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PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
Light and heavy particles on a fluctuating surface: Bunchwise balance,
irreducible sequences, and local density-height correlations
Samvit Mahapatra ,*Kabir Ramola ,†and Mustansir Barma‡
TIFR Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
(Received 29 June 2020; accepted 16 October 2020; published 24 November 2020)
We study the early-time and coarsening dynamics in the light-heavy model, a system consisting of two species
of particles (light and heavy) coupled to a fluctuating surface (described by tilt fields). The dynamics of particles
and tilts are coupled through local update rules, and are known to lead to different ordered and disordered
steady-state phases depending on the microscopic rates. We introduce a generalized balance mechanism in
nonequilibrium systems, namely, bunchwise balance, in which incoming and outgoing transition currents are
balanced between groups of configurations. This allows us to exactly determine the steady state in a subspace
of the phase diagram of this model. We introduce the concept of irreducible sequences of interfaces and bends
in this model. These sequences are nonlocal, and we show that they provide a coarsening length scale in the
ordered phases at late times. Finally, we propose a local correlation function (S) that has a direct relation to the
number of irreducible sequences, and is able to distinguish between several phases of this system through its
coarsening properties. Starting from a totally disordered initial configuration, Sdisplays an initial linear rise and
a broad maximum. As the system evolves toward the ordered steady states, Sfurther exhibits power-law decays
at late times that encode coarsening properties of the approach to the ordered phases. Focusing on early-time
dynamics, we posit coupled mean field evolution equations governing the particles and tilts, which at short times
are well approximated by a set of linearized equations, which we solve analytically. Beyond a timescale set by
an ultraviolet (lattice) cutoff and preceding the onset of coarsening, our linearized theory predicts the existence
of an intermediate diffusive (power-law) stretch, which we also find in simulations of the ordered regime of the
system.
DOI: 10.1103/PhysRevResearch.2.043279
I. INTRODUCTION
Phase separation, coarsening, and dynamical arrest in inter-
acting nonequilibrium systems arise in a variety of contexts in
physics and biology. Examples include turbulent mixtures [1],
driven granular materials [2], constrained systems at low tem-
peratures [3,4], as well as soft matter and biological systems
[5]. Hard-core particle systems that model several types of
materials often display glassy dynamics and provide examples
of unusually slow coarsening toward phase separation, how-
ever, they remain hard to characterize theoretically. Simple
models of confined hard particles also display other nontriv-
ial behavior such as anomalous transport properties [6]. For
instance, driven hard-core particles in one dimension serve
as useful models for transport along channels and surfaces,
and have a long history of study [7]. Their coarse-grained
properties have been related to the Kardar-Parisi-Zhang (KPZ)
and Burgers equations that also describe the hydrodynamics
of surfaces and compressible fluids [8].
*samvit.mahapatra@gmail.com
†kramola@tifrh.res.in
‡barma@tifrh.res.in
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
A. Fluctuating local drives: Hard-core particles
on a fluctuating surface
While globally driven systems have been well studied, in-
teresting effects arise in systems with a fluctuating local drive,
which have been of considerable recent interest in connection
with active particle systems [9]. These systems display new
nonequilibrium properties such as motility-induced phase sep-
aration (MIPS) [10] and even complete dynamical arrest [11].
However, several key questions about the dynamics of such
systems remain open. In this context, one-dimensional models
of particles coupled to a fluctuating surface have proved to be
useful tools to study and characterize the behavior of locally
driven nonequilibrium systems [12–14]. Even though many
such models are built using simple local rules, an exact deter-
mination of their nonequilibrium steady states has been hard,
with only a few known cases. In this paper, we introduce the-
oretical approaches to study the out-of-equilibrium behavior
of the light-heavy (LH) model, a simple lattice model con-
sisting of two species of hard-core particles interacting with a
fluctuating surface. This model (discussed in detail below) is
known to exhibit several nonequilibrium steady-state phases
[12,15–20]. In particular, we present methods which allow us
to exactly determine the nonequilibrium steady state within a
subspace of parameters of this model, and to derive results on
its early and late-time dynamics.
Theoretically, the LH model is interesting as it has char-
acteristics of a class of multispecies nonequilibrium systems.
2643-1564/2020/2(4)/043279(23) 043279-1 Published by the American Physical Society
MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
Moreover, the sorts of cooperative effects exhibited by the
LH model, namely, propagating waves and clustering remi-
niscent of phase separation, arise in several physical settings,
as discussed below. For instance, waves and separately clus-
tering have been observed on the membrane of a cell as a
result of the coupling between the membrane and the actin
cytoskeleton [21]. A continuum model which couples the
membrane and protein dynamics has been used to explain
the formation of protrusions and protein segregation on the
membrane [22]. The effect of active inclusions in an interface
was studied in [23], motivated by the formation of orga-
nized dynamical structures within the membrane of crawling
cells. If particles on the membrane are passive and do not
act back on the actomyosin, the result is a clustered state
of a different sort with signatures of fluctuation-dominated
phase ordering (FDPO) [24], in which long-range order co-
exists with macroscopic fluctuations. Propagating waves and
clustering are also found in a system of active pumps with
a two-way interaction with a fluid membrane [25]. A very
different context is provided by the problem of a colloidal
crystal which is sedimenting through a viscous fluid. The
problem involves the interplay between two coupled fields,
namely, the overall particle density, and the tilt field which
governs the local direction of settling, with respect to grav-
ity. A study of a lattice model of the system shows that
the system displays macroscopic phase separation in steady
state [12,15].
The local rules of the LH model were first defined in [15].
It was initially studied within a restricted subspace in [15,16],
and was later studied more generally in [17–20]. The model
describes light (L) and heavy (H) particles interacting with
the local slopes (tilts) of a fluctuating surface. The particle-
surface coupling arises as follows: surface slopes provide a
dynamically evolving bias which guides particle movement,
while the back-action of Land Hparticles on the surface
affects its evolution. In spite of its simplicity, the system
exhibits a rich phase diagram with a disordered phase and
several types of ordered phases. The disordered phase arises
when the back-action of particles on the surface goes op-
posite to the action of surface slopes on the particles. The
dynamics is then dominated by mixed-mode kinematic waves.
These waves constitute a generalization of kinematic waves
familiar in single-component driven systems [26,27]. A recent
numerical study [20] shows that the decay of these waves in
one dimension typifies new universality classes beyond that
of single-component systems [28–31], with distinct dynamic
exponents and scaling functions [32–37].
On the other hand, when the back-action of the particles on
the surface is in consonance with the action of surface slopes
on the particles, there is a tendency to form large Hparticle
clusters residing on large sloping segments. At late times,
these segments show coarsening behavior, ultimately resulting
in the formation of one of three ordered phases, in all of which
Hparticles are segregated from Lparticles. The three phases
differ from each other in the macroscopic shape of the part
of the surface which holds the Lparticles. There are strong
variations in the slow approaches to the respective steady
states coming from differences in the dynamics of coarsening,
distinct from other instances of slow dynamics, for example,
in constrained systems in one dimension [38,39]. Finally, the
transition boundary between disordered and ordered phases
exhibits FDPO [14,40].
B. Summary of main results
In this paper, we use three theoretical constructs to uncover
the dynamic properties of the LH model: a balance condition
which allows us to obtain exact steady states; a configuration-
wise irreducible sequence whose mean length tracks growing
length scales; and a local cross-correlation function which,
surprisingly, is able to capture nonlocal features such as
coarsening. The significance of our approaches and results
extends beyond the model considered in this paper and should
be useful in the study of several other driven systems. Be-
low we discuss our principal results on the evolution and
coarsening properties of the density and tilt fields in the LH
model.
We uncover a subspace in the phase diagram of this model,
where all configurations of the system occur with equal prob-
ability in the steady state. We show that this occurs due to
a general balance condition, namely bunchwise balance,in
which for every configuration, the incoming probability cur-
rent from a bunch (or group) of incoming transitions is exactly
balanced by the outgoing current from another uniquely spec-
ified group. This mechanism generalizes the pairwise balance
mechanism that leads to the determination of the exact steady
state in several driven, diffusive systems [41] including in
the asymmetric simple exclusion process (ASEP), as well as
within a smaller subspace in the LH model [16]. A recent
submission has independently discussed the application of the
bunchwise balance condition to the zero range process and
related systems [42].
We also show that the dynamics of the system can be
described in terms of an irreducible sequence, defined as
follows: any configuration of the LH model can be specified
by the locations and lengths of domains of Land Hparticles,
and separately, of up and down tilts. More succinctly, this is
encoded in a sequence which specifies the locations of walls
which separate domains of Land Hparticles, and of up and
down tilts. On eliminating those walls of one species (either
particles or tilts) which do not enclose a wall of the other, we
arrive at the irreducible sequence (IS). The IS helps to prove
that bunchwise balance indeed holds for the steady state in the
relevant subspace. It also helps to characterize the dynamics,
as the number of elements in the IS provides a quantitative
measure of coarsening during the approach to the steady
state.
In the second half of the paper we focus specifically on
alocal cross-correlation function S defined for each con-
figuration, as well as its counterpart Saveraged over initial
conditions, that is, surprisingly, able to capture the nonlocal
evolution and coarsening properties of the system. This occurs
because Sis directly related to the number of irreducible
sequences. It measures the correlation between the gradient of
one species (say the particles) and the density of the other (the
local slope), though there is a reciprocity in the definition. S
allows us to probe the evolution of the system at all timescales,
ranging from very small (less than a single time step) to
very large (during coarsening, as the system approaches an
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ordered steady state). Interestingly, the cross-correlation func-
tion has a meaning at both local and global levels. At the
local level, in any configuration Sis nonzero at those locations
of the system where dynamic evolution is possible. At the
global level, the averaged correlation Sprovides a precise
measure of the degree of coarsening as the system evolves
toward an ordered state. We provide numerical evidence that
Sis able to distinguish and characterize the several nontrivial
phases that occur in the LH model, through its coarsening
properties.
We derive the exact early-time evolution of Sfor totally
disordered initial conditions, and determine the slope which
describes the early-time linear growth. We use the result to
show that the subspace determined from the bunchwise bal-
ance condition in fact exhausts the set of all states with equal
weights for all configurations. We next discuss a mean field
theory for the evolution of the density and tilt fields. It de-
scribes the evolution of Sat early times surprisingly well, and
reproduces the exact slopes at early times. At later times, this
approximate theory departs from numerical results, reflecting
a buildup of correlations that are ignored in the mean field
treatment.
This paper is organized as follows. In Sec. II we introduce
the well-studied light-heavy (LH) model and discuss known
results as well as different regions in the phase diagram in de-
tail. In Sec. III we describe the bunchwise balance mechanism
that produces an equiprobable steady state in the LH model.
In Sec. IV we introduce the local cross-correlation function
Sand describe its observed properties in various regions of
the phase diagram. In Sec. Vwe derive the exact early-time
behavior of Sup to linear order in time, beginning from a
totally disordered initial configuration. In Sec. VI we derive
a mean field theory for the coupled evolution of the density
and tilt fields in the system, which we use to characterize the
early-time behavior of Sbeyond the linear regime. In Sec. VII
we study the coarsening properties of the system using Sand
provide numerical evidence that such local cross correlations
encode information about the large length scale coarsening
in this system. Finally, we conclude in Sec. VIII and provide
directions for further investigation related to the methods and
results discussed in this paper.
II. LIGHT-HEAVY MODEL
A. Model
The light-heavy (LH) model is a lattice system of hard-
core particles with stochastic dynamics on a one-dimensional
fluctuating landscape [17–19]. There are two sublattices, with
one sublattice containing particles represented by σand the
other containing tilts represented by τ. The particles can be
of two types, light particles ◦(L) or heavy particles •(H).
The tilts can be in one of two states, an up tilt or a
down tilt . Every configuration of tilts constitutes a discrete
surface whose local slopes are given by these tilts. We label
the locations of particles by integers j, and those of tilts by
half-integers j+1
2(see Fig. 1). Since there are two types of
particles and two types of tilts, we may represent the state σj
of particles on each site and the state τj+1
2of the tilts on each
site by Ising variables that take the values ±1. We assign these
FIG. 1. Representations of a configuration of the LH model.
(a) The array of light (◦)andheavy (•) particles occupy integer
locations jof a lattice whereas up ()anddown () tilts occupy
half-integer locations j+1
2. (b) The array of tilts generates a discrete
one-dimensional surface as illustrated.
as follows:
σj=−1
+1
Light ◦
Heavy •;τj+1
2=−1
+1
Up
Down .
(1)
A typical LH configuration can be represented either as
an array of particles and tilts, or as particles residing on the
discrete surface, as shown in Fig. 1. We assume that there are
an even number of Nsys sites on each sub-lattice, with periodic
boundary conditions. We consider configurations with num-
bers of light and heavy particles N◦and N•, respectively, in
a lattice size of Nsys. The numbers of Land Hparticles are
conserved, but may differ. We also have N◦+N•=Nsys.Itis
convenient to define σ0=(N•−N◦)/Nsys, such that the mean
densities of light and heavy particles are given by ρ(◦)=1−σ0
2
and ρ(•)=1+σ0
2, respectively. Similarly, Nand Nrepre-
sent the total number of up and down tilts, respectively. We
further take the numbers of up and down tilts to be equal, thus,
N=N=Nsys /2. In this work we deal only with surfaces
with no overall slope, nevertheless some of our results may
also be useful for surfaces with finite overall slopes.
The update rules for the model couple the dynamics of
particle and tilt sublattices [Fig. 2(a)]. They allow exchanges
between neighboring Land Hparticles. Likewise, exchanges
between neighboring up and down tilts are also allowed,
which take a local hill to a valley, and vice versa. The up-
date rules involve transition probabilities controlled by the
parameters a,b, and b, which introduce biases between the
forward and reverse updates. The parameter amodulates
the exchanges between neighboring Land Hparticles [rules
(i) and (ii) in Fig. 2(a)] depending on the sign of the tilt
between them. In a microscopic time step [t,t+t], the
forward transitions in rules (i) and (ii) occur with a probability
(1
2+a)t, whereas the reverse transitions occur with a proba-
bility ( 1
2−a)t. The parameters band bmodulate exchanges
between neighboring up and down tilts [rules (iii) and (iv)
in Fig. 2(a)], depending on whether Lor Hparticles occur
in-between. Alternatively, the tilt exchange rates embody the
propensities of the particles to push the surface upwards or
downwards, according to the signs of b,b.
043279-3
MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
Disordered
SS
P
P
SS
IP
S
FDPO
F
F
F
P
P
P
S
S
S
S
IPS
b
b
a
F
F
FF
PP
P
S
S
S
S
S
(a) (b)
1
2+a
1
2−a
1
2+b
1
2−b
1
2+b
1
2−b
(i)
(ii)
(iii)
(iv)
(i)
(ii)
(iii)
(iv)
(v)
SPS
IPS
FPS
FDPO
DIS
(c)
FIG. 2. (a) Update rules for the LH model. The first set of rules (i) and (ii) involve the exchange of neighboring particles, and thus generate
particle currents. The forward and backward transitions occur with rates 1
2+aand 1
2−a, respectively. The second set of rules (iii) and (iv)
dictate the interaction of particles with the surface, and correspondingly generate tilt currents across the LH lattice. (b) The three-dimensional
phase diagram of the LH model in the space of update parameters a,b,b. The bunchwise balance condition 2a+b+b=0representsa
plane (shown in red) that lies in the disordered phase (light yellow) of the phase diagram. Although the parameter aranges from 0 to 1
2,the
section for an intermediate value of a=0.3 is depicted as the exterior face in the figure. (c) Schematic plots of the steady-state profiles with H
particles shown in color, for different phases of the LH model. Profiles (i), (ii), and (iii) are the ordered phases SPS, IPS, and FPS, respectively,
with differing extents of phase separation between the tilts, while the particles are always perfectly phase separated. The order-disorder phase
boundary in (iv) exhibits fluctuation-dominated phase ordering (FDPO). Long-range order prevails in FDPO, in contrast to the disordered
regime (v).
It is apparent that reversing the signs of a,b,bis the
same as interchanging the up and down tilts. In this paper,
we assume a>0, which implies that the Hparticles tend to
slide downwards by displacing the Lparticles upwards. The
values of b(and b) can be positive, negative, or zero, and thus
the L(or H) particles may push the surface either upwards for
b>0(orb<0), downwards for b<0(orb>0), or to an
equal extent for b=−b, or not at all for b=0(orb=0).
Several earlier studies of models in different physical con-
texts are included in the LH model as special cases. First,
this model was defined in the context of sedimenting colloidal
crystals [15], in which case the two species represent gradients
of the longitudinal and shear strains. However, only the case
b=b>0 was studied in [15]andshowntogiverisetoan
exceptionally robust sort of phase separation. Second, when
b+b=0, the surface is pushed to an equal extent by L
and Hparticles irrespective of their placements. Thus, the
problem reduces to that of Hparticles sliding down a fluctuat-
ing interface, which itself evolves autonomously according to
KPZ dynamics (if b=−bis nonzero) or Edwards-Wilkinson
dynamics (if b=−b=0) [14,40,43]. Third, the problem of a
single active slider on a fluctuating surface was studied in [44]
as a model of membrane proteins that activate cytoskeletal
growth. This model corresponds to the case of a single H
particle in the LH model.
B. Phase diagram
Figure 2(b) shows that within the three-dimensional pa-
rameter space (a,b,b), the system in steady state exhibits
several ordered and disordered phases. In all the ordered
phases SPS, IPS, and FPS [profiles (i)–(iii) in Fig. 2(c)], the L
and Hparticles are completely phase separated. These consti-
tute pure phases, as typically only Lor Hparticles are present
in the bulk of each phase. However, the extent to which the
up and down tilts are ordered differs from one phase to an-
other, leading to different forms of the height profile of the
surface at the macroscopic level. Further, the phase separated
regions in the tilt profile behave as reservoirs, generating tilt
currents whose uniform flow throughout the system causes the
steady-state landscape to drift downwards collectively. The
drift velocities depend on the magnitudes of tilt current for
the different ordered phases. Below, we briefly summarize the
general features of the different steady-state phases, namely,
the nature of their phase separation and overall movement of
their landscapes.
Strong phase separation (SPS) (b+b>0, band b>0).
In this phase, the Lparticles push the surface upwards, while
the Hparticles push the surface downwards [12,15,18]. Like
the particles, the tilts also cleanly phase separate into two
coexisting ordered regions of up and down tilts [profile (i) in
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LIGHT AND HEAVY PARTICLES ON A FLUCTUATING … PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
Fig. 2(c)]. The steady-state configurations comprise a single
macroscopic tilt valley with Hparticles, and a macroscopic
hill with Lparticles. The complete state is thus near perfectly
ordered, except at the interfaces between ordered regions.
The system is essentially static in steady state, apart from
an exponentially small tilt current ∼exp(−λNsys ), where λ
is a constant which depends on the update parameters. The
relaxation to the steady state in this regime is logarithmically
slow [15,45].
Infinitesimal current with phase separation (IPS) (b+b>
0, bor b=0). Only one species, i.e., either Lor Hparticles
push the surface upward or downward respectively [17–19].
The tilts are separated into three coexisting regions; two of
these regions are perfectly ordered, whereas the third region is
disordered. For the case b=0, the two ordered regions reside
within the Hcluster, forming a single macroscopic valley,
while the disordered region spans the Lcluster [profile (ii)
in Fig. 2(c)]. The behavior of the disordered tilt region can be
mapped to an open chain symmetric exclusion process (SEP)
[46] with up and down tilts always fixed at the two ends.
This is seen easily by identifying the up tilts with particles
and down tilts with holes in the SEP. Consequently, the up-tilt
density in the disordered region varies linearly with a gradient
∼1/Nsys , giving rise to an “infinitesimal” tilt current which
scales as ∼1/Nsys . Since the mean current is uniform in the
steady state, this translates to a downward drift of the system
with an average velocity ∼1/Nsys. For the case b=0, b>0,
the same features follow, except that the Land Hparticles are
interchanged and the system drifts upward.
Finite current with phase separation (FPS) (b+b>0, bor
b<0). Both Land Hparticles push the surface in the same
direction, but to unequal extents [17–19]. The tilt profile is
again made up of three distinct regions. However, in contrast
to IPS, two of these phases are imperfectly ordered, i.e., are
not pure phases, while the third phase is disordered. For the
case b<0, the two ordered phases form a valley beneath
the Hparticles with smaller slopes, because of imperfect or-
dering. The disordered phase resides along with the Lcluster
[profile (iii) in Fig. 2(c)], and maps to an asymmetric simple
exclusion process (ASEP) in the maximal current phase [47].
This induces a uniform tilt current that remains finite in the
thermodynamic limit, and depends on the update parameters.
The finite current causes the system to drift downwards with
a constant velocity. Other than Land Hparticles having inter-
changed, the same features follow for the case b<0.
Fluctuation-dominated phase ordering (FDPO)
(b+b=0). This is the order-disorder separatrix, and
shows fluctuation-dominated phase ordering [profile (iv) in
Fig. 2(c)]. In this regime, both Land Hparticles push the
surface to an equal extent, and thereby cause the surface
to evolve as if it is autonomous with KPZ dynamics [17].
Thus, the surface is completely disordered, and drifts
downwards with a finite rate. The particles behave as passive
scalars, directed by the dynamics of the fluctuating surface.
This results in a statistical state which exhibits FDPO,
in which long-range order coexists with extremely large
fluctuations, leading to a characteristic cusp in the scaled
two-point correlation function. The largest particle clusters
are macroscopic with typical size of order ∼Nsys. However,
these macroscopic clusters reorganize continuously in time.
The properties of FDPO have been studied in detail in
[14,40,43,48,49]. It has been invoked in the study of a variety
of equilibrium and nonequilibrium systems, including active
nematics and vibrated rods [50], actin-stirred membranes
[24], inelastically colliding particles [51], as well as Ising
models with long-range interactions [52].
Disordered phase (b+b<0). In the disordered phases,
the surface-pushing tendencies of the Land Hparticles op-
pose the tendency of the Hparticles to drift downwards
[16]. This does not allow large, ordered structures to form
throughout the system, and both particle and tilt profiles are
completely disordered; the steady state is characterized by the
absence of long-range correlations [profile (v) in Fig. 2(c)].
Kinematic waves propagate across the particle and tilt sublat-
tices in steady state [16], and the nature of their decay changes
across the disordered phase [20], giving rise to several new
dynamical universality classes [32–37].
It has recently been hypothesized [52] that the phase tran-
sition from the disordered phase to an ordered phase in the
LH model, across the FDPO transition locus, is a mixed-order
transition, which shows a discontinuity of the order parameter
as well as a divergent correlation length at the transition. Such
a mixed-order transition occurs across the FDPO locus in a
one-dimensional (1D) Ising model with short- and truncated
long-ranged interactions [52]. In the LH model, recall that
all the ordered phases, FPS, IPS, and SPS, display complete
phase segregation of Land Hparticles, implying maximum
order. In the disordered phase, there is no L-Hsegregation
on large scales, implying that the order parameter vanishes.
Thus, the order parameter shows a strong discontinuity across
the transition. While the divergence of the correlation length
remains to be established, the trends displayed by a local
cross correlation function reported below (Secs. IV and VI)
are consistent with this scenario.
1. Subspaces with exactly known steady states
There are two special subspaces within the phase diagram,
where the steady state of the system can be determined ex-
actly. The first subspace resides within the SPS phase. Here,
the system satisfies the condition of detailed balance. The
steady-state measure is ∼exp (−βH), for a Hamiltonian H
defined with potential energies assigned to each Hparticle, in
proportion to their vertical heights. Detailed balance in SPS
was first established in [15] on the locus b=bwhich ensures
interchange symmetry between σand τ. This was later gener-
alized to the case of variable particle densities in [18]. In this
paper we show there is a second subspace in the disordered
phase which is given by the condition R=2a+b+b=0.
In Sec. III, we establish equiprobability measure in this sub-
space using a new condition known as bunchwise balance,
substantially enlarging the earlier result which was proved on
the locus b=b=−a[16].
III. BUNCHWISE BALANCE
In this section we discuss a region in the phase diagram
of the LH model in which the steady state is described by an
equiprobable measure over all accessible configurations of the
system. We show that this occurs due to a mechanism in which
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the incoming probability currents into a given configuration
from a group of configurations, are balanced by outgoing tran-
sition currents to another group of configurations. We term
this condition “bunchwise balance” and show that it occurs
when
R=2a+b+b=0.(2)
The above condition maps out a plane in the phase diagram of
the LH model, as shown in Fig. 2(b).WeprovethatEq.(2)
is a sufficient condition for this balance to hold, implying
that in steady state, all configurations are sampled with equal
probability. In Sec. V, we also establish this is a necessary
condition for equiprobability of the steady-state measure over
all accessible configurations.
A. Steady-state balance
We begin by analyzing the evolution equation of a general
Markov process, represented as a master equation of the form
d
dt p(c,t)=
c
M(c,c)p(c,t).(3)
The elements of the evolution matrix are given by the micro-
scopic transition rates between configurations, for example,
the rates of particle and tilt updates described in Sec. II.The
matrix elements are then explicitly given by
M(c,c)=rc←cfor c= c,M(c,c)=−
c
rc→c,
(4)
where rc→cis the transition rate from configuration cto
configuration c. Here, rc→cand rc←crepresent the same
transition rate. The outgoing probability current from configu-
ration c→cand the incoming current from c←c are given
by
jc→c=p(c,t)rc→c,jc←c =p(c,t)rc←c .(5)
The net incoming and outgoing probability currents from
every configuration cper unit time are then given by
jout(c,t)=
c
p(c,t)rc→c,jin(c,t)=
c
p(c,t)rc←c .
(6)
In steady state, the net incoming and outgoing probability
currents at any time are equal for every configuration and
hence the probability of occurrence of a configuration p(c,t)
becomes independent of time, i.e., jin (c,t)=jout (c,t). There-
fore, in steady state
c
p(c,t)rc→c=
c
p(c,t)rc←c .(7)
We refer to this condition as steady-state balance, as all tran-
sitions in and out of any configuration cneed to be summed
over in order for the above balance condition to hold.
There are several ways in which the steady-state balance
condition in Eq. (7) can be achieved. We focus on three cases
that occur in the LH model: (i) Detailed balance in which
the forward and reverse probability currents between any two
c
.
.
.
c1
c2
j∈Bin
cj
.
.
.
i∈Bout
.
.
.
c1
c2
ci
.
.
.
.
.
..
.
.
.
.
.
.
.
.
FIG. 3. A schematic illustration of the bunchwise balance mech-
anism. For every configuration c, the incoming probability currents
from a group of configurations {c
j}with j∈Bin (the incoming
bunch), are balanced by outgoing currents to another uniquely iden-
tified group of configurations {c
i}with i∈Bout (the outgoing bunch).
configurations are equal. This is given by
jc→c=jc←c.Detailed
Balance (8)
(ii) Pairwise balance in which the incoming current from one
configuration is balanced by the outgoing current to another
uniquely identified configuration [41]. This is given by
jc→c=jc←c.Pairwise
Balance (9)
Finally, we introduce (iii) bunchwise balance in which in-
coming currents from a group of configurations are balanced
by outgoing currents to another uniquely identified group of
configurations. This mechanism is illustrated in Fig. 3and is
given by the condition
i∈Bout
jc→c
i=
j∈Bin
jc←c
j.Bunchwise
Balance (10)
Above, the sum iis over outgoing transitions belonging to a
“bunch” Bout whereas the sum jis over incoming transitions
belonging to a uniquely identified “bunch” Bin, such that
the probabilities from these bunches are exactly equal. Each
nonzero current jc→c
ioccurs in one and only one outgoing
bunch, and likewise each current jc←c
joccurs once and only
once in an incoming bunch. Each of the balance mechanisms
discussed above appear in the LH model, and have been sum-
marized in Fig. 4.
In some systems, in addition to the above condition, the
microscopic rates also balance as
c
rc→c=
c
rc←c.(11)
This special condition holds in a wide class of systems includ-
ing the LH model in some ranges of the parameter space as we
show below. It is easy to see that Eq. (11) along with Eq. (7)
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ccc
cc
Bunchwise Balance
a=0,b=0,b =0
a=0,b=0,b =0
2a+b+b=0
in the LH Model
in the SEP
Detailed Balance
Pairwise Balance
in the ASE
P
c1
c2
c2
c1
c
FIG. 4. A schematic illustration of the different balance mecha-
nisms that are displayed by the LH model. (i) Detailed balance in
the simple exclusion process (SEP) in which the forward and reverse
probability currents between any two configurations are equal. (ii)
Pairwise balance in the asymmetric simple exclusion process (ASEP)
in which the incoming current from one configuration is balanced
by the outgoing current to another uniquely identified configura-
tion and (iii) bunchwise balance in the light-heavy (LH) model in
which incoming currents from a group of configurations are bal-
anced by outgoing currents to another uniquely identified group of
configurations.
yield the time-independent solution p(c,t)=1
Ncwhere Nc
represents the total number of configurations, i.e., all config-
urations occur with equal probability. Since the steady state
of the Markovian dynamics governed by Eqs. (3) and (4)is
unique, at large times the system converges to an equiprobable
measure over configurations.
B. Interfaces and bends in the LH model
Consider a configuration cin the LH model, represented as
an ordered list of occupied and unoccupied sites as well as up
and down tilts on the bonds
c≡···◦◦•••◦···.(12)
The evolution rules of the LH model described in Sec. II only
allow local updates at the interfaces that separate occupied
and unoccupied sites, as well as at bends that separate regions
of positive and negative tilts. In order to study the dynamics
of the system, it is therefore convenient to parametrize the
configurations in the LH model in terms of variables that
describe these interfaces and bends. In order to parametrize
σjon the sites j, we introduce interface variables Ij+1
2on the
bonds j+1
2of the lattice. The state of the interface at each
bond is determined by the density variables σjand σj+1on
the adjacent sites as
Ij+1
2=(σj+1−σj)/2.(13)
We note that the particle configurations are now represented
by variables that can take on three values 0,±1 at every bond
in comparison with the site occupation variables which were
represented by two states −1 or 1. However, the interface
variables satisfy constraints that ensure the one-to-one map-
ping between the two variables, namely, that any interface
can only be followed by a zero interface or an interface with
opposite sign. The periodic boundary conditions of the system
ensure that there are equal numbers of positive and negative
interfaces. For simplicity of presentation, we represent the
three states of an interface as
Ij+1
2=⎧
⎨
⎩
1≡(
0≡.
−1≡)
(14)
Similarly, in order to parametrize the tilts τj+1
2,weintro-
duce bend variables Bjon the sites jof the lattice. The state
of the bend at each site is determined by the tilt variables τj−1
2
and τj+1
2on the adjacent bonds as
Bj=τj−1
2−τj+1
22.(15)
Once again, these new variables can take on three values 0,±1
at every site, with a constraint that a bend can only be followed
by a zero bend or a bend with the opposite sign. The periodic
boundary conditions of the system ensure that there are equal
numbers of positive and negative bends. Again, for simplicity
of presentation, we represent the three states of a bend at every
site as
Bj=⎧
⎨
⎩
1≡
0≡.
−1≡
(16)
Above, we have represented both interfaces and bends with
values of 0 with the same “.” symbol, since no local updates
are possible at these positions. We note that the above map-
ping from a configuration of densities and tilts to interfaces
and bends {{σ},{τ}}→{{I},{B}}is one to one and invert-
ible. In terms of these new interface and bend variables, the
configuration in Eq. (12) simplifies to
c≡......(.....)... . (17)
Conveniently, the only transitions in and out of this con-
figuration coccur through the updates of these “brackets”
representing the interfaces and bends that uniquely specify c.
The particle updates described in Sec. II convert the interfaces
at a bond to a triad centered on the same bond as (→()
(and ) →)(), respectively. Similarly, the tilt updates convert
the bends at a site to a triad centered on the same site as
→ and →, respectively. The motion of interfaces
and bends through the system thus proceeds through their
creation and annihilation at adjacent locations, with open and
closed brackets of the same type annihilating if they reach the
same position. We also note that the transition rates for the
interfaces depend on the enclosing bends since, depending on
the local slope, particles are biased to move either leftwards or
rightwards. Similarly, depending on whether a site contains a
light or heavy particle, the update of bends of a particular sign
are favored over the other.
C. Pairwise balance in the ASEP
Pairwise balance is a condition found to hold in the steady
state of several driven diffusive systems [41], including the
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MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
locus b=b=−ain the LH model [16]. It can be used to
exactly determine the steady state of a well-known nonequi-
librium model describing the biased motion of particles in one
dimension, namely, the asymmetric simple exclusion process
(ASEP). The ASEP is a special case of the LH model obtained
when all the tilts point in the same direction. Below we use the
ASEP as an example to illustrate the condition of pairwise
balance, and also to illustrate the bracket notation that we
use in analyzing the more general LH model. When all the
tilts in the LH model are equal (all positive or all negative),
and the particles do not influence the surface evolution (i.e.,
b=b=0), the dynamics of the particles in the LH model can
be mapped onto the ASEP, with right and left hopping rates
p=1
2+aand q=1
2−a, respectively. The ASEP displays
an equiprobable steady state, satisfying Eq. (11) in two ways
as we show below: detailed balance when p=qwhen the
model reduces to the simple exclusion process (SEP), and
pairwise balance when p= q.
We begin with the simplest case when there is no left-right
bias, i.e., a=0,b=0,b=0. In this case the transition
rates between any two configurations cand care given by
rc→c=rc←c=1
2.(18)
These rates trivially satisfy Eq. (11), implying an
equiprobable steady state. This steady-state solution along
with the above rates then implies the detailed balance
condition given in Eq. (8).
We next consider the case of the ASEP with a bias, i.e.,
a= 0,b=0,b=0. In this case, the forward and reverse
rates between any two configurations are biased as 1
2±a,
violating detailed balance. However, the microscopic rates are
still balanced as we show below. Consider a general config-
uration of the ASEP, described by only density variables {σ}
and correspondingly only by interface variables {I}as
c≡... (...)..(.....)... . (19)
To uniquely classify all the transitions of this configuration,
we begin at the origin and look at the enclosing interfaces.
Suppose moving rightwards from the origin the first bracket
encountered is (, we move forward until the next ) is en-
countered. Whereas if the first bracket encountered is ), we
move backward until the next ( is encountered. In this manner,
every transition from this configuration, parametrized by the
interfaces, has a unique corresponding pair. An ( interface, as
shown in Fig. 5, represents a hole-particle interface which can
be updated resulting in a configuration cwith rate rc→c=
1
2−a. The reverse transition occurs with a rate rc←c=1
2+a.
In order to find a corresponding unique transition with the
same rate, we consider the corresponding ) interface in the
pair. This represents a particle-hole interface which can be
updated, with rate rc→c =1
2+a. Crucially, the reverse tran-
sition occurs with a rate rc←c =1
2−abalancing the original
transition. Similarly, the reverse transitions corresponding to
this pair are balanced as well. Therefore, the transition rates
within each such () pair are balanced.
This procedure can be extended to every interface in the
configuration c, therefore, corresponding to any outgoing tran-
sition from configuration c→c, one can identify a unique
incoming transition c←c with the same microscopic rate,
1
2+a
1
2−a
1
2+a1
2−a
c≡... ...
c←c c →c
c←cc →c
()
............. .....
FIG. 5. A schematic illustration of the pairwise balance condi-
tion in the ASEP model. The black arrows represent transitions out of
the configuration c, while the green arrows represent transitions into
the configuration. In this case, each transition to a configuration cis
balanced by a corresponding incoming transition from configuration
c with the same rate. The configuration in terms of interfaces is
represented below. Transitions between interface pairs () are pairwise
balanced.
i.e.,
rc→c=rc←c.(20)
Therefore, in this case as well Eq. (11) is satisfied, implying
an equiprobable steady state. This steady-state solution along
with the above rates then implies the pairwise balance condi-
tion for probability currents as given in Eq. (9).
D. Bunchwise balance in the LH model
Finally, we consider the case of the LH model with a=
0,b= 0,b= 0. As opposed to the ASEP, in this case the
transition rates for the interfaces are affected by the enclosing
bends. For interfaces that are enclosed by (i.e., lie within
consecutive bends) of the type , the transition rates are given
by
(21)
whereas, when an interface is enclosed by bends, the tran-
sition rates are given by
(22)
Similarly, for bends enclosed by interfaces of the type (), the
transition rates are given by
(23)
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LIGHT AND HEAVY PARTICLES ON A FLUCTUATING … PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
1
2−a
c≡...
c→c1
c←c1
1
2+b
1
2−a
c→c2
c←c2
1
2+b
(...... ......
Pairwise
)
......
FIG. 6. A schematic illustration of the bunchwise balance con-
dition in the LH model. The black arrows represent transitions out
of the configuration c, while the green arrows represent transitions
into the configuration. In this case, the transition rates to the con-
figurations c
1and c
2are balanced by incoming transitions from c
1
and c
2when the condition 2a+b+bis satisfied. The configuration
in terms of interfaces and bends is represented below. Transitions
between uninterrupted pairs of bends (or interfaces) are pairwise
balanced. This sequence is an irreducible sequence of type +≡().
whereas, when a bend is enclosed by )( interfaces, the transi-
tion rates are given by
(24)
We note that the above rules apply even when the enclosing
brackets are adjacent to the bracket being updated. In this case,
part of the resulting triad falls outside the enclosing brackets.
E. Irreducible sequences
Consider a general configuration of the LH model as shown
in Fig. 6, described by both density and tilt variables {{σ},{τ}}
and correspondingly by both interface and bend variables
{{I},{B}}as
c≡... (...()...)... . (25)
In order to classify the transitions of this configuration into
groups, we begin at the origin and look at the enclosing
interfaces and bends. Suppose traversing rightwards, the first
bracket encountered is , we move forward until the next is
encountered. Whereas if the first bracket encountered is ,we
move backward until the next is encountered. Any bracket
encountered during this traversal is added to the group. For ex-
ample, it is possible to encounter an interface: ( or ). We apply
this procedure recursively moving forward for a ( bracket, and
backward for ) brackets, adding any interface or bend brackets
encountered into the group. This procedure terminates when
all the interface and bend brackets that are encountered have
been paired, i.e. are closed by a corresponding bracket within
the group. We note that these groups can contain sequences of
any length such as ()() ...()()since once an open bracket
is encountered, the group of transitions is not closed until the
corresponding is encountered. Similarly, the rest of the inter-
faces and bends in the configuration can be uniquely classified
into groups in this manner. Once again, this classification is
unique, as every bracket belongs to the smallest group of
closed brackets that contain it.
We next show that the transition rates into and out
of each such group are balanced when 2a+b+b=0.
This occurs via two mechanisms: pairwise balance and
bunchwise balance. We begin by noticing that any uninter-
rupted pair of interfaces (...) or bends ...obeys pairwise
balance within itself. The proof for both these cases pro-
ceeds exactly as for the ASEP described above. Therefore,
within each group, we can balance the transitions between
these reducible pairs for any value of the transition rates
a,b, and b. Next, consider a general group of transitions
represented by its sequence of interfaces and bends, for
example, ...(......()()() ......). We can reduce this
sequence by pairing and eliminating the brackets representing
transitions that satisfy pairwise balance, i.e., uninterrupted
sequences of () and . Proceeding in this manner, we arrive
at an irreducible sequence ...(......). It is easy to see that
there are only two types of irreducible sequences that arise,
which we label +and −:
+≡(......)...,−≡...(......).(26)
These represent groups of transitions that do not balance pair-
wise within themselves. We show below that these transitions
are instead balanced as a group of four. Let us focus on the
first case +≡(......)..., which is the case represented
in Fig. 6. The proof for the −case proceeds analogously.
Using Eq. (22), the transition out of the first bracket c→c
1
occurs with a rate 1
2−a.UsingEq.(23), the reverse transition
corresponding to the second bracket c←c
1occurs with a rate
1
2+b.UsingEq.(21), the transition out of the third bracket
c→c
2occurs with a rate 1
2−a. Finally, using Eq. (24), the
reverse transition corresponding to the last bracket c←c
2
occurs with a rate 1
2+b. The net incoming and outgoing
transition rates between these pairs of forward and reverse
transitions are given by
rc←c
1+rc←c
2−rc→c
1−rc→c
2=2a+b+b.(27)
Thus, the net incoming transition rate from these four tran-
sitions is R=2a+b+b. It is easy to show that the four
corresponding reverse transitions also yield a net incom-
ing transition rate of R. Since reducible pairs of brackets
are pairwise balanced, these two groups of four represent
all the unbalanced transitions associated with an irreducible
sequence. Therefore, the net incoming transition rate for a
sequence of type +is 2R. Proceeding analogously for the type
−sequences, the net incoming rate can be shown to be −2R.
We can interpret this asymmetry as follows, starting from a
configuration with equal numbers of +and −irreducible se-
quences, for R>0 the system evolves toward a state in which
the number of +sequences is larger than the −sequences
in the steady state. Thus, when R>0, the +sequences are
“favored,” whereas the −sequences are “unfavored.” This
scenario is reversed when R<0. Importantly, when the con-
dition R=2a+b+b=0 is satisfied, the transition rates for
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MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
both types of irreducible sequences balance as
rc→c
1+rc→c
2=rc←c
1+rc←c
2.(28)
Therefore, when R=2a+b+b=0, the LH model satisfies
the rate balance in Eq. (11) and thus all configurations occur
with equal probability in steady state. Therefore, this leads to
the bunchwise balance condition for the probability currents
in steady state
jc→c
1+jc→c
2=jc←c
1+jc←c
2,(29)
which is a special case of the general bunchwise balance
condition given in Eq. (10), with bunches containing two
configurations each. Finally, it is also easy to show that when
b=b=−a, the transitions within this group obey pairwise
balance, as balance can now be achieved by pairing one of the
interfaces and one of the bends that specify each irreducible
string.
The equiprobable steady state when bunchwise balance is
satisfied leads to a product measure state in the thermody-
namic limit. Product measure implies that local configurations
occur with a probability independent of their neighborhood in
the steady state. The bunchwise balance condition in Eq. (2)
exhausts all possibilities in the LH parameter space for prod-
uct measure in the steady state. This is confirmed by an
exact calculation in Sec. V, where we have enumerated every
possible lattice transition, starting from a totally disordered
(product measure) initial condition. We point out that a re-
cent submission uses bunchwise balance (called multibalance
in that paper) to find the steady states of several lattice
models [42].
IV. LOCAL CROSS-CORRELATION FUNCTION
In this section, we introduce the local cross-correlation
function Sand its disorder-averaged version S, which mea-
sures the average correlation across the lattice between
particle (or tilt) sites with the gradients of their neighboring
tilt (or particle) sites. Although it is a local quantity, Scaptures
important aspects of the early-time dynamics as well as later
time dynamics, including coarsening toward ordered steady-
state phases.
In terms of the discrete space-lattice variables for parti-
cles and tilts, the local cross correlation between the particle
and tilt sublattices can be expressed in two equivalent forms:
Sσ∇τ≡Sτ∇σ. This has been discussed previously in [53]. In
any configuration {σj,τj+1
2}, we define the two forms at any
time tas
Sσ∇τ(t)=
Nsys
j=1
1
2τj−1
2−τj+1
2σj,
Sτ∇σ(t)=
Nsys
j=1
1
2(σj+1−σj)τj+1
2.(30)
The form Sσ∇τmeasures the cross correlation between triads
comprised by an individual particle and its two neighboring
tilts. Likewise, the other form Sτ∇σmeasures the cross cor-
relation between an individual tilt and its two neighboring
particles. The two forms are always equal for any lattice
configuration with periodic boundary conditions. This can
be easily seen by reordering the pairs of constituent σ−τ
product terms typified in Eq. (30). We represent either Sσ∇τ
or Sτ∇σat time tby S≡S(t). We next define the disorder-
averaged correlation function S≡S(t)as
S(t)=1
Nsys Sσ∇τ(t)= 1
Nsys Sτ∇σ(t),(31)
where the average ...is performed over multiple evolu-
tions starting from different disordered initial configurations.
Across the lattice at any instant in time, Sσ∇τcounts triads
•and ◦as +1, and counts triads ◦and •as
−1. Likewise, the triads ◦•and •◦are counted by Sτ∇σ
as +1, and triads •◦and ◦•are counted as −1. When the
lattice update parameters a,b, and bare all positive, triads
counted by the two forms as +1 are kinetically “favored,”
and occur more frequently; whereas triads counted as −1are
kinetically “unfavored,” occurring relatively less frequently.
Therefore, for a,b,b>0, S(t) evolves as a positive-valued
function; its magnitude can be interpreted as a measure of the
relative local “satisfaction” of the favored triads at time t.On
the other hand, if either or both b,bare negative, there is a
possibility that S(t) can evolve as a negative-valued function
of tsince the triads counted as −1 will now occur in relatively
greater numbers.
At this point, it is important to note an interesting con-
nection between S, and the irreducible sequences defined in
Sec. III.Wehave
S=2N+−2N−,(32)
where N+and N−denote the number of irreducible sequences
of types +≡()and −≡(), respectively. This can be
shown easily as reducible pairs do not contribute to S.
The form of the function S(t) captures important aspects of
the evolution from the initial state to the final steady state. In
this paper, we choose the initial state to be totally disordered;
the final steady state can be ordered, disordered, or the separa-
trix between ordered and disordered, depending on the values
of band b,fora>0 as detailed in Sec. II. For the totally
disordered initial state, we have S(0) =0 since all forms of
triads (favored or unfavored) are equally likely. Below we
provide an overview of the behavior of S(t) in the ordered and
disordered regimes, bringing out also the distinctive features
at early and late times.
Ordered regime. As discussed in Sec. II, there are several
types of ordered phases. The light and heavy particles in all
these ordered steady states are phase separated, and typical
configurations have large stretches with “inactive” triads such
as ••or ◦, which are constituted by pairs of identical
particles or tilts, and do not contribute to S(t). Typically, there
are only a small number A∼O(1) of active triads (that can
be updated), and these occur at the cluster boundaries. Such
triads are the only contributors to the dynamics of the sys-
tem, and also to S(t). For all ordered phases of a finite-sized
system, the saturation value is Sss ≈A/Nsys , which vanishes
in the thermodynamic limit Nsys →∞. Below, we indepen-
dently discuss the distinct steady-state phases (SPS, IPS, and
FPS) within the ordered regime.
SPS. The function S(t) rises linearly with a positive slope
at early time, until about t0.5 (plot 1 in Fig. 7). The linear
growth results from the swift settling of particles and tilts
043279-10
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FIG. 7. Early-time evolution of Sleading to the ordered, dis-
ordered, and FDPO steady-state phases, beginning from an initial
ensemble of random particle and tilt configurations (totally disor-
dered state). Plots 1,2,and3lead to SPS, IPS, and FPS phases
in steady state, respectively. Plot 4is the order-disorder boundary
(FDPO), and plots 5,6,7,and8correspond to the disordered
phase. In Plot 7,S(t) does not evolve suggesting that the com-
pletely disordered state persists up to and includes the steady
state. In this case, parameters a,b,bsatisfy the relation 2a+
b+b=0, derived using the condition of bunchwise balance in
Sec. III. Inset: Expanded portion showing the growth of S(t)at
small times. The system size used here is Nsys =512. The param-
eter values used for (a,b,b) are, respectively, 1(0.4,0.2,0.2), 2
(0.4,0.2,0), 3(0.4,0.2,−0.1), 4(0.4,0.2,−0.2), 5(0.4,0.2,−0.3),
6(0.3,−0.1,−0.1), 7(0.3,−0.3,−0.3), and 8(0.3,−0.5,−0.5).
into locally satisfied triads, momentarily unhindered by exclu-
sion interactions. Over time, the effects of exclusion become
more prominent, and compete with the fulfillment of local
satisfaction, at which point the linear profile of Sflattens to
attain a broad maximum. At later times t>100, exclusion
effects promote the formation of quasistable structures across
the lattice, whose slow relaxation, i.e., coarsening behavior
toward a steady state, now dominates the dynamics. As will
be discussed further in Sec. VII,Sis a local operator which is
a global counter of these coarsening structures. We see from
plot 1 in Fig. 8that Sdecays as ∼1/log tat later times,
reflecting the slow coarsening process, which proceeds by
activation in SPS. In the steady state of a finite-sized system,
the number of active triads is A=2, constituted by •
and ◦. Hence, we expect the SPS profile to saturate in
the steady state at Sss 2/Nsys . Due to the slow logarithmic
relaxation, however, we do not observe saturation over the
timescales shown in Fig. 8.
IPS.Atshorttimes,Sbehaves as in SPS, growing linearly
and reaching a maximum (plot 2 in Fig. 7). Following the
maximum, S(t) decays as a power law ∼t−φwith φ0.5
(plot 2 in Fig. 8). For a finite-sized lattice, we observe that
S(t) saturates to a constant, finite value Sss ≈A/Nsys , with
A2.
FPS. In this phase Sbehaves as in the IPS, growing linearly
at short times (plot 3 in Fig. 7), and decaying as a power law
with φ0.5(plot3inFig.8). In steady state, the saturation
value of S(t) is larger than in IPS: Sss ≈A/Nsys , with A2.5.
FIG. 8. Comparison of the early and late-time behavior of Sas
the system evolves toward different steady-state phases beginning
from a random initial state. The parameter values chosen here are
the same as in plots 1–5in Fig. 7. The plots 1,2,3,and4lead
to SPS, IPS, FPS, and FDPO phases, respectively. The different
features shown by S(t) have been labeled as (a) early rise, (b) broad
maximum, (c) decay during coarsening, and (d) saturation at steady
state. At late times, the system coarsens very slowly for SPS leading
to S(t)∼1/log t, whereas for FPS and IPS, the coarsening proceeds
faster and S(t) decays as a power law ∼t−φwith φ0.5. For FDPO,
S(t) decays with φ0.11. In the cases of IPS, FPS, and FDPO, the
steady state is reached, and S(t) reaches a saturation value Sss,which
decreases as the system size increases. In the case of SPS, the decay
of S(t) is so slow that the steady state is not reached in the time of
the simulation. In plot 5,S(t) evolves to a disordered steady state
which saturates at a finite value at long times. The system size used
here is Nsys =512.
FDPO. This surface is the separatrix between the ordered
and disordered phases in the parameter space of the system.
Here Sgrows linearly and attains a maximum (plot 4 in Fig. 7)
as in the ordered phases discussed above. Further, S(t)shows
a slow decay with φ0.11 (plot 4 in Fig. 8) to a steady state
with fluctuating long-range order [14,40,43,49]. For a finite
system, the number of active triads in steady state depends
on the system size: A∼N1−μ
sys . The saturation value is thus
Sss ≈A/Nsys ∼N−μ
sys with μ0.17, which tends to zero in
the thermodynamic limit.
Disordered regime. Initially, S(t) grows linearly with t,
but the slope can be positive, negative, or zero (plots 5–8 in
Fig. 7), depending on the sign of the combination 2a+b+b.
Further, in this regime the extrema are less prominent since
there is no coarsening. Particularly significant is the zero slope
case in plot 7 of Fig. 7, which corresponds to dS/dt =0,
implying that S(t) does not evolve. This suggests that the
steady state is completely disordered if R=2a+b+bis
zero, as was proved in Sec. III using the condition of bunch-
wise balance. When Ris nonzero, Srelaxes to a constant
steady state value Sss (plot 5 in Fig. 8). Recall that in the
disordered phase, the settling tendency of an Hparticle in a
local valley or an Lparticle on a hill (at a rate set by a)is
opposed by the unsettling of the valley or hill as soon as the
particle arrives (set by −bor −b). Qualitatively, the value
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MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
of Sss is a measure of settling: Sss >0forR>0 indicates
that the majority of Hand Lparticles find themselves well
settled; likewise, for R<0 we observe Sss <0, implying that
the majority are unsettled in this case. Close to R=0, we have
checked that the value of Sss is proportional to R. However,
it varies in a strongly nonlinear manner and vanishes contin-
uously as parameter values approach the transition (FDPO)
locus. Further, the time taken to reach the constant value
increases and appears to diverge as the locus of transitions
is approached. This behavior is consistent with the hypothesis
of a diverging correlation length at a mixed-order transition,
as discussed in Sec. II.
V. EXACT CALCULATION OF EARLY-TIME
SLOPE OF S(t)
In this section, we derive the exact early-time behavior for
the evolution of Swithin a short time interval [0,t]. We
perform this computation starting from a totally disordered
initial state. In this disordered state, the occupation probability
of each site is independent of the others, with only an overall
density and tilt imposed. For large systems this implies that
the particle and tilt densities are statistically homogeneous
across the lattice. This allows us to calculate the early-time
behavior exactly, by considering transitions at early time to
be independent of each other. Finally, we consider the con-
tributions to Sfrom every possible transition out of a given
configuration, and perform a disorder average over all initial
conditions.
It is easy to show from the definition in Eq. (31) that
starting with a totally disordered initial configuration at t=0,
we have S(t)=0. For a given configuration c, the change in
Sup to first order in time tis given by
S=
T
pc(T)ST,(33)
where pc(T) represents the probability of occurrence of a
particular transition T in the configuration c, which we label
by the state of the triad involved in the update. Note that
these transitions T only involve triads that can be updated. The
sum is over all possible transitions out of the configuration.
This form assumes that the initial updates are uncorrelated
as they occur at a sufficient distance away from each other
and therefore do not have any effect on each other. As the
probability pc(T) ∝rTt, where rTis the microscopic rate of
the transition, the probability of occurrence of two transitions
within a short distance (3) of each other leads to a higher-
order term of O(t2/Nsys ).
To compute the change in the disorder averaged Sto lowest
order in time, we need to consider the neighborhood of each
triad that is updated. Since three sites are involved in every
update, the sites immediately adjacent to the triad are also
involved in the computation of S, i.e., we need to consider a
quintet associated with each update. We therefore consider all
transitions associated with five consecutive sites in the system,
and perform an average over all initial conditions. The change
in Scan then be written in the form
S=1
Nsys
Tpc(T)ST= 1
Nsys
T
p(T)ST,(34)
where STrepresents the average change in Sfor a given
transition T over all possible initial conditions of the system.
Above, we have used the fact that for configurations drawn
from the totally disordered state, the transition probabilities
are independent of the configuration with pc(T) ≡p(T). Next,
we enumerate the contributions to Sfrom all possible tran-
sitions. The sum over these contributions yields the exact
early-time behavior of S.
As an example, we consider the contribution to Sfrom a
transition involving a particle exchange across a tilt, labeled
as T ≡•◦−→ ◦•, which occurs at a rate ( 1
2+a). We
note that for any configuration c, the state of the triad and its
position uniquely specify the transition, and we have dropped
the position indices for brevity. The probability of occurrence
of such a triad in a disordered initial state can be computed
from the individual occupation probabilities of each member
of the triad. For a disordered configuration, we have p(•◦)=
p(•)p()p(◦). Next, the individual probabilities are given by
their mean densities in the disordered (product measure) state
p(◦)=1−σ0
2,p(•)=1+σ0
2,p()=1
2.(35)
Since this transition can occur anywhere in the system, the
probability that this transition occurs in the first time step
[0,t]is
p(T) =Nsys1−σ2
0
81
2+at.(36)
Next, we consider the possible changes in Sthat such a tran-
sition can cause. To do this, we need to consider the state
of the system on the quintet associated with the transition
site. It is easy to see that there are only three possibilities
that produce a nonzero change to Sfor such a transition,
namely, (i) •◦, (ii) •◦, and (iii) •◦.The
contributions from each of these cases to Sare (i) +4, (ii) +2,
and (iii) +2, respectively. Summing over all three cases we
arrive at the contribution to S=S/Nsys from this transition
T≡•◦, labeled as (1)
S(1) =1
2+a1−σ2
0
4t,(37)
as stated in the first row of Table I. Next, all the transitions that
can occur in the system can be treated in the same manner.
There are eight possible states of a given triad, centered either
on a particle site or a tilt site. The contributions from each
of these cases have been summarized in Table I. Finally,
summing over the contributions from all possible transitions,
we arrive at the following exact behavior of Sat early time:
S=1−σ2
0
2(2a+b+b)t.(38)
We note that the slope of dS/dt has the same factor R=
(2a+b+b) that appears in the bunchwise balance con-
dition, and therefore when the system exhibits bunchwise
balance, the correlation Sdoes not evolve in time. In Fig. 9
we display the early-time behavior of Sobtained from simu-
lations for various densities and microscopic rates in the LH
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LIGHT AND HEAVY PARTICLES ON A FLUCTUATING … PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
TABLE I. The various possible transitions associated with triads
centered on a tilt site or a particle site, along with their respective
contributions to the early-time slope of dS/dt.
Transition (T) Transition rate Contribution to S/t
(1) •◦−→ ◦•1
2+a+1
2+a1−σ2
0
4
(2) ◦•−→ •◦1
2−a−1
2−a1−σ2
0
4
(3) ◦•−→ •◦1
2+a+1
2+a1−σ2
0
4
(4) •◦−→ ◦•1
2−a−1
2−a1−σ2
0
4
(5) ◦−→ ◦1
2−b−1
2−b1−σ2
0
4
(6) ◦−→ ◦1
2+b+1
2+b1−σ2
0
4
(7) •−→ •1
2−b−1
2−b1−σ2
0
4
(8) •−→ •1
2+b+1
2+b1−σ2
0
4
model, displaying a linear rise and collapse consistent with
Eq. (38).
Finally, we present an argument to show that R=2a+
b+b=0 is a necessary and sufficient condition for product
measure in the steady state. We have shown in Sec. III that
R=0 implies product measure through bunchwise balance.
To prove that product measure implies R=0, we assume the
contrary, i.e., that Ris nonzero. However, we have derived the
exact evolution of Sstarting from a product measure initial
condition in Eq. (38). We therefore have dS
dt ∝R, which is
nonzero. Thus, Smust change from its initial value of 0.
But this is not possible if the state is product measure, as
product-measure states have S=0. Therefore, the condition
R=0 implies product measure and the locus of the bunchwise
balance condition exhausts all possibilities in the LH parame-
ter space for product measure in the steady state.
VI. EARLY-TIME EVOLUTION OF SFROM
MEAN FIELD THEORY
In this section, we study the time evolution of S(t) within
a linearized mean field approximation which neglects correla-
tions between different sites. We will see that it describes the
early-time behavior very well, while at late times, it displays
instabilities that identify the passage to ordered states.
Let us define fluctuation variables δσj=σj−σ0and
δτj+1
2=τj+1
2−τ0for the particles and tilts, respectively,
about their mean densities σ0,τ0, further restricting ourselves
to the case of an equal number of up and down tilts (τ0=0).
In terms of δσ and δτ, the function S(t) defined in Eq. (31)
FIG. 9. Collapse of the early-time evolution of S(t)/K0for dif-
ferent values of parameters a,b,band particle densities ρ(•)=
1+σ0
2. Here, K0=1−σ2
0
2(2a+b+b) is the exact early-time slope of
dS/dt derived in Eq. (38). The system size used here is Nsys =512.
can be expressed as
S(t)=1
Nsys
Nsys
j=1
1
2[δσj+1(t)−δσ j(t)]δτj+1
2(t).(39)
Within the mean field approximation, the fluctuation variables
on different lattice sites are taken to be uncorrelated. Thus,
Scan be approximated as 1
Nsys Nsys
j=1
1
2δσj+1−δσjδτj+1/2.
Furthermore, with the expectation that nonlinear effects would
not feature prominently at very early time, we focus on deriv-
ing the evolution of Sfrom the linearized mean field equations
governing δσ and δτ.
In preceding studies [15,16,18,20,53], the LH model has
been analyzed in the continuum, using hydrodynamic mean
field equations describing coarse-grained density and tilt
fields. However, while the continuum approximation may be
justified for large-distance, long-time properties, it fails to
describe the early-time behavior even qualitatively. Hence, we
deal with the linearized mean field equations on a discrete lat-
tice, and find that our results reproduce the principal features
of S(t), observed in simulations at early times, aside from
matching the initial slope which was exactly determined in
Sec. V.
A. Lattice mean field equations
Recalling that the number of heavy particles and the num-
ber of up tilts both are conserved, we write the following
discrete continuity equations:
∂tσj=Jσ(j−1,j)−Jσ(j,j+1),∂
tτj+1
2=Jτj−1
2,j+1
2−Jτj+1
2,j+3
2.(40)
Here, the terms Jσ,τ(i,i+1) represent the resultant particle or tilt currents from ito i+1, where istands for jor j+1
2.The
currents Jσ,τ can be calculated within the mean field approximation. We have
Jσ(j,j+1)=a
2τj+1
2{(1+σj)(1−σj+1)+(1+σj+1)(1−σj)}+νσ
2(σj−σj+1)+ησ
j(t),
Jτj+1
2,j−1
2=1+σj
2(b+b)−b1+τj+1
21−τj−1
2+1+τj−1
21−τj+1
2+ντ
2τj+1
2−τj−1
2+ητ
j+1
2
(t),
(41)
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MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
where the terms a
2τj+1
2{...}and [ 1+σj
2(b+b)−b]{...}are
the “systematic” contributions to the currents originating from
the update rules. These rules also generate diffusive terms
νσ
2(σj−σj+1) and ντ
2(τj+1
2−τj−1
2); their coefficients νσand
ντare proportional to the frequency of particle and tilt updates
at every time step. Finally, the currents Jσ,τ also include phe-
nomenologically added noise terms ησand ητ. We consider
the noise terms to be delta correlated with the correla-
tors ησ
j(t)ησ
j(t)=Dσδj,jδ(t−t) and ητ
j+1
2
(t)ητ
j+1
2
(t)=
Dτδj,jδ(t−t), where Dσand Dτare the respective strengths.
We insert the current expressions Jσ,τ in Eq. (41) into the
continuity equations for σand τ[Eq. (40)], by retaining only
the linear terms from the systematic parts of Jσ,τ . Reexpress-
ing the resulting equations in terms of fluctuation variables
δσ and δτ, we arrive at the following linearized, mean field
equations:
∂tδσj=a1−σ2
0δτj−1
2−δτj+1
2
+νσ
2{2δσj−δσ j+1−δσj−1}+fσ
j(t) (42)
and
∂tδτj+1
2=−
b+b
2{δσj−δσ j+1}
+ντ
22δτj+1
2−δτj+3
2−δτj−1
2+fτ
j+1
2
(t),(43)
where fσand fτare negative discrete gradients of the noise
terms: fσ
j=ηρ
j−1−ηρ
jand fτ
j+1
2=ητ
j−1
2−ητ
j+1
2
.Italsoproves
expedient to define height fields for particles {hσ
j}and tilts
{hτ
j+1
2}as follows:
hσ
j=
j
j=1
σj,hτ
j+1
2=
j
j=1
τj+1
2.(44)
Evidently, we have σj=hσ
j−hσ
j−1and τj+1
2=hτ
j+1
2−hτ
j−1
2
.
The corresponding relations in discrete Fourier space are
δσk(t)=
hσ
k(t)(1 −eik ),
δτk(t)=
hτ
k(t)(1 −eik ),(45)
where we have defined the Fourier variables as
δσk=Nsys
j=1eik j δσjand
δτk=Nsys
j=1eik(j+1
2)δτj+1
2;
hσ
k=Nsys
j=1eik j hσ
jand
hτ
k=Nsys
j=1eik(j+1
2)hτ
j+1
2
where
k=2π
Nsys m(m∈Z).
B. Solving the linearized mean field equations
The coupled, linearized equations [Eqs. (42) and (43)] can
be solved by going to Fourier space. We write them as a matrix
equation in the following form:
∂t
δσk
δτk=M
δσk
δτk+
fσ
k
fτ
k,(46)
where the diagonal and off-diagonal elements of matrix M
involve the diffusive and drift terms, respectively,
M=2νσsin2k
2i2a1−σ2
0sin k
2
−i(b+b)sin k
22ντsin2k
2.(47)
It is straightforward to find the eigenmodes
Pkand
Qkof
M, and their corresponding eigenvalues λ±
k. If the update fre-
quencies are equal for particles and tilts, as in our simulations,
then νσ=ντ=νis the effective diffusion constant for both
the eigenmodes. The eigenvalue expressions then reduce to
λ±
k=2νsin2k
2±csin k
2,(48)
and the corresponding eigenmodes are
Pk=1
C
δσk+
δτk,
Qk=−1
C
δσk+
δτk.(49)
The constants cand Care related to the off-diagonal elements
of matrix M, and are defined as c=√2a(1 −σ2
0)(b+b)
and C=√2a(1 −σ2
0)/(b+b). Here, Cis a proportionality
constant which governs the relative admixture of
δσkand
δτk
in the eigenmodes.
The constant cchanges in an important way across the
phase boundary, associated with the fact that it is real, imagi-
nary, or zero, according to whether b+bis positive, negative,
or zero. Depending on whether cis real or imaginary, the sys-
tem evolves into an ordered state or a disordered state. In the
ordered regime, cis real and represents an instability; fluctua-
tions in particle and tilt densities grow into instabilities which
are curbed by nonlinearities that are neglected in the linearized
theory. By contrast, in the disordered regime, cis imaginary
and its magnitude represents a speed; the fluctuations do not
grow, but move as mixed-mode kinematic waves with speeds
given by the magnitudes of the eigenvalues, i.e., c. The case
when cis zero (b+b=0) defines the order-disorder phase
boundary. In this case, the linear coupling vanishes in Eq. (43).
Hence, the tilt field evolves autonomously, directing the evo-
lution of the particle field, an example of a passive scalar
problem [54,55].
The evolution equations of the eigenmodes
Pkand
Qkare
∂t
Pk=λ+
k
Pk+
fP
k,∂
t
Qk=λ−
k
Qk+
fQ
k.(50)
In Fourier space, we define the height fields
hP
kand
hQ
kfor the
eigenmodes, in analogy with
hσ,τ
kdefined earlier just below
Eq. (45) for particles and tilts. Integrating the equations for
hP
k
and
hQ
k, we find
hP
k(t)=
hP
k(0)e−λ+
kt+e−λ+
ktt
0
dteλ+
ktηP
k(t),
hQ
k(t)=
hQ
k(0)e−λ−
kt+e−λ−
ktt
0
dteλ−
ktηQ
k(t).(51)
The right-hand side involves the initial condition and an inte-
gration over the noise. For a randomly chosen initial configu-
ration, the initial particle and tilt profiles are delta correlated,
i.e., δσj(0) δσj(0)=Mσδj,jand δτj+1
2(0) δτj+1
2(0)=
Mτδj,j, where Mσ=1−σ2
0and Mτ=1 are the correla-
tion strengths computed for the random initial configuration.
This leads to
Pk(0)
Qk(0)=Nsys MPQ δk,−kwith MPQ =
(1
C2Mσ+Mτ). The correlator (i)
hP
k(0)
hQ
k(0)can be derived
from the eigenmode correlator
Pk(0)
Qk(0), and the correla-
tor (ii) ηP
k(t)ηQ
k(t)comprises a linear combination of the
noise correlators for the fluctuation variables: ησ
j(t)ησ
j(t)
043279-14
LIGHT AND HEAVY PARTICLES ON A FLUCTUATING … PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
and ητ
j+1
2
(t)ητ
j+1
2
(t). The expressions of correlators (i) and
(ii) are, respectively,
hP
k(0)
hQ
k(0)=Nsys MPQ
(1 −eik )(1 −e−ik)δk,−k,(52)
ηP
k(t)ηQ
k(t)=Nsys DPQ δk,−kδ(t−t),(53)
where DPQ =(1
C2Dσ+Dτ).
C. Evolution of S
Within the mean field approximation, the local cross-
correlation function can be expressed in the factorized form
S(t)=1
Nsys Nsys
j=1
1
2δσj+1−δσjδτj+1
2. Keeping terms to lin-
ear order, we may write S(t) in terms of the Fourier variables
δσkand
δτk,wehave
S(t)=1
2N3
sys
Nsys
j=1
π
k=−π
π
k=−π
e−ik j e−ikj
×{e−ik −1}e−ik/2
δσk(t)
δτk(t).(54)
We further write the time-dependent correlator
δσk(t)
δτk(t)in terms of the height fields
hP,Q
k(t). Noting
that the terms
hP
k
hP
−kand
hQ
k
hQ
−kdo not contribute to S(t)
since they are even functions of k,wearriveatthefollowing
expression:
S(t)=C
8N2
sys
π
k=−π
(eik/2−e−ik/2)3
×
hP
k(t)
hQ
−k(t)−
hQ
k(t)
hP
−k(t).(55)
Expressing
hP,Q
k(t) in terms of the initial condition and
noise evolutions through Eq. (51), we see that S(t) can be
written as the sum of two terms S1(t)+S2(t). A detailed
derivation of these expressions is presented in Appendix. Each
of the two terms has a distinct physical origin. S1[Eq. (A2)]
involves the random initial configuration through the correla-
tor (i)
hP
k(0)
hQ
−k(0), while S2[Eq. (A3)] involves the noise
through the correlator (ii) ηP
k(t)ηQ
−k(t). In the continuum
limit, we find that S(t) is a linear combination of two integrals
I1(t) and I2(t), derived from S1and S2, respectively. Their
forms are presented in Eqs. (A5) and (A6). The resulting
form of S(t)inEq.(A4) involves strengths for the random
initial state, Mσ=1−σ2
0and Mτ=1. We further consider
the noises ησand ητto be similarly distