# Effect of relative timescale on a system of particles sliding on a
fluctuating energy landscape: Exact derivation of product measure condition

Chandradip Khamrai and Sakuntala Chatterjee  Department of Physics of Complex
Systems, S. N. Bose National Centre for Basic Sciences, Block JD, Sector 3,
Salt Lake, Kolkata 700106, India.

###### Abstract

We consider a system of hardcore particles advected by a fluctuating potential
energy landscape, whose dynamics is in turn affected by the particles. Earlier
studies have shown that as a result of two-way coupling between the landscape
and the particles, the system shows an interesting phase diagram as the
coupling parameters are varied. The phase diagram consists of various
different kinds of ordered phases and a disordered phase. We introduce a
relative timescale $\omega$ between the particle and landscape dynamics, and
study its effect on the steady state properties. We find there exists a
critical value $\omega=\omega_{c}$ when all configurations of the system are
equally likely in the steady state. We prove this result exactly in a discrete
lattice system and obtain an exact expression for $\omega_{c}$ in terms of the
coupling parameters of the system. We show that $\omega_{c}$ is finite in the
disordered phase, diverges at the boundary between the ordered and disordered
phase, and is undefined in the ordered phase. We also derive $\omega_{c}$ from
a coarse-grained level description of the system using linear hydrodynamics.
We start with the assumption that there is a specific value $\omega^{\ast}$ of
the relative timescale when correlations in the system vanish, and mean-field
theory gives exact expressions for the current Jacobian matrix $A$ and
compressibility matrix $K$. Our exact calculations show that Onsager-type
current symmetry relation $AK=KA^{T}$ can be satisfied if and only if
$\omega^{\ast}=\omega_{c}$ . Our coarse-grained model calculations can be
easily generalized to other coupled systems.

##  I Introduction

Models of coupled driven systems are often used to study several complex
phenomena in physics and biology [1, 2, 3]. Examples include particles
advected by turbulent fluid [4, 5, 6, 7], shape deformation of cell membrane
by curvature sensitive membrane proteins [8, 9, 10, 11, 12, 13, 14],
sedimenting colloidal crystal through a dissipative medium [2, 15], etc. Also,
recently therehas been a surge of interest in coupled nonequilibrium systems
with multiple conserved quantities [16, 17, 18, 19] since it was shown using
the tools of nonlinear fluctuating hydrodynamics [20, 21, 22, 16] that such
systems often show unusual dynamical universality classes, apart from the
well-known diffusive and Kardar-Parisi-Zhang (KPZ) class. In particular, it
was shown that the dynamical exponent which characterizes the universality
class can be expressed as the Kepler ratio of successive numbers of the
Fibonacci sequence [16]. Although most of these models follow rather simple
dynamical rules, predicting the state of the system in the long-time limit is
often a challenging task.

In this paper, we consider a lattice model of “light” and “heavy” particles
moving on a fluctuating energy landscape in one dimension. The heavy (light)
particles prefer to move towards the local energy minima (maxima), and the
shape of the local landscape gets modified by the presence of these heavy and
light particles. This model is known as the light-heavy model or LH model, and
was introduced in [23]. It was shown that by varying the nature of coupling
between the particle motion and the landscape dynamics, a rich phase diagram
can be obtained, consisting of various kinds of ordered phases and a
disordered phase [23]. The static and dynamic properties of these phases were
characterized in [24, 25, 26]. While the ordered phases show long-range order
among the particles and in the landscape, in the disordered phase only short-
range correlations exist in the system. However, the exact form of these
short-range correlations is not known in general. Often these correlations are
quite strong, because of which simple analytical approximations such as mean-
field theory fail. In the absence of any exact knowledge of the nonequilibrium
steady state, and breaking down of mean-field approximations, it becomes
almost impossible to perform any exact, or even approximate analytical
calculation in the presence of these short-range correlations.

In an attempt to understand these short-ranged correlations better, in this
work we introduce a relative time-scale in the coupled dynamics of particle
slide and landscape movement. This means compared to the landscape movement,
the particle movement now happens faster, or slower. If the particles move
much faster than the landscape, then the particles get enough time to find
their preferred locations on the landscape. This is not possible in the
opposite limit when the landscape movement happens faster. Therefore, the
choice of the relative time-scale has a strong effect on the nature of the
steady state [27, 28, 29, 30, 31] and hence on the short-ranged correlations
described above. We investigate this effect in this work, using exact
calculations and Monte Carlo simulations.

In our Monte Carlo simulations, performed on a discrete lattice model, we
include the relative timescale in the following way. We define a parameter
$\omega$ which denotes the average number of times the landscape moves in
between two particle movements. $\omega=1$ corresponds to the case when one
particle movement is followed by one landscape movement, on average. In all
earlier studies of the LH model, $\omega=1$ was considered [23, 24, 25, 26].
Here, we consider any arbitrary $\omega$ within the $0<\omega<\infty$ range.
For $\omega\ll 1$, we have a landscape that moves much slower compared to the
particles, and for $\omega\gg 1$, the landscape moves more rapidly compared to
particle movement. We numerically measure various different short-range
correlations in the system as a function of $\omega$. Surprisingly, we find a
critical value $\omega=\omega_{c}$ at which all correlations in the system
vanish, and the steady state is simply given by equiprobable measure. We
explain this surprising result from exact calculation generalizing a formalism
introduced by Mahapatra et al. for the lattice model [32]. We show that for
every discrete configuration, it is possible to identify a bunch of incoming
and outgoing transitions whose rates exactly balance each other for
$\omega=\omega_{c}$ . This condition, “bunchwise balance” [32], ensures that
in the steady state, all configurations occur with equal probability. We
derive an exact expression for $\omega_{c}$ in terms of the coupling
parameters between the particle and landscape dynamics.

