$
(40)

Note that the above equation can be interpreted as the FP equation of a
drifted random walker moving on an infinite lattice with velocity
$\mu=-2k_{o}$ and diffusion constant $\mathscr{D}=\gamma$. For large $|i-j|$
and $t$, the Green’s function has the following scaling form

$\displaystyle
G_{i,j}(t)\simeq\frac{1}{\sqrt{t}}\mathcal{G}\left(\frac{i-j+2k_{o}t}{\sqrt{t}}\right),~{}~{}\text{where}~{}\mathcal{G}(z)=\frac{1}{\sqrt{4\pi\gamma}}e^{-\frac{z^{2}}{4\gamma}}.$
(41)

Using this form for the Green’s function in Eq. (39) and simplifying one gets

$\displaystyle\mathcal{M}_{i,\ell}$
$\displaystyle=\frac{k_{o}^{2}}{2\pi\gamma\beta_{0}^{2}}\int_{0}^{\infty}dt^{\prime}\frac{1}{t^{\prime}}\mathcal{G}\left(\frac{i-\ell+2k_{o}t^{\prime}}{\sqrt{t^{\prime}}}\right)\mathcal{G}\left(\frac{i-\ell+2k_{o}t^{\prime}}{\sqrt{t^{\prime}}}\right).$
(42)
$\displaystyle=\frac{k_{o}^{2}}{\pi\gamma\beta_{0}^{2}}e^{-\frac{2(i-\ell)k_{o}}{\gamma}}~{}K_{0}\left[\frac{2|i-\ell|k_{o}}{\gamma}\right],$
(43)

where $K_{0}(z)$ is modified Bessel function of second kind of zeroth order.
Inserting the expressions of the current in LE from Eq. (26) in Eq. (34), we
get the following expressions of the average currents

$\displaystyle\begin{split}j^{(h)}_{i,i+1}(t)&\approx
2\tau_{0}-k_{o}(\tilde{h}_{i}+\tilde{h}_{i+1})-\gamma\nabla_{i}\tilde{h}_{i},\\\
j^{(e)}_{i,i+1}(t)&\approx-\tau_{0}^{2}-k_{o}^{2}h_{0}(\tilde{h}_{i}+\tilde{h}_{i+1})-\gamma\nabla_{i}\tilde{e}_{i}-\beta_{0}^{2}\sum_{\ell}\mathcal{M}_{i,\ell}\nabla_{\ell}T_{\ell}(t).\end{split}$
(44)

Further inserting these expressions of the average currents in the continuity
equations (22) we get

$\displaystyle\begin{split}\partial_{t}\tilde{h}_{i}(t)&=k_{o}\nabla_{i}(\tilde{h}_{i}+\tilde{h}_{i-1})+\gamma\Delta_{i}\tilde{h}_{i},\\\
\partial_{t}\tilde{e}_{i}(t)&=k_{o}^{2}h_{0}\nabla_{i}(\tilde{h}_{i}+\tilde{h}_{i-1})+\gamma\Delta_{i}\tilde{e}_{i}+2\beta_{0}^{2}\nabla_{i}\sum_{\ell}\mathcal{M}_{i,\ell}\left(\tau_{0}\nabla_{\ell}\tilde{h}_{\ell}+\nabla_{\ell}\tilde{e}_{\ell}\right),\end{split}$
(45)

where we have used the relation between $\tilde{T}_{\ell}$ with
$\tilde{e}_{\ell}$ and $\tilde{h}_{\ell}$ from Eq. (25) and neglected terms
involving higher order in deviations. Comparing these equations with Eqs.
(27), we observe that the evolution equation for the ‘pressure’ field did not
get modified while the equation for the energy field got modified after
incorporating the contribution from the deviation $P_{d}$ from the local
equilibrium distribution [see Eq. (12)]. In terms of the local ‘pressure’
deviation field $\tilde{\tau}_{i}=\tau_{i}-\tau_{0}$ and the local temperature
deviation field $\tilde{T}_{i}=T_{i}-T_{0}$, these equations read

$\displaystyle\partial_{t}\tilde{\tau}_{i}(t)$
$\displaystyle=k_{o}\nabla_{i}(\tilde{\tau}_{i}+\tilde{\tau}_{i-1})+\gamma\Delta_{i}\tilde{\tau}_{i},$
(46) $\displaystyle\partial_{t}\tilde{T}_{i}(t)$
$\displaystyle=\gamma\Delta_{i}\tilde{T}_{i}+2\beta_{0}^{2}\nabla_{i}\sum_{\ell}\mathcal{M}_{i,\ell}\nabla_{\ell}\tilde{T}_{\ell}(t).$
(47)

where we have used the equations of state in Eq. 25.

###  3.3 Continuum limit

We take continuum limit as discussed around Eq. (24) by making the
transformation $i\to x=i\epsilon$ and $t\to s=\bar{\epsilon}t$. Let us first
focus on the ‘pressure’ field equation (46). Observe that the field
$\tilde{\tau}_{i}(t)$ has a ballistic propagation with velocity $\mu=-2k_{o}$.
This suggests us to expect the following scaling form
$\tilde{\tau}_{i}(t)\to\tilde{\mathcal{T}}((i+2k_{o}t)\epsilon,\bar{\epsilon}t)$
for the pressure field. This scaling density function evolves according to

$\displaystyle\partial_{s}\tilde{\mathcal{T}}(z,s)$
$\displaystyle=\frac{\epsilon^{2}}{\bar{\epsilon}}\gamma\partial_{x}^{2}\tilde{\mathcal{T}}(z,s),~{}~{}~{}\text{with}~{}~{}z=(i+2k_{o}t)\epsilon$
(48)

Choosing diffusive space-time scaling $\bar{\epsilon}\sim\epsilon^{2}$, we
find that the ‘pressure’ field moves ballistically and spreads diffusively at
the ballistic front.

For the temperature field, we consider the scaling form
$\tilde{T}_{i}(t)\to\tilde{T}(i\epsilon,\bar{\epsilon}t)$ and get,

$\displaystyle\partial_{s}\tilde{T}(x,s)$
$\displaystyle=\frac{\epsilon^{2}}{\bar{\epsilon}}\gamma\partial_{x}^{2}\tilde{T}(x,s)+\frac{\epsilon^{3/2}}{\bar{\epsilon}}\frac{k_{o}^{3/2}}{\sqrt{\pi\gamma}}\partial_{x}\int
dy\frac{\theta(y-x)}{\sqrt{y-x}}\partial_{y}\tilde{T}(y,s).$ (49)

To arrive at the above equation we have used the below limit

$\displaystyle\lim_{\epsilon\to
0}\epsilon^{-1/2}2\beta_{0}^{2}\mathcal{M}_{i=x/\epsilon,\ell=y/\epsilon}\rightarrow\frac{k_{o}^{3/2}}{\sqrt{\pi\gamma}}\frac{\theta(y-x)}{\sqrt{y-x}},$
(50)

where $\theta(x)$ is Heaviside theta function. From this equation one can get
different hydrodynamic evolutions, depending on the choice of the space-time
scaling for the coarse graining.

  * •

Ballistic space-time scaling i.e. $\bar{\epsilon}=\epsilon$: In this case one
finds

$\displaystyle\partial_{s}\tilde{T}(x,s)$
$\displaystyle=\epsilon{\gamma}\partial_{x}^{2}\tilde{T}(x,s)+\sqrt{\epsilon}~{}\frac{k_{o}^{3/2}}{\sqrt{\pi\gamma}}\partial_{x}\int
dy\frac{\theta(y-x)}{\sqrt{y-x}}\partial_{y}\tilde{T}(y,s)\approx 0$ (51)

which indicates that the temperature profile does not evolve.

  * •

Super-diffusive scaling i.e. $\bar{\epsilon}=\epsilon^{3/2}$: For this case
one finds

$\displaystyle\partial_{s}\tilde{T}(x,s)$
$\displaystyle=\frac{k_{o}^{3/2}}{\sqrt{\pi\gamma}}\left[\partial_{x}\int
dy\frac{\theta(y-x)}{\sqrt{y-x}}\partial_{y}\tilde{T}(y,s)+\sqrt{\frac{\epsilon}{\epsilon_{c}}}\partial_{x}^{2}\tilde{T}(x,s)\right],$
(52)
$\displaystyle~{}~{}~{}\text{with}~{}~{}~{}\epsilon_{c}={\pi}\left(\frac{k_{o}}{\gamma}\right)^{3}.$
(53)

This equation implies that the local temperature field (equivalently the
energy density field) performs super-diffusion on large length and time
scales. The diffusive correction suggests a crossover from diffusive evolution
at shorter space-time scale to super-diffusive evolution on larger space-time
scale. Such a crossover can be observed through the system size scaling of the
stationary current in the open system set-up. In the following section we
demonstrate this.

##  4 Study in open system set-up and crossover from diffusive to anomalous
transport

In this section we consider the open set-up in which we attach two Langevin
reservoirs of different temperatures $T_{L}$ and $T_{R}$ at the two ends. The
dynamics in Eq. (1) is modified to

$\displaystyle\begin{split}\dot{\eta}_{i}=&~{}V^{\prime}(\eta_{i+1})-V^{\prime}(\eta_{i-1})\\\
&+\text{stochastic~{}exchange~{}$\eta$~{}between~{}neighbouring
sites~{}at~{}rate~{}}\gamma\\\ &+\delta_{i,1}\left(-\lambda
V^{\prime}(\eta_{1})+\sqrt{2\lambda T_{L}}\zeta_{L}(t)\right)\\\
&+\delta_{i,L}\left(-\lambda V^{\prime}(\eta_{L})+\sqrt{2\lambda
T_{R}}\zeta_{R}(t)\right)\end{split}$ (54)

with fixed boundary conditions $\eta_{0}=\eta_{N+1}=0$. Here $\zeta_{L,R}(t)$
are mean zero and unit variance white Gaussian noises and $\lambda$ is the
strength of the dissipation into the bath (which we assume to be a constant of
$O(1)$.). The FP equation now reads

$\displaystyle\partial_{t}P(\vec{\eta})=\mathcal{L}P(\vec{\eta}),~{}~{}\text{with}~{}~{}\mathcal{L}=\mathcal{L}_{\ell}+\mathcal{L}_{ex}+\mathcal{L}_{b}.$
(55)

The Liouvillian part $\mathcal{L}_{\ell}$ is given in Eq. (10) and for the
stochastic exchange part $\mathcal{L}_{ex}$ in Eq. (11), the summation now
runs from $i=1$ to $(N-1)$. The boundary part $\mathcal{L}_{b}$ is given by

$\displaystyle\mathcal{L}_{b}P(\vec{\eta},t)=\lambda
T_{L}\partial_{\eta_{1}}^{2}P+\lambda\partial_{\eta_{1}}V^{\prime}(\eta_{1})P+\lambda
T_{R}\partial_{\eta_{N}}^{2}P+\lambda\partial_{\eta_{N}}V^{\prime}(\eta_{N})P.$
(56)

For $T_{L}=T_{R}=T_{0}$ the dynamics in Eq. (54) takes the system to the
global equilibrium state described by

$\displaystyle P_{GE}(\\{\eta_{i}\\})=\prod_{i=1}^{N}\sqrt{\frac{k_{o}}{2\pi
T_{0}}}e^{-\frac{k_{o}}{2T_{0}}\eta_{i}^{2}},$ (57)

which implies $e_{0}=\langle\hat{e}_{i}\rangle_{P_{GE}}=T_{0}/2$ and
$h_{0}=\langle\hat{h}_{i}\rangle_{P_{GE}}=0,~{}\forall i$.

When $T_{L}\neq T_{R}$, we write approximate solution of the FP equation as
sum of local equilibrium distribution $P_{LE}$ plus a deviation from it as in
Eqs. (12) to (16). We further assume $\delta T=T_{L}-T_{R}$ is small, hence
the local equilibrium is slightly deviated from an underlying GE described by
the distribution $P_{GE}$ in Eq. (57) with $T_{0}=\frac{T_{L}+T_{R}}{2}$. The
expression of $\Phi_{LE}(\vec{\eta},t)$ remains same as in Eq. (17). However,
the expression of $\Phi(\vec{\eta},t)$ gets slightly modified due the presence
of boundary currents from the baths and it now reads as

$\displaystyle\begin{split}\Phi(\vec{\eta},t)&=\sum_{i=1}^{N-1}\left[-\beta_{0}^{2}\nabla_{i}T_{i}\hat{\mathscr{Y}}^{(e)}_{i,i+1}+\nabla_{i}\left(\frac{\tau_{i}}{T_{i}}\right)\hat{\mathscr{Y}}^{(h)}_{i,i+1}\right]\\\
&+\lambda
k_{o}\left[\frac{\beta_{1}-\beta_{L}}{\beta_{L}}\left(k_{o}\beta_{1}\left(\hat{h}_{1}+\frac{\tau_{1}}{k_{o}}\right)^{2}-1\right)+\beta_{1}\tau_{1}\left(\hat{h}_{1}+\frac{\tau_{1}}{k_{o}}\right)\right],\\\
&+\lambda
k_{o}\left[\frac{\beta_{N}-\beta_{R}}{\beta_{R}}\left(k_{o}\beta_{N}\left(\hat{h}_{N}+\frac{\tau_{N}}{k_{o}}\right)^{2}-1\right)+\beta_{N}\tau_{N}\left(\hat{h}_{N}+\frac{\tau_{N}}{k_{o}}\right)\right],\end{split}$
(58)

where $\beta_{L}=1/T_{L}$ and $\beta_{R}=1/T_{R}$.

Following the steps as done in sections 3.1, 3.2 and 3.2.1, one arrives at the
same equations as in Eq. (45):

$\displaystyle\partial_{t}\tilde{h}_{i}(t)$
$\displaystyle=k_{o}\nabla_{i}(\tilde{h}_{i}+\tilde{h}_{i-1})+\gamma\Delta_{i}\tilde{h}_{i},$
(59) $\displaystyle\partial_{t}\tilde{e}_{i}(t)$
$\displaystyle=\gamma\Delta_{i}\tilde{e}_{i}+2\beta_{0}^{2}\nabla_{i}\sum_{\ell}\mathcal{M}_{i,\ell}~{}\nabla_{\ell}\tilde{e}_{\ell},$
(60)

with the kernel $\mathcal{M}_{i,\ell}$ given in Eq. (39) and
$\beta_{0}=T_{0}^{-1}=\frac{2}{T_{L}+T_{R}}$. Note unlike Eq. (45) there are
no terms depending on the ‘volume’ field in Eq. (60). This is because
$h_{0}=\langle\hat{h}\rangle_{GE}=0$ (hence $\tau_{0}=0$) in the GE as can be
seen from the GE distribution in Eq. (57). The boundary currents in the
expression of $\Phi(\vec{\eta},t)$ in Eq. (58) do not contribute at the
leading order in system size. The main difference with the previous case is
that the boundary conditions now are

$\displaystyle{h}_{i=0}$ $\displaystyle=0,~{}{h}_{i=N+1}=0,$ (61)
$\displaystyle{e}_{i=0}$
$\displaystyle={e}_{L}=\frac{T_{L}}{2},~{}{e}_{i=N+1}=\frac{T_{R}}{2},$ (62)

which imply $\tilde{h}_{i=0}=0,~{}\tilde{h}_{i=N+1}=0$ and
$\tilde{e}_{i=0}=\frac{T_{L}-T_{R}}{4},~{}\tilde{e}_{i=N+1}=\frac{T_{R}-T_{L}}{4}$.
To evaluate the kernel $\mathcal{M}_{i,\ell}$ given in Eq. (39), one needs to
solve the Green’s function equation (40) with absorbing boundary conditions
$G_{i,j}=0$ for $i$ or $j$ equal to $0$ and $N+1$. In the scaling limit (as in
Eq. (41), the Green’s function is given by

$\displaystyle G_{i,j}(t)=\frac{e^{-\frac{k_{o}(i-j)}{2\gamma
t}}e^{-\frac{2k_{o}^{2}t}{2\gamma}}}{\sqrt{4\pi\gamma
t}}\sum_{p=-\infty}^{\infty}\left[e^{-\frac{(i-j+2pN)^{2}}{4\gamma
t}}-e^{-\frac{(i+j+2pN)^{2}}{4\gamma t}}\right],$ (63)

using which in Eq. (39) one can show

$\displaystyle\lim_{N\to\infty}\sqrt{N}2\beta_{0}^{2}~{}\mathcal{M}_{i=xN,\ell=yN}\rightarrow\frac{k_{o}^{3/2}}{\sqrt{\pi\gamma}}\frac{\theta(y-x)}{\sqrt{y-x}}.$
(64)

Note in the continuum limit, we get the same position space representation of
the kernel as in the infinite chain case studied in the previous section,
however with different boundary conditions. Following the same method as done
in this paper, a similar kernel was obtained for the harmonic chain with
momentum exchange model for different boundary conditions [23, 8].

To take continuum limit as discussed in sec. 3.3 we once again make the
transformation $i\to x=i\epsilon$ and $t\to s=\bar{\epsilon}t$. For the
‘pressure’ field we choose $\epsilon=N^{-1}$ and $\bar{\epsilon}=N^{-2}$ since
one expects the scaling form
$\tilde{\tau}_{i}(t)\to\tilde{\mathcal{T}}((i+2k_{o}t)\epsilon,\bar{\epsilon}t)$.
Such choice provides diffusive evolution of $\tilde{\mathcal{T}}(z,s)$ as
given in Eqs. (48). For the evolution of the temperature field, we again
expect super-diffusive scaling
$\tilde{T}_{i}(t)\to\tilde{T}(i\epsilon,\bar{\epsilon}t)$ with
$\epsilon=N^{-1},~{}\bar{\epsilon}=N^{-3/2}$. We get super-diffusive evolution
as given in Eq. (52).

The boundary conditions in Eq. (61) implies that the ‘volume’ profile decays
to zero everywhere in the non-equilibrium steady state (NESS). On the other
hand, the temperature profile $T_{ss}(x)=T_{0}+\delta T\Psi(x)$ in the steady
state satisfies

$\displaystyle\frac{k_{o}^{3/2}}{\sqrt{\pi\gamma}}\partial_{x}\int_{0}^{1}dy\frac{\theta(y-x)}{\sqrt{y-x}}\partial_{y}\Psi(y)+\sqrt{\frac{1}{N}}\gamma\partial_{x}^{2}\Psi(x)=0,~{}~{}\text{for}~{}~{}0\leq
x\leq 1,$ (65)

with $\Psi(0)=1/2$ and $\Psi(1)=-1/2$. The energy current in the steady state
can be read off from Eq. (44) with $h_{0}=0$ and is given by
$J_{ss}=-\gamma\nabla_{i}{e}_{i}+2\beta_{0}^{2}\sum_{\ell}\mathcal{M}_{i,\ell}\nabla_{\ell}e_{\ell}$
which in the continuum limit can be expressed in terms of the temperature
profile using $2e(x)=T_{ss}(x)=T_{0}+\delta T\Psi(x)$. It reads as

$\displaystyle\begin{split}J_{ss}&=-\frac{1}{N}\frac{\gamma}{2}\partial_{x}\Psi_{ss}(x)-\frac{1}{\sqrt{N}}\frac{k_{o}^{3/2}}{2\sqrt{\pi\gamma}}\int_{0}^{1}dy\frac{\theta(y-x)}{\sqrt{y-x}}\partial_{y}\Psi_{ss}(y).\end{split}$
(66)

The equation (65) ensures that the stationary current $J_{ss}$ is $x$
independent. Hence integrating both sides of Eq. (67) with respect to $x$ over
$[0,1]$, one gets

$\displaystyle\begin{split}J_{ss}&=\frac{\delta
T}{2}\left[\frac{1}{\sqrt{N}}\frac{k_{o}^{3/2}C}{\sqrt{\pi\gamma}}+\frac{1}{N}\gamma\right],~{}~{}\text{with}~{}~{}C=\int_{0}^{1}dx\int_{0}^{1}dy\frac{\theta(y-x)}{\sqrt{y-x}}\partial_{y}\Psi(y).\end{split}$
(67)

At this stage, few comments are in order.

  * –

Anomalous scaling: From Eq. (67) we see that the stationary current decays
anomalously with as $\sim\frac{1}{\sqrt{N}}$ at the leading order in $N$. This
anomalous scaling was obtained previously in [17] and [18] using methods
different from the one presented here. The $O(1/N)$ term provides a diffusive
correction to the anomalous scaling.

  * –

Non-local Fourier’s law: The expression of the current in Eq. (67) is a non-
local linear response relation, which is drastically different from the usual
Fourier’s law. In local Fourier’s law, the current at any point $x$ in the
system is directly proportional to the local derivative of the temperature
profile. On the other hand, in the non-local version, the current at any point
$x$ gets contribution from the derivative of the temperature profile at other
points also. Such non-local generalisation of Fourier’s law was also obtained
in few other systems [7, 24, 9, 8, 25, 23]. The non-local generalization
implies non-local generalization of the heat diffusion equation as we have
obtained in Eq. (52). Such a generalization of the heat diffusion equation was
obtained for the HCVE model in [18] by computing the microscopic two-point
correlations $\langle\eta_{i}\eta_{j}\rangle_{P_{ss}}$ in the steady state
described the steady state distribution $P_{ss}(\vec{\eta})$. In this paper we
have re-derived the same equation using a different method along with a
diffusive correction.

  * –

Nonlinear temperature profile $\Psi(x)$: Neglecting the diffusive part one can
solve the non-local part of Eq. (65) and the solution is given by
$\Psi(x)=\sqrt{1-x}-1/2$ [18]. Using this solution in Eq. (67) one finds
$C=\frac{\pi}{2}$. Unlike the (purely) diffusive case, the temperature profile
$\Psi(x)$ is non-linear and singular (has diverging derivative at the right
boundary). Similar non-linear and singular temperature profiles were also
obtained in different contexts, such as in momentum exchange model [23, 24,
8], in hard-point gas [26, 27], in non-linear chains [28], in graphene layers
[29, 30] and in nanotubes [31].

Figure 2: Plot of $J_{ss}$ vs. $N$ for $\gamma=1$ and different values of
$k_{o}$ i.e. for different values of $N_{c}$ as given by Eq. (68). The symbols
are obtained by solving the equations of the two-point correlations
$\langle\eta_{i}\eta_{j}\rangle_{P_{ss}}$ in NESS numerically for different
$N$. In (a) we observe diffusive scaling as one has $N\ll N_{c}=159155$. In
(b), the value of $N_{c}=218$. We observe a change in the exponent of the
system size scaling from $0.75$ to $0.62$. In principle, one should have a
crossover from diffusive (exponent $1$) to anomalous (exponent $0.5$)
behaviour. For that one needs to have a sufficiently large $N_{c}$ so that one
has sufficiently large $N$ even for $N<N_{c}$ to observe the true diffusive
scaling and then one should be able to evaluate stationary currents $J_{ss}$
for $N$ much large than $N_{c}$ to observe the true anomalous scaling.
Numerically this is very hard to achieve. Instead, in (c) we choose
$k_{o}=1.0$ so that one observes only the anomalous scaling because $N_{c}=1$.

  * –

Diffusive to anomalous crossover: Putting the value $C=\pi/2$ in the
expression of $J_{ss}$ in Eq. (67) one rewrites

$\displaystyle\begin{split}J_{ss}&=\frac{T_{L}-T_{R}}{4}\frac{1}{\sqrt{N}}\frac{k_{o}^{3/2}\sqrt{\pi}}{\sqrt{2\gamma}}\left[1+\sqrt{\frac{N_{c}}{N}}~{}\right],~{}~{}\text{with}~{}~{}N_{c}={\frac{4}{\pi}}\left(\frac{\gamma}{k_{o}}\right)^{3}.\end{split}$
(68)

For fixed $k_{o}$ and $\gamma$ this expression suggests a crossover from
diffusive scaling $\sim\frac{1}{N}$ for $N\ll N_{c}$ to anomalous scaling
$\sim\frac{1}{\sqrt{N}}$ for $N\gg N_{c}$ as the system size $N$ is increased.
We have numerically verified this crossover in fig. 2 where we plot $J_{ss}$
versus $N$ for three choices of the parameters $k_{o}$ and $\gamma$ such that
we have three scenarios of $N_{c}$ being very small, very large and
intermediate. For very large $N_{c}$ we observe only diffusive scaling in fig.
2(a) within the system sizes that were possible to study numerically. Whereas
in fig. 2(b) we observe purely anomalous scaling $\sim\frac{1}{\sqrt{N}}$
because $N_{c}$ is very small. In fig. 2(c), we observe a sort of crossover as
manifested by the change in the exponent of the system-size scaling of
$J_{ss}$ though not from pure diffusive scaling exponent to the correct
anomalous scaling exponent. For that one requires to compute $J_{ss}$ for very
large $N$ along with large $N_{c}$.

  * –

Anomalous to diffusive crossover: We end this section by making the following
remark: If one considers $k_{o}$ to be system size dependent as
$k_{o}=N^{-\alpha}$, then for $0<\alpha<\frac{1}{3}$, one finds anomalous
scaling for the stationary current $J_{ss}\sim\frac{1}{N^{(1+3\alpha)/2}}$ and
for $\alpha\geq\frac{1}{3}$ the transport becomes diffusive with
$J_{ss}\sim\frac{1}{N}$. This crossover by tuning the strength $k_{o}$ of the
interaction was predicted previously in [32].

##  5 Conclusion

In this paper we have derived macroscopic linearized hydrodynamics for the two
conserved quantities present in the HCVE model. Assuming a slowly varying
(both in space and time) LE state that is slightly deviated from a underlying
GE state, we study the evolution of the average conserved field densities.
This is achieved by asking what equations the temperature and the ‘pressure’
fields (characterizing the LE state) should satisfy to linear order in
deviations from their GE values. Approximating the solution of the FP equation
by the LE distribution yields linear diffusive hydrodynamics in which one
neglects the space-time correlations of the currents corresponding to the
conserved fields. In order to include the contributions from such correlations
in the linear response regime, we estimate the correction to the local
equilibrium distribution in the solution of the relevant FP equation. Such
correction naturally produce contributions to the average currents as space-
time integrals of the certain current-current correlations. Interestingly, we
find that these current-current correlations involves mainly the non-linear
parts of the currents when written in terms of the deviations of the conserved
fields from their GE values. To compute such correlations, we invoke
fluctuating hydrodynamics equations for the ‘volume’ field written at a
mesoscopic scale. We finally obtain drift-diffusion equation for the ‘volume’
field and super-diffusion equation for the non-convective part of the energy
field to linear order in deviations i.e. in the LR regime.

Our calculation, in addition, also provides the diffusive correction to the
super-diffusion equation which allows us to study a crossover from diffusive
to super-diffusive transport. In particular, our analysis allows us to
identify a length scale $N_{c}$ which depends on the microscopic parameters.
Below this length scale, one would observe a diffusive transport and above
this length scale the super-diffusive transport sets in. The physical picture
is the following: as smaller length scale the conserved quantities dissipate
through diffusion. However at larger length scales the correlations among the
hydrodynamic currents starts providing dominant channels for transport and as
a result one observes a crossover from a diffusive transport to anomalous
transport. We have demonstrated this crossover through the system size scaling
of the NESS current in the system when connected to two reservoirs of
different temperatures at the two ends. Since the HCVE model dynamics is
linear (due to harmonic potential), we believe the linearized macroscopic
hydrodynamics is exact in the sense there will be no non-linear corrections.
However, one can expect to get non-linear corrections both local and non-local
in other models such as Fermi-Past-Ulam-Tsingou model. It would be interesting
to see how super-diffusion and higher order corrections appear following the
formalism presented in this paper. Applying this formalism to non-linear
hamiltonian models in open set-up requires to solve NFHD equations in bounded
domain with appropriate boundary conditions which, to our knowledge, are not
known. It would be interesting to investigate such cases. Often an interesting
problem that people consider in systems permitting hydrodynamics description
is to observe the evolution from a domain wall initial condition. In systems
exhibiting anomalous transport, one should solve the super-diffusion equation
for such problems. We believe our result will be useful in such contexts.

##  6 Acknowledgement

The author would like thank Abhishek Dhar for useful comments on the
manuscript. He also acknowledges the support of the core research grant no.
CRG/2021/002455 and MATRICS grant MTR/2021/000350 from the Science and
Engineering Research Board (SERB), Department of Science and Technology,
Government of India. A.K. also acknowledges support from the Department of
Atomic Energy, Government of India, under project no. 19P1112R&D.

##  Appendix A NFHD for the HCVE model

In this appendix we discuss the NFHD theory for the HCVE model as given in
[15]. We start with the conservation equations (22), which in the hydrodynamic
limit can be written as

$\displaystyle\begin{split}\partial_{t}h_{i}(t)&=-2k_{o}\nabla_{i}h_{i}(t)+\gamma\Delta_{i}^{2}h_{i}(t),\\\
\partial_{t}e_{i}(t)&=-k_{o}^{2}\nabla_{i}h_{i}(t)^{2}+\gamma\Delta_{i}^{2}e_{i}(t).\end{split}$
(69)

Writing $h_{i}(t)=h_{0}+\tilde{h}_{i}(t)$ and
$e_{i}(t)=e_{0}+\tilde{e}_{i}(t)$ and expanding the currents up to second
order in the deviations $\tilde{h}_{i}$ and $\tilde{e}_{i}$ from the global
equilibrium values, one gets

$\displaystyle\partial_{t}\tilde{\bm{u}}_{i}+\nabla_{i}\left[A\tilde{\bm{u}}_{i}+\frac{1}{2}\left(\begin{array}[]{c}\tilde{\bm{u}}_{i}^{T}~{}H^{(h)}~{}\tilde{\bm{u}}_{i}\\\
\tilde{\bm{u}}_{i}^{T}H^{(e)}\tilde{\bm{u}}_{i}\end{array}\right)\right]=0,~{}~{}\text{with,}~{}\tilde{\bm{u}}_{i}=\left(\begin{array}[]{c}\tilde{h}_{i}\\\
\tilde{e}_{i}\end{array}\right)$ (74)

where for the Harmonic potential $V(\eta)=k_{o}\eta^{2}/2$,

$\displaystyle A$ $\displaystyle=\left(\begin{array}[]{cc}-2k_{o}&0\\\
-k_{o}^{2}h_{0}&0\end{array}\right),~{}~{}H^{(h)}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),~{}~{}\text{and}~{}~{}H^{(e)}=\left(\begin{array}[]{cc}-2k_{o}^{2}&0\\\
0&0\end{array}\right).$ (81)

Adding diffusion and noise terms, one gets the non-linear fluctuating
hydrodynamic equations

$\displaystyle\partial_{t}\hat{\tilde{\bm{u}}}_{i}+$
$\displaystyle\nabla_{i}\left(A\hat{\tilde{\bm{u}}}_{i}+\frac{1}{2}\left(\begin{array}[]{c}~{}\hat{\tilde{\bm{u}}}_{i}^{T}H^{(h)}~{}\hat{\tilde{\bm{u}}}_{i}\\\
\hat{\tilde{\bm{u}}}_{i}^{T}~{}H^{(e)}~{}\hat{\tilde{\bm{u}}}_{i}\end{array}\right)-\nabla_{i}\tilde{D}\hat{\tilde{\bm{u}}}_{i}+\tilde{B}\tilde{\bm{\xi}}_{i}\right)=0,$
(84)

where $\tilde{D}=\tilde{D}^{T}>0$ is the diffusion matrix, and
$\xi_{i}^{(\alpha)}(t)$ for $\alpha=1,2$ are white Gaussian noises with zero
mean and covariance

$\displaystyle\
\langle\xi_{i}^{(\alpha)}(t)\xi_{i^{\prime}}^{(\alpha^{\prime})}(t^{\prime})\rangle=\delta_{\alpha\alpha^{\prime}}\delta_{i,i^{\prime}}\delta(t-t^{\prime}).$
(85)

The strength of the noise $\tilde{B}\tilde{B}^{T}$ is related to the diffusion
matrix as $\tilde{D}\tilde{C}+\tilde{C}\tilde{D}=\tilde{B}\tilde{B}^{T}$,
where $\tilde{C}$ is the susceptibility matrix given by

$\displaystyle\langle\hat{\tilde{u}}_{\alpha}(i,t)\hat{\tilde{u}}_{\alpha^{\prime}}(i^{\prime},t)\rangle_{P_{GE}}=\tilde{C}_{\alpha,\alpha^{\prime}}\delta_{i,i^{\prime}}.$
(86)

Following [4, 5, 15], one next decomposes the fields $\tilde{\bm{u}}$ into
normal modes ${\bm{\phi}}$ using the transformation

$\displaystyle\hat{{\bm{\phi}}}=R\hat{\tilde{\bm{u}}},$ (87)

where the matrix $R$ has the properties

$\displaystyle RAR^{-1}=\left(\begin{array}[]{cc}-2k_{o}&0\\\
0&0\end{array}\right),~{}~{}\text{and}~{}~{}RCR^{T}=1.$ (90)

For our case $R$ is explicitly given by

$\displaystyle R=\left(\begin{array}[]{cc}-\sqrt{k_{o}\beta_{0}}&0\\\
\sqrt{2}\beta_{0}\tau_{0}&\sqrt{2}\beta_{0}\end{array}\right),$ (93)

which implies

$\displaystyle\hat{{\bm{\phi}}}_{i}=R\hat{\tilde{\bm{u}}}_{i}=\left(\begin{array}[]{c}-\sqrt{k_{o}\beta_{0}}~{}\hat{\tilde{h}}_{i}\\\
\sqrt{2}\beta_{0}~{}(\hat{\tilde{e}}_{i}+\tau_{0}\hat{\tilde{h}}_{i})\end{array}\right)=\left(\begin{array}[]{c}-\sqrt{k_{o}\beta_{0}}~{}\hat{\tilde{h}}_{i}\\\
\sqrt{2}\beta_{0}\hat{\tilde{\theta}}_{i}\end{array}\right),$ (98)

where we have written
$\hat{\tilde{\theta}}_{i}=(\hat{\tilde{e}}_{i}+\tau_{0}\hat{\tilde{h}}_{i})$.
Note, $\langle\hat{\tilde{\theta}}_{i}\rangle_{P_{LE}}=\tilde{T}_{i}/2$
according to Eq. (25). The HD equations (84) can now be written in terms of
$\phi^{(1)}_{i}(t)$ and $\phi^{(2)}_{i}(t)$ as

$\displaystyle\partial_{t}\hat{\phi}^{(1)}_{i}(t)$
$\displaystyle+\nabla_{i}\left(-2k_{0}\hat{\phi}^{(1)}_{i}(t)-\nabla_{i}(D_{11}\hat{\phi}^{(1)}_{i}(t)+D_{12}\hat{\phi}^{(2)}_{i}(t))+(B{\bm{\xi}})^{(1)}\right)=0,$
(99) $\displaystyle\partial_{t}\hat{\phi}^{(2)}_{i}(t)$
$\displaystyle+\nabla_{i}\left(-\sqrt{2}k_{0}\hat{\phi}^{(1)}_{i}(t)^{2}-\nabla_{i}(D_{21}\hat{\phi}^{(1)}_{i}(t)+D_{22}\hat{\phi}^{(2)}_{i}(t))+(B{\bm{\xi}})^{(2)}\right)=0,$
(100)

where $D=R\tilde{D}R^{-1}>0$ and $B=R\tilde{B}$. The matrices $D$ and $B$ now
satisfies

$\displaystyle D+D^{T}=BB^{T}.$ (101)

Using Eq. (98), one can express these equations in terms of $\hat{\tilde{h}}$
and $\hat{\tilde{\theta}}$ as

$\displaystyle\partial_{t}\hat{\tilde{h}}_{i}(t)$
$\displaystyle+\nabla_{i}\left[-2k_{o}\hat{\tilde{h}}_{i}(t)-\nabla_{i}\left(D_{11}\hat{\tilde{h}}_{i}(t)-\frac{\sqrt{2\beta_{0}}D_{12}}{\sqrt{k_{o}}}\hat{\tilde{\theta}}_{i}(t)\right)-\frac{(B{\bm{\xi}})^{(1)}}{\sqrt{k_{o}\beta_{0}}}\right]=0,$
(102) $\displaystyle\partial_{t}\hat{\tilde{\theta}}_{i}$
$\displaystyle+\nabla_{i}\left[-k_{o}^{2}\hat{\tilde{h}}_{i}(t)^{2}-\nabla_{i}\left(-\frac{D_{21}}{\sqrt{2k_{o}}\beta_{0}}\hat{\tilde{h}}_{i}(t)+D_{22}\hat{\tilde{\theta}}_{i}(t)\right)+\frac{(B{\bm{\xi}})^{(2)}}{\sqrt{2}\beta_{0}}\right]=0,$
(103)

Note in order for the above equations to be stable one is required to choose
$D_{12}=D_{21}=0$. Hence, the matrix $D$ is a diagonal matrix and consequently
by Eq. (101) the matrix $B$ is also diagonal. The fluctuating hydrodynamic
equations now look like

$\displaystyle\partial_{t}\hat{\tilde{h}}_{i}(t)+\nabla_{i}\left[-2k_{o}\hat{\tilde{h}}_{i}(t)-\nabla_{i}\left(D_{11}\hat{\tilde{h}}_{i}(t)\right)+\frac{B_{11}{\xi}^{(1)}}{\sqrt{k_{o}\beta_{0}}}\right]=0,$
(104)
$\displaystyle\partial_{t}\hat{\tilde{\theta}}_{i}+\nabla_{i}\left[-k_{o}^{2}\hat{\tilde{h}}_{i}(t)^{2}-\nabla_{i}\left(D_{22}\hat{\tilde{\theta}}_{i}(t)\right)+\frac{B_{22}{\xi}^{(2)}}{\sqrt{2}\beta_{0}}\right]=0,$
(105)

Note Eq. (104) is of the same form at the Eq. (37), except the diffusion and
the noise terms are introduced phenomenologically and not explicitly known. In
the next section, we provide a heuristic derivation of Eq. (37) with explicit
dissipation and noise terms.

One of the main prediction of the NFHD theory is the space-time dependence of
the correlations

$\displaystyle
S_{\alpha,\alpha^{\prime}}(i,t)=\langle\hat{g}_{\alpha}(i,t)\hat{g}_{\alpha^{\prime}}(0,0)\rangle-\langle\hat{g}_{\alpha}(i,t)\rangle\langle\hat{g}_{\alpha^{\prime}}(0,0)\rangle$
(106)

where $\hat{g}_{1}(i,t)=\hat{h}_{i}(\vec{\eta}(t))$ and
$\hat{g}_{2}(i,t)=\hat{e}_{i}(\vec{\eta}(t))$. The space-time correlation of
the normal mode fields can be obtained by transforming the matrix $S(i,t)$ as
$S^{({\bm{\phi}})}(i,t)=RS(i,t)R^{T}$. Since the mode $\phi^{(1)}_{i}(t)$ gets
linearly separated from the non-moving mode $\phi^{(1)}_{i}(t)$ with time, on
sufficiently large space-time scales, the matrix $S^{({\bm{\phi}})}(i,t)$ is
approximately diagonal i..e.
$S^{({\bm{\phi}})}_{\alpha,\alpha^{\prime}}(i,t)\simeq\delta_{\alpha,\alpha^{\prime}}\mathscr{F}_{\alpha}(i/N,t)$.
In [15] it was argued that for $V(\eta)=k_{o}\eta^{2}/2$, the sound peak
$\mathscr{F}_{1}(i/N,t)$ asymptotically possess diffusive scaling and is
described by a Gaussian whereas the heat peak possess anomalous scaling and is
described by a $\frac{3}{2}-$Lévy function. For other choices of potentials
one may get KPZ scaling for the sound mode and $\frac{5}{3}-$Lévy heat mode. A
complete classification of the scaling behaviors of the modes for general
potential $V(\eta)$ is given in [15].

##  Appendix B Heuristic derivation of Eq. (37)

From Eq. (3) and Eq. (5) we rewrite the equation for the ‘volume’ field
explicitly as

$\displaystyle\partial_{t}\hat{h}_{i}$
$\displaystyle=k_{o}\nabla_{i}(\hat{h}_{i}+\tilde{h}_{i-1})+\gamma\Delta_{i}\hat{h}_{i}+\nabla_{i}\left(\frac{d\mathcal{Z}_{i+1/2}}{dt}\right),$
(107) with
$\displaystyle~{}\mathcal{Z}_{i+1/2}(t)=\int_{0}^{t}dt^{\prime}~{}[\hat{h}_{i+1}(t^{\prime})-\hat{h}_{i}(t^{\prime})]~{}\left(\frac{dN_{i+1/2}(t^{\prime})}{dt^{\prime}}-\gamma\right),$
(108)

where $N_{i+1/2}(t)$ represents the Poisson process describing the exchanges
that are happening at the bond $(i,i+1)$ with rate $\gamma$. More precisely,
$N_{i+1/2}$ counts the number of exchange events happened till time $t$ on the
bond $(i,i+1)$.

Since we are interested in a slowly evolving LE picture in which the conserved
fields are slowly varying both over space and time, one can imagine
appreciable evolution to occur over time scale that is much larger than the
microscopic time scales of the system. Over $\delta t$ time, there are many
independent exchange events whose cumulative effect can be obtained following
the idea of central limit theorem. Hence we can write

$\displaystyle\delta\mathcal{Z}_{i+1/2}$
$\displaystyle=\mathcal{Z}_{i+1/2}(t+\delta t)-\mathcal{Z}_{i+1/2}(t),$ (109)
$\displaystyle=\sum_{k=N_{i+1/2}(t)}^{N_{i+1/2}(t+\delta
t)}[\hat{h}_{i+1}(t_{k})-\hat{h}_{i}(t_{k})]-\gamma\int_{t}^{t+\delta
t}dt^{\prime}[\hat{h}_{i+1}(t^{\prime})-\hat{h}_{i}(t^{\prime})]$ (110)

where the set of times $\\{t_{k}\\}$ represent the times at which exchanges
have occurred in the time interval $t$ to $t+\delta t$. Assuming the field
$\hat{h}_{i}(t)$ is changing slowly, for small $\delta t$ one can approximate
the above expression as

$\displaystyle\delta\mathcal{Z}_{i+1/2}$
$\displaystyle\simeq[\hat{h}_{i+1}(t)-\hat{h}_{i}(t)]~{}\left(N_{i+1/2}(\delta
t)-\gamma\delta t\right),$ (111)

It can be shown that

$\displaystyle\langle\delta\mathcal{Z}_{i+1/2}\rangle$ $\displaystyle=0$ (112)
$\displaystyle\langle\delta\mathcal{Z}_{i+1/2}^{2}\rangle_{c}$
$\displaystyle=\gamma\delta
t\langle(\hat{h}_{i+1}(t)-\hat{h}_{i}(t))^{2}\rangle_{c,P},$ (113)
$\displaystyle\simeq\gamma\delta
t\langle(\hat{h}_{i+1}(t)-\hat{h}_{i}(t))^{2}\rangle_{c,P_{GE}}+O(\nabla_{i}\tilde{u}_{i}),$
(114) $\displaystyle=2\gamma\delta
t\left(\frac{T_{0}}{k_{o}}+h_{0}^{2}\right)+O(\nabla_{i}\tilde{u}_{i}),$ (115)
$\displaystyle\langle\delta\mathcal{Z}_{i+1/2}^{m}\rangle_{c}$
$\displaystyle\simeq O(\nabla_{i}\tilde{u}_{i}),~{}~{}\text{for}~{}~{}m>2,$
(116)

where $\langle\hat{o}\rangle_{c,P}$ represents cumulants of $\hat{o}$
evaluated with respect to the distribution $P$. Hence, for $\delta t\to 0$,
one can write
$\frac{d\mathcal{Z}_{i+1/2}}{dt}=\sqrt{\mathscr{B}}~{}\xi_{i+1/2}(t)$ with
$\mathscr{B}=2\gamma\left(\frac{T_{0}}{k_{o}}+h_{0}^{2}\right)$ at the leading
order in deviations. Using this in Eq. (107) gives rise to Eq. (37).

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