To gain further insights into the above striking result, we consider the
coarse-grained level description of the discrete LH model using hydrodynamics
and examine the implications of the relative timescale there. In particular,
for the LH model, the dynamical rules are such that the local particle density
and local height gradient of the landscape remain conserved. Therefore, one
can start from a set of two coupled continuity equations for these two
conserved quantities, and within linear hydrodynamic approximation, one can
solve these equations by diagonalizing the current Jacobian matrix $A$ and
constructing the normal modes [33]. The stationary fluctuations of the
conserved variables are captured in the covariance matrix or compressibility
matrix $K$. For a wide class of coupled driven systems with multiple conserved
quantities, it was shown in [34] that the product $AK$ of the Jacobian and
compressibility matrix must be symmetric, which is the nonequilibrium analog
of Onsager’s reciprocity relation. However, to explicitly construct the matrix
elements of $A$ and $K$, an exact knowledge of the stationary measure is
required. We make an assumption that there exists a special value of the
relative timescale $\omega=\omega^{\ast}$ where the steady state of the system
satisfies product measure and all correlations vanish. Under this assumption,
we can explicitly construct the matrices $A$ and $K$. Our calculations show
that the symmetry condition is satisfied by the product $AK$ if and only if
$\omega^{\ast}$ is equal to $\omega_{c}$ . Thus, not only for the discrete
lattice model, even at a coarse-grained level, we have been able to exactly
calculate the expression for the critical value of the relative timescale
which supports the product measure stationary state.

##  II Model description

The LH model, which is a short form for the light-heavy model, is defined on a
lattice whose sites can be either occupied by an $H$ (heavy) particle or an
$L$ (light) particle. The lattice bonds can have an orientation of $\pm\pi/4$.
We denote the site occupancy by a variable $\eta_{j}$ , which takes the value
$+1$ or $-1$ if the site $j$ is occupied by an $H$ particle or an $L$
particle, respectively. We also use the filled circle $\bullet$ to represent
an $H$-occupied site and empty circle $\circ$ to represent an $L$-occupied
site. Similarly, the bond orientations are denoted by the variable
$\tau_{j+\frac{1}{2}}$ which takes the value $-1$ if the bond between sites
$j$ and $(j+1)$ is an upslope bond, i.e., it has orientation $\pi/4$. For a
downslope bond with orientation $-\pi/4$, the value of $\tau_{j+\frac{1}{2}}$
is $1$. The upslope and downslope bonds are also symbolically represented as
$/$ and $\backslash$, respectively. With each lattice site, we associate a
“height” variable, defined as $h_{i}=-\sum_{j=1}^{i-1}\tau_{j+\frac{1}{2}}$ ,
as illustrated in Fig.1. From this figure, it is also clear that a local hill
is represented by $/\backslash$ and a local valley by $\backslash/$.

Figure 1: A typical configuration in the LH model

The dynamics of the particles and the landscape are coupled in the following
way. An $H$ particle can exchange position with an $L$ particle on its
neighboring site, with the exchange rate depending on the slope of the
intervening bond. $H$ particles tend to slide downward and $L$ particles show
the opposite tendency. We define the particle sliding rates as follows:

$\begin{split}W(H\setminus L\rightarrow L\setminus H)&=D+a,\\\ W(L\setminus
H\rightarrow H\setminus L)&=D-a,\\\ W(H\hskip 1.99997pt/\hskip
1.99997ptL\rightarrow L\hskip 1.99997pt/\hskip 1.99997ptH)&=D-a,\\\ W(L\hskip
1.99997pt/\hskip 1.99997ptH\rightarrow H\hskip 1.99997pt/\hskip
1.99997ptL)&=D+a,\end{split}$ (1)

with $0<a\leq D$. In Fig. 2, we pictorially show these transitions. No other
transitions are possible for the particles.

Figure 2: Transition rules for the particles in the LH model

The steady state particle current, defined as the rate of net flux of H
particles across a bond, can be formally written as

$(D+a)[P(H\backslash L)-P(L/H)]+(D-a)[P(H/L)-P(L\backslash H)],$ (2)

where $P(H\backslash L)$ denotes the probability of occurrence of the local
configuration $H\backslash L$, etc. The landscape evolves by exchanging slopes
between adjacent bonds, but the exchange rate depends on the occupancy of the
site between those two bonds:

$\begin{split}W(/\hskip 1.00006ptH\hskip
1.00006pt\setminus\rightarrow\setminus\hskip 1.00006ptH\hskip
1.00006pt/)&=\omega(E+b),\\\ W(\setminus\hskip 1.00006ptH\hskip
1.00006pt/\rightarrow/\hskip 1.00006ptH\hskip
1.00006pt\setminus)&=\omega(E-b),\\\ W(/\hskip 1.99997ptL\hskip
1.99997pt\setminus\rightarrow\setminus\hskip 1.99997ptL\hskip
1.99997pt/)&=\omega(E-b^{\prime}),\\\ W(\setminus\hskip 1.99997ptL\hskip
1.99997pt/\rightarrow/\hskip 1.99997ptL\hskip
1.99997pt\setminus)&=\omega(E+b^{\prime}),\end{split}$ (3)

where the parameter $\omega$ controls the relative timescale between the
landscape movement and the particle movement. We explain below how, in our
simulations, $\omega$ introduces an update ratio of the particles and the
landscape. Both the parameters $b$ and $b^{\prime}$ in Eq. 3 are bounded
between $-E$ to $E$. In Fig. 3 we pictorially show these transitions.

Figure 3: Dynamical rules for the landscape movement in the LH model

These landscape transitions result in a net upward or downward velocity of the
landscape or, equivalently, a net rightward or leftward movement of the
upslope bond through the system. The landscape current, defined as the rate at
which an upslope bond moves through the system from left to right, is given by

$\omega(E+b)P(/H\backslash)-\omega(E-b)P(\backslash
H/)+\omega(E-b^{\prime})P(/L\backslash)-\omega(E+b^{\prime})P(\backslash L/).$
(4)

From the dynamical rules shown in Eqs. 1 and 3, it is clear that the local
dynamics conserves the number of $H$ particles and $L$ particles, as well as
the number of upslope and downslope bonds, in the system. We consider a
lattice of size $N$ and a total number of $N_{up}$ upslope bonds and
$(N-N_{up})$ downslope bonds. We denote the total number of $H$ particles as
$N_{H}$; the number of $L$ particles then becomes $(N-N_{H})$. We denote the
density of $H$ particles as $\rho=N_{H}/N$ and the overall tilt of the
landscape as $m=N_{up}/N$. We assume a periodic boundary condition on the
lattice.

In all earlier studies of the LH model [23, 24, 25, 26, 32], $\omega=1$ was
assumed, but here we consider any arbitrary non-negative value of $\omega$.
From Eq. 3, it might appear that $\omega$ can be simply reabsorbed into the
rate parameters and our present model then becomes identical to the previously
considered version of the LH model [23, 24, 25, 26, 32]. But this is not the
case. In particular, varying $\omega$ is not the same as varying
$b,b^{\prime}$ for fixed $E$, as was done previously [23, 24, 25, 26]. In the
rates $\omega(E\pm b)$ and $\omega(E\pm b^{\prime})$, if we absorb $\omega$
into $E,b$ and $b^{\prime}$ then as we vary $\omega$, it would mean varying
all these three rescaled parameters $E,b$ and $b^{\prime}$ proportionately. To
ensure that rates are non-negative, rescaled $b$ and $b^{\prime}$ should stay
bounded between rescaled $[-E,E]$, i.e. as $\omega$ is varied, the allowed
range of variation of $b,b^{\prime}$ must also change.

This is how we incorporate $\omega$ in our simulation algorithm. One Monte
Carlo (MC) time-step consists of $N$ number of update trials. Before each
update trial, we choose with probability $\omega/(1+\omega)$ to carry out a
landscape movement, and with probability $1/(1+\omega)$ to carry out a
particle movement. If a landscape movement is decided, then the transition
takes place with rate $(E\pm b)$ or $(E\pm b^{\prime})$ depending on the local
configuration (see Eq. 3). If a particle movement is decided, then it is
performed with rate $D\pm a$, as shown in Eq. 1. Clearly, for large $\omega$,
the landscape gets a chance to be updated much more frequently than the
particles. For our choice of $D=E$, $\omega$ denotes average number of times
the landscape moves in between two particle movements. $\omega\ll 1$ indicates
a slowly moving landscape and $\omega\gg 1$ represents a fast moving
landscape.

For $\omega=1$, some of the co-authors had constructed the phase diagram for
the system in an earlier work [23]. For any $a>0$, the ordered and disordered
phases were mapped out in the $bb^{\prime}$ plane. It was found that for
$(b+b^{\prime})>0$, the system is in various different kinds of ordered
phases, where both particles and the landscape show long-range order. For
$(b+b^{\prime})<0$, the system is in disordered phase, but short-range
correlations still develop in the system. For $(b+b^{\prime})=0$, it follows
from Eq. 3 that the landscape dynamics no longer depends on the particles, and
the coupling becomes one way. This is a special limit and the system shows an
unusual type of ordering for the particles here, where long-range order and
strong fluctuations are simultaneously present [6, 7, 35].

##  III Simulation results on short-range correlations

Starting from a random initial configuration, we perform $\sim N^{2}$ MC steps
to reach the steady state and all our measurements are conducted in the steady
state. We define the following nearest neighbor correlations. Consider
$Prob(\eta_{i}=1,\eta_{i+1}=1)$, which represents the joint probability that
two consecutive lattice sites are occupied by $H$ particles. Similarly,
$Prob(\tau_{i-\frac{1}{2}}=-1,\tau_{i+\frac{1}{2}}=-1)$ is the joint
probability that there are two successive upslope bonds. In a similar way, one
can define $Prob(\eta_{i}=1,\tau_{i+\frac{1}{2}}=-1)$ and
$Prob(\tau_{i-\frac{1}{2}}=-1,\eta_{i}=1)$. The two-point correlation
functions can be defined from the joint probabilities. For example, the
nearest neighbor correlation between two heavy particles is defined as
$Prob(\eta_{i}=1,\eta_{i+1}=1)-(N_{H}/N)^{2}$. In Figs. 4(a) and 4(b), we plot
the two-point correlations as a function of $\omega$. Surprisingly, we find
that all four correlations vanish at a precise value of $\omega$. We denote
this special point as $\omega_{c}$. For different choices of rate parameters
$a,b,b^{\prime}$, the value of $\omega_{c}$ is also different.

Figure 4: Simulation results for two-point and three-point correlations. Top
row corresponds to $b=-0.1$, $b^{\prime}=-0.4$ and $a=0.5$. Bottom row is for
$b=0.1$, $b^{\prime}=-0.35$ and $a=0.375$. For all panels, $N=1024$,
$N_{H}/N=\rho=0.5$, $N_{up}/N=m=0.5$.

We also measure the variation of three-point correlations with $\omega$. While
it is possible to define many different three-point correlations in the
system, we present here data for only four different kinds, which appear in
Eq. 3. In Figs. 4(c) and 4(d) we present the data. Here, up-heavy-down refers
to
$Prob(\tau_{i-\frac{1}{2}}=-1,\eta_{i}=1,\tau_{i+\frac{1}{2}}=1)-(\frac{N_{up}}{N})(1-\frac{N_{up}}{N})(\frac{N_{H}}{N})$
and other three correlations can also be similarly defined. We find that at
exactly the same $\omega=\omega_{c}$ all three-point correlations vanish. We
have verified it for a few other correlations too (data not shown here). To
explain the above results, we generalize the formalism introduced in [32] and
show that at $\omega=\omega_{c}$ the system indeed satisfies equiprobable
measure and no correlations exist in the system.

##  IV Exact calculation for $\omega_{c}$ for discrete lattice model

In this section, we generalise the formalism introduced in [32] to prove that
at $\omega=\omega_{c}$, the system can be characterized by equiprobable
measure in steady state. In other words, for a lattice of size $N$ with
$N_{H}$ number of $H$ particles, $(N-N_{H})$ number of $L$ particles, $N_{up}$
number of upslope bonds, and $(N-N_{up})$ number of downslope bonds, the total
number of possible configurations is
$\mathcal{N}=\binom{N}{N_{H}}\times\binom{N}{N_{H}}$ and for
$\omega=\omega_{c}$, each configuration occurs with probability
$1/\mathcal{N}$ in the steady state. Below, We exactly calculate an expression
for $\omega_{c}$ in terms of the rate parameters $a,b,b^{\prime}$.

The configurations in the LH model are discrete which are specified by the
occupancy of the lattice sites and orientation or tilt of the lattice bonds.
Let $p_{c}(t)$ is the probability to find the system in a particular
configuration $c$ at time $t$. The time-evolution of $p_{c}(t)$ is governed by
the master equation [36]

$\frac{dp_{c}(t)}{dt}=\sum_{c^{\prime}\neq c}\bigl{[}r_{c^{\prime}\rightarrow
c}p_{c^{\prime}}(t)-r_{c\rightarrow c^{\prime}}p_{c}(t)\bigr{]}.$ (5)

Here, $r_{c^{\prime}\rightarrow c}$ is the incoming transition rate from any
other configuration $c^{\prime}$ to $c$ and $r_{c\rightarrow c^{\prime}}$ is
the outgoing rate. In the steady state, the right-hand side of the above
equation must vanish. Now, transition between two configurations are possible
either via particle movement or via landscape movement. Particle movement
involves exchange of positions between an $H$ and $L$ particle at two
neighboring sites. A landscape movement happens when a neighboring pair of
upslope and downslope bonds exchange their orientations. Let us consider the
quantity $(\eta_{j+1}-\eta_{j})/2$ which can take the values $\pm 1$ or $0$.
For a local configuration of the form $L/H$ or $L\backslash H$ the value is
$+1$, for $H\backslash L$ or $H/L$ the value is $-1$ and for any other case
where both neighboring sites are occupied by particles of the same species,
the value is zero. Clearly, no particle transition takes place if the value is
zero.

###  IV.1 Mapping of configuration to sequence of brackets and dots

At this stage, we introduce an alternative way to represent a configuration,
following the method introduced in [32]. Instead of explicitly specifying the
site occupancies, we use the symbol open parenthesis ‘$($’ if
$(\eta_{j+1}-\eta_{j})/2$ is $+1$, a closed parenthesis ‘$)$’ if it is $-1$,
and a dot ‘.’ if it is zero. Similarly, the quantity
$(\tau_{j-\frac{1}{2}}-\tau_{j+\frac{1}{2}})/2$ takes the value $+1$ for a
local valley, $-1$ for a local hill, and zero for all other cases where no
transitions are possible. We use an open angular bracket ‘$<$’ for a local
valley, a closed angular bracket ‘$>$’ for a local hill and a dot ‘.’ for
other cases. In Table 1 we summarize the mapping.

Variable | Value | Local Configurations | Name | Symbol
---|---|---|---|---
$\frac{1}{2}\bigl{(}\eta_{j+1}-\eta_{j}\bigr{)}$ | 1 | $\circ/\bullet\hskip 4.26773pt;\hskip 4.26773pt\circ\setminus\bullet$ | open parenthesis | (
0 | $\circ/\circ\hskip 4.26773pt;\hskip 4.26773pt\circ\setminus\circ\hskip 4.26773pt;\hskip 4.26773pt\bullet\setminus\bullet\hskip 4.26773pt;\hskip 4.26773pt\bullet/\bullet$ | dot | $\cdot$
-1 | $\bullet/\circ\hskip 4.26773pt;\hskip 4.26773pt\bullet\setminus\circ$ | closed parenthesis | )
$\frac{1}{2}\bigl{(}\tau_{j-\frac{1}{2}}-\tau_{j+\frac{1}{2}}\bigr{)}$ | 1 | $\backslash\circ/\hskip 4.26773pt;\hskip 4.26773pt\backslash\bullet/$ | open angular bracket | $\langle$
0 | $/\bullet/\hskip 4.26773pt;\hskip 4.26773pt/\circ/\hskip 4.26773pt;\hskip 4.26773pt\backslash\bullet\backslash\hskip 4.26773pt;\hskip 4.26773pt\backslash\circ\backslash$ | dot | $\cdot$
-1 | $/\circ\backslash\hskip 4.26773pt;\hskip 4.26773pt/\bullet\backslash$ | closed angular bracket | $\rangle$
Table 1: Representation of configuration in the LH model as a sequence of
brackets and dots

Using the above method, one can map any configuration in the LH model as a
unique sequence of parentheses, angular brackets and dots. In Fig. 5 we
illustrate this process. The sequence follows some constraints though. Two
open (closed) parentheses can not occur in succession, there must be a closed
(open) parenthesis between them. Similar restriction holds for angular
brackets too. Due to periodic boundary condition, each configuration must have
equal number of open and closed parentheses, and an equal number of open and
closed angular brackets. However, the total number of parentheses may or may
not be equal to the total number of angular brackets in a configuration.
Moreover, the dynamics does not conserve these total numbers and they can
change with time, even though $N_{H}$ and $N_{up}$ remain conserved.

Figure 5: Description of a configuration in terms of parentheses and bends and
different type of segments arising in it.

###  IV.2 Grouping brackets and dots into segments

In the steady state, the total incoming flux to any configuration $c$ must be
balanced by the total outgoing flux from $c$. Here we classify all incoming
and outgoing transitions for a general configuration $c$ into groups. For
this, we follow the protocol introduced in [32]. We start from any random
point in the sequence of parentheses, angular brackets and dots which
represents configuration $c$, and move rightward through the sequence. Every
time we encounter an open (closed) parenthesis or open (closed) angular
bracket, we move towards right (left), and keep moving in that direction until
we find a closed (open) bracket of the same type. While doing so, if we
encounter other type of open (closed) unpaired bracket, we move in rightward
direction (leftward direction) until we get a closed (open) bracket of that
type. This procedure is continued until a segment of minimum size is
identified, where corresponding to each open bracket, there is a closed
bracket of the same type. After a segment has been completed, we start a new
segment from the current position and repeat the process. This method divides
a sequence into a few non-overlapping segments and also ensures that this
segmentation is unique since the protocol ensures that every bracket belongs
to the smallest group of open and closed brackets [32]. In Fig. 5 we
illustrate this method. Next, we show how within each segment the steady state
flux balance condition is satisfied.

###  IV.3 Pairwise balance

The simplest possible segment contains only one pair of open and closed
brackets of the same type, i.e. either $(...)$ or $<....>$. Let us first
consider $(...)$ which represents a local configuration $c$ where only
particle movements are possible, and no landscape movement is allowed. As we
illustrate in Fig. 6 for every such configuration $c$, one can identify a
unique pair of configurations $c^{\prime}$ and $c^{\prime\prime}$, such that
$r_{c^{\prime}\to c}=r_{c\to c^{\prime\prime}}$. Absence of any angular
brackets rules out the existence of local hills and valleys in $c$. For
example, if we assume all lattice bonds are downhill, then $c$ can be entered
(exited) by an upward (downward) slide $H$ particle at the left (right) end.
Thus for every incoming transition, there exists an outgoing transition of the
very same type. This condition is known as the ‘pairwise balance’ [37, 32].
More precisely, if $r_{c^{\prime}\to c}=r_{c\to c^{\prime\prime}}$ is
satisfied, then $p_{c^{\prime}}=p_{c}$ indeed makes the right hand side of Eq.
5 zero for any values of $a,b,b^{\prime}$. In a similar way, one can show that
pairwise balance with equiprobable measure condition is satisfied for all
$<...>$ segments.

Figure 6: Pairwise balance for a sequence enclosed by a pair of parentheses.
For every incoming transition, there is an outgoing transition of the same
rate.

Even for a more general type of segment, where both parentheses and angular
brackets are present, pairwise balance condition can be satisfied as long as
the leftmost opening bracket and the rightmost closing bracket are of the same
type. To show this we use the method of reduction introduced in [32] where a
given segment can be shortened by eliminating those pairs of brackets which
represent transitions that satisfy pairwise balance. For example, consider the
segment $(..<..)..(..>..)$. It can be easily shown that the pair $)..($,
appearing in the middle, satisfies pairwise balance. The proof runs along
exactly the same lines as illustrated in the previous paragraph for $(..)$
segment. After eliminating this pair, the segment is reduced to $(..<..>..)$
and after eliminating the pair of angular brackets $<...>$ which obviously
satisfy pairwise balance, we are left with $(..)$, another pairwise balanced
segment. In Fig. 7 we explicitly show, without using the method of reduction,
that indeed the local configuration represented by the above sequence segment
satisfies pairwise balance. In a similar way, one can show that pairwise
balance is satisfied for the segment $<..(..>..<..)..>$.

Figure 7: Pairwise balance for a local configuration $c$ represented by
$(\,..\langle..)\,..(\,..\rangle..)\,$

###  IV.4 Bunchwise balance

However, if a segment has leftmost opening bracket and rightmost closing
bracket of opposite types, then it can not be reduced completely. After
eliminating all pairwise balanced brackets like $<..>$, $>..<$, $(..)$ and
$)..($ which may have been present in the middle of the segment, we are left
with either $(..<..)..>$ or $<..(...>..)$, which can not be reduced any
further. These are known as irreducible sequence of $+$ type and $-$ type,
respectively. They do not satisfy pairwise balance [32]. However, both these
irreducible sequences represent local configurations that can be reached in
four different ways, two of which involve particle exchange and the other two
involve landscape movement. In Fig. 8 we explicitly show this for $+$ type
irreducible sequence. The particle transitions are always of the same type,
and if one landscape transition consists of heavy (light) hill flipping to a
heavy (light) valley, then the other one corresponds to light (heavy) valley
flipping to a light (heavy) hill. The configurations can be exited by four
transitions, which are exactly the reverse transitions of the incoming ones.
If equiprobable measure has to be satisfied in steady state, then the sum of
all incoming transition rates must be equal to the sum of all outgoing
transition rates. This leads to the condition of ‘bunchwise balance’

$2(D+a)+\omega(E+b)+\omega(E+b^{\prime})=2(D-a)+\omega(E-b)+\omega(E-b^{\prime})$
(6)

which gives

$\omega=\omega_{c}=-\frac{2a}{b+b^{\prime}}.$ (7)

For $-$ type irreducible sequence also, condition of bunchwise balance gives
the exactly same expression for the critical time scale. In the special case
of $\omega_{c}=1$, the bunchwise balance condition simplifies further. In this
case, the total incoming (outgoing) transition rate through particle exchange
becomes equal to the total outgoing (incoming) transition rate through slope
update, such that $2(D+a)=(E-b)+(E-b^{\prime})$ which gives
$2a+b+b^{\prime}=0$, the condition derived in [32].

Figure 8: Incoming and outgoing transitions for a plus type irreducible
sequence.

Eq. 7 shows that it is only in the disordered phase, i.e. for
$(b+b^{\prime})<0$ one has a positive, finite value of $\omega_{c}$. As one
approaches the $(b+b^{\prime})=0$ line, the value of $\omega_{c}$ becomes
infinitely large. This is expected, since the particles show long range order
here and product measure cannot be satisfied. For $(b+b^{\prime})>0$ both
particles and the landscape show long range order and $\omega_{c}$ takes
unphysical negative values here. In Fig. 9 we compare our exact result with
our simulation observation and find excellent agreement.

Figure 9: Comparison between theory and simulations. Discrete points show
$\omega_{c}$ measured from simulations and lines show exact result from Eq. 7.
Here, $\rho=m=0.5$ are used.

##  V Relative time-scale in hydrodynamics

In the previous section, an exact expression for $\omega_{c}$ was derived for
a discrete lattice model using the condition of flux balance in steady state.
In this section, we present an alternative derivation in the continuum limit
where we coarse-grain the LH model and use hydodynamic expansion. As discussed
in Sec. II the dynamical rules are such that the number of $H$ particles and
the number of upslope bonds are locally conserved. We denote the local density
of $H$ particles as $\rho(x,t)$ and that of upslope bonds as $m(x,t)$. As the
particles move around and the landscape fluctuates, these densities change and
their time-evolution is described by the following continuity equations

$\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial J_{\rho}(\rho,m)}{\partial x}$ (8)
$\displaystyle\frac{\partial m}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial J_{m}(\rho,m)}{\partial x}$ (9)

where $J_{\rho}$ and $J_{m}$ are local currents of $H$ particles and upslope
bonds, respectively. Due to coupled dynamics, these currents depend on both
$\rho$ and $m$. Here, we have used the assumption of local equilibrium, i.e.
local currents depend on $x,t$ only through their dependence on $\rho,m$ and
do not have any explicit space-time dependence. Then the continuity equations
can be written as

$\displaystyle\frac{\partial\rho}{\partial t}+\frac{\partial
J_{\rho}}{\partial\rho}\frac{\partial\rho}{\partial x}+\frac{\partial
J_{\rho}}{\partial m}\frac{\partial m}{\partial x}=0$ (10)
$\displaystyle\frac{\partial m}{\partial t}+\frac{\partial
J_{m}}{\partial\rho}\frac{\partial m}{\partial x}+\frac{\partial
J_{m}}{\partial m}\frac{\partial m}{\partial x}=0.$ (11)

By defining a two dimensional vector $\vec{\psi}(x,t)$, whose components are
$\rho(x,t)$ and $m(x,t)$, the above equations can be written in a more compact
form as

$\frac{\partial\vec{\psi}}{\partial t}+{A}\frac{\partial\vec{\psi}}{\partial
x}=0$ (12)

where $A$ is the current Jacobian defined as

${A}=\begin{pmatrix}\frac{\partial J_{\rho}}{\partial\rho}&\frac{\partial
J_{\rho}}{\partial m}\\\ \frac{\partial J_{m}}{\partial\rho}&\frac{\partial
J_{m}}{\partial m}\end{pmatrix}.$ (13)

Expanding components of $\vec{\psi}(x,t)$ around their conserved global values
$\psi_{\alpha}(x,t)=\psi_{\alpha}^{0}+u_{\alpha}(x,t)$ and retaining only
linear terms in $u_{\alpha}$ we get

$\frac{\partial\vec{u}}{\partial
t}+{\bm{A}}^{0}\frac{\partial\vec{u}}{\partial x}=0$ (14)

where ${A}^{0}$ is a $2\times 2$ matrix whose elements are those of $\bm{A}$
evaluated at $\rho_{0}=N_{H}/N$ and $m_{0}=N_{up}/N$. To solve this equation,
we diagonalize ${A}^{0}$ and obtain the normal modes. However, this requires
explicit knowledge of $J_{\rho}$ and $J_{m}$ as a function of $\rho$ and $m$.
From Eqs. 2 and 4 in Sec. II it follows that the exact expressions for certain
short ranged correlations are needed, which is generally hard to obtain. At
this stage, we make an assumption that there exists a specific
$\omega=\omega^{\ast}$ for which product measure holds and these correlations
factorize. Then from Eqs. 2 and 4 we can write $J_{\rho}$ and $J_{m}$ as
(scaled by a factor of ($1+\omega^{\ast}$))

$\begin{split}J_{\rho}&=2a\rho(1-\rho)(1-2m)\\\
J_{m}&=\omega^{\ast}2m(1-m)\bigl{[}\rho(b+b^{\prime})-b^{\prime}\bigr{]}.\end{split}$
(15)

The current jacobian matrix $A^{0}_{\alpha\beta}$ = $\dfrac{\partial
J_{\alpha}}{\partial\rho_{\beta}}$ then has the form

$\centering{A^{0}}=\begin{bmatrix}2a(1-2m_{0})(1-2\rho_{0})&-4a\rho_{0}(1-\rho_{0})\vspace{0.15cm}\\\
2\omega^{\ast}m_{0}(1-m_{0})(b+b^{\prime})&\omega^{\ast}(1-2m_{0})\bigl{(}2\rho_{0}(b+b^{\prime})-2b^{\prime}\bigr{)}\end{bmatrix}.\@add@centering$
(16)

which has real, distinct eigenvalues. Therefore, we have a system of strictly
hyperbolic conservation laws. For such systems it was shown in [34] using
time-reversal invariance that the product of current Jacobian matrix and the
compressibility matrix must be symmetric,

${A}^{0}K=K({A}^{0})^{T}$ (17)

where $({A}^{0})^{T}$ is transpose of ${A}^{0}$.

Now in a canonical ensemble which we use in the simulations, the global
conservation of $N_{H}$ and $N_{up}$ does give rise to correlations in a
finite size system, even when equiprobable measure is satisfied in steady
state. Product measure condition, which we used in writing Eq. 15, actually
holds for a grand canonical ensemble with a fluctuating $N_{H}$ and $N_{up}$.
These fluctuations are characterized by a symmetric covariance matrix (also
known as compresssibility matrix) $K$ whose elements are [16, 17, 18]

$K=\frac{1}{N}\begin{pmatrix}\langle(N_{H}-\rho
N)^{2}\rangle&\langle(N_{H}-\rho N)(N_{up}-mN)\rangle\\\
\langle(N_{up}-mN)(N_{H}-\rho
N)\rangle&\langle(N_{up}-mN)^{2}\rangle\end{pmatrix}$ (18)

In the case when product measure is satisfied, $K$ becomes

$K=\begin{pmatrix}\rho_{0}(1-\rho_{0})&0\\\ 0&m_{0}(1-m_{0}).\end{pmatrix}$
(19)

Substituting forms of ${A^{0}}$ and $K$ in Eq.17 we get

$\centering{A}^{0}K-K({A}^{0})^{T}=\begin{bmatrix}0&-2\rho_{0}(1-\rho_{0})m_{0}(1-m_{0})\bigl{(}2a+\omega^{\ast}(b+b^{\prime})\bigr{)}\vspace{0.15cm}\\\
2\rho_{0}(1-\rho_{0})m_{0}(1-m_{0})\bigl{(}2a+\omega^{\ast}(b+b^{\prime})\bigr{)}&0\end{bmatrix}\@add@centering$
(20)

The right hand side of Eq. 20 vanishes only when
$\omega^{\ast}=-\dfrac{2a}{(b+b^{\prime})}$, which matches exactly with
$\omega_{c}$ derived in sec.IV.

##  VI Conclusions

To gain insights into the behavior of coupled driven systems, one useful tool
is to make one system evolve faster or slower than the other, by introducing a
relative timescale in their dynamics. This method has been used in quite a few
recent studies [27, 28, 29, 30, 31] and various interesting effects of the
relative timescale were observed. In [29, 30], noninteracting particles
sliding on a fluctuating landscape were considered in the limit of one-way
coupling, where the landscape dynamics remains unaffected by the particles,
and it was found that depending on whether the particles move faster or slower
compared to the landscape, the scaling description of several steady state
correlations changes. In [31], a single particle was considered on a
fluctuating energy landscape in the presence of a two-way coupling, and
qualitatively different long-time behavior were observed, depending on the
landscape to slider timescale ratio. In this work, we have examined the effect
of introducing a relative timescale $\omega$ in a coupled time evolution of
hard-core particles on a fluctuating landscape. We find a critical value
$\omega_{c}$ of the relative timescale at which all configurations are equally
likely in the steady state. We prove this result exactly for a discrete
lattice model, as well as for a coarse-grained model. Our proof for the case
of a discrete lattice model relies on generalizing a formalism introduced in
[32], according to which, we break up each configuration into nonoverlapping
segments and identify the incoming and outgoing transitions for each segment.
For certain segments, we find that for each incoming transition, there exists
an outgoing transition of exactly the same type. This condition is known as
pairwise balance [37] and it follows from Eq. 5 that the equiprobable measure
is satisfied in the steady state. However, there are some special types of
segments where the incoming and outgoing transitions are exactly the reverse
of each other. For these cases, the equiprobable measure can be satisfied if
and only if the total incoming rate and outgoing rate are equated, which
yields the $\omega=\omega_{c}$ criterion. For $\omega\neq\omega_{c}$, the
equiprobable measure does not hold and local correlations are nonzero in
general. It was not possible to calculate these correlations exactly, but
their qualitative behavior can be explained by following the line of argument
presented in [32]. In Appendixes A and B, we have included detailed
discussions of this. Our study opens up the intriguing possibility to control
the stationary measure of a coupled driven system by tuning its relative
timescale. This idea can be implemented for other model systems as well. In
particular, our calculation in the coarse-grained model is quite generic and,
for any other system which allows hydrodynamic expansion, similar calculations
can readily be performed. It will be of interest to see if such calculations
yield a critical timescale for other systems. Finally, for systems with $n>2$
conserved modes, there can be $(n-1)$ relative timescales. How their interplay
affects the correlation functions and whether it allows for a product measure
steady state are interesting questions and offer a promising direction for
future research.

##  VII Acknowledgments

We acknowledge useful discussions with Gunter M. Schütz. C.K. acknowledges a
research fellowship (Grant No. 09/0575(12571)/2021-EMR-I) from the Council of
Scientific and Industrial Research (CSIR), India.

##  Appendix A Variation of correlations with $\omega$

As shown in Fig.4, different local short-ranged correlations show different
natures of variation with $\omega$. In case of three point correlations,
$/$$\circ$$\backslash$ and $\backslash$$\bullet$$/$ start from a positive
value, decrease with $\omega$, pass through zero at $\omega=\omega_{c}$ and
become negative for larger $\omega$. The other three point correlations,
$/$$\bullet$$\backslash$ and $\backslash$$\circ$$/$ show exactly the opposite
trend. They are negative for $\omega<\omega_{c}$ and positive for
$\omega>\omega_{c}$. These variations can be easily explained by considering
how the bunchwise balance of incoming and outgoing transition rates is
violated when $\omega\neq\omega_{c}$. As shown in Fig.8, the sum total of all
incoming transition rates for a $+$ type irreducible sequence is
$2(D+a)+\omega(2E+b+b^{\prime})$, which is greater than the sum of all
outgoing transition rates $2(D-a)+\omega(2E-b-b^{\prime})$, if
$\omega<\omega_{c}$. For a $-$ type sequence, similarly, one can show that the
total outgoing rate is larger than the incoming ones. Therefore, for
$\omega<\omega_{c}$ the steady state probability to find a $+$ type sequence
is more than $-$ type. As a result, those correlations which occur in $+$ type
sequences are positive for $\omega<\omega_{c}$ and negative for
$\omega>\omega_{c}$, while those correlations which are absent from $+$ type
sequences but can be found in $-$ type sequences, show the opposite trend.

At $\omega=\omega_{c}$ the $+$ type and $-$ type irreducible sequences are
equally probable. The following quantity, known as cross-correlation function
captures this clearly [32]. This is defined as

${\mathcal{S}}(\\{\eta,\tau\\})=\sum_{j=1}^{N}\frac{1}{2}(\eta_{j+1}-\eta_{j})\tau_{j+\frac{1}{2}}$
(21)

which depends on the specific configuration $\\{\eta,\tau\\}$. The cross-
correlation function can also be defined as
${\mathcal{S^{\prime}}}(\\{\eta,\tau\\})=\sum_{j=1}^{N}\frac{1}{2}\bigl{(}\tau_{j-\frac{1}{2}}-\tau_{j+\frac{1}{2}}\bigr{)}\eta_{j}$
which has the same value as ${\mathcal{S}}(\\{\eta,\tau\\})$ for any
configuration with periodic boundary conditions. ${\mathcal{S}}$ and
${\mathcal{S^{\prime}}}$ give a value of $0$ when we calculate them only over
a parenthesis pair or a bend pair. A segment that satisfies pairwise balance
has a vanishing contribution towards the sum in Eq. 21. The only nonzero
contribution comes from irreducible sequences of $+$ and $-$ type, which
contribute $2$ and $-2$, respectively. If $N_{+}(\\{\eta,\tau\\})$ and
$N_{-}(\\{\eta,\tau\\})$ denote the number of $+$ type and $-$ type
irreducible sequences in a configuration, then

${\mathcal{S}}(\\{\eta,\tau\\})=2[N_{+}(\\{\eta,\tau\\})-N_{-}(\\{\eta,\tau\\})].$
(22)

When averaged over steady state ensemble, we get
$S=\langle{\mathcal{S}}(\\{\eta,\tau\\})\rangle=2(\langle N_{+}\rangle-\langle
N_{-}\rangle)$, where $\langle N_{\pm}\rangle$ are the average number of $+$
type and $-$ type irreducible sequences in steady state. In Fig. 10 we plot
$S$ (scaled by system size) as a function of the relative time-scale and find
that for $\omega<\omega_{c}$ when $+$ type sequences are more probable, $S$ is
positive, while for $\omega>\omega_{c}$, when $-$ type sequences are more
probable, $S$ is negative. This is consistent with the above argument.

Figure 10: Variation of $S$ with relative time-scale for
$b=-0.1,b^{\prime}=-0.4,a=0.5$, $\rho_{0}=m_{0}=0.5$ and $N=1024$.

##  Appendix B height fluctuations and cluster size distributions

To further probe the effect of relative time-scale on the steady state of the
system, we define two ‘height’ variables for site occupancy and bond
orientation as follows.

$h_{\eta}(i)=\sum_{j=1}^{i}\eta_{j}\;\;\;\;\;\;{\text{and}}\;\;\;\;\;\;h_{\tau}(i)=-\sum_{j=1}^{i}\tau_{j+1/2}.$
(23)

Let ${W}_{\eta}$ and ${W}_{\tau}$ be the width of the corresponding height
profiles, defined as

${W}_{\eta}=\biggl{[}\frac{1}{L}\Bigl{\langle}\sum_{i=1}^{L}\bigl{[}h_{\eta}(i)-\overline{h_{\eta}}\hskip
2.84544pt\bigr{]}^{2}\Bigr{\rangle}\biggr{]}^{1/2}\hskip 5.69046pt;\hskip
5.69046pt{W}_{\tau}=\biggl{[}\frac{1}{L}\Bigl{\langle}\sum_{i=1}^{L}\bigl{[}h_{\tau}(i)-\overline{h_{\tau}}\hskip
2.84544pt\bigr{]}^{2}\Bigr{\rangle}\biggr{]}^{1/2}$ (24)

where the overhead bar denotes averaging over all lattice points in a given
configuration and the angular brackets denote averaging over different steady
state configurations. In Fig. 11 we plot ${W}_{\eta}$ and ${W}_{\tau}$ for
different $\omega$ values and find two opposite trends. While ${W}_{\eta}$
decreases with $\omega$ (purple points), ${W}_{\tau}$ increases (green points)
and they intersect at $\omega=\omega_{c}$. To explain this behavior, we note
that the tendency of $H$ particles to slide down to the local valleys is
exactly balanced by their tendency to destabilize the valleys by landscape
movement at $\omega=\omega_{c}$. This gives rise to equiprobable measure. For
$\omega$ smaller than $\omega_{c}$, the landscape movement is slower and the
$H$ particles get a chance to form local clusters around the valleys. Although
this heterogeneity in particle occupancy does not give rise to long range
order in the system, the width ${W}_{\eta}$ is still larger than its product
measure value. To verify this argument, we explicitly measure the cluster size
distribution of $H$ particles (Fig. 12 ) and show that for
$\omega<\omega_{c}$, medium and large size clusters are more probable than the
product measure case. Simiarly, for $\omega>\omega_{c}$ the landscape moves
faster and heavy valleys are quickly destabilized. This makes it more
difficult for the $H$ particle to clump together and isolated $H$ particles
are found more frequently now. This brings down ${W}_{\eta}$. In a similar
way, variation of ${W}_{\tau}$ can also be explained.

Figure 11: Variation of $W_{\eta}$ and $W_{\tau}$ with $\omega$ for for
$b=-0.1,b^{\prime}=-0.4$ and $a=0.5$ with $\rho_{0}=m_{0}=0.5$.  Figure 12:
Cluster size distribution of particles at different $\omega$ for
$b=-0.1,b^{\prime}=-0.4$ and $a=0.5$ with $\rho_{0}=m_{0}=0.5$. The solid line
represents $P(n)$ for product measure.

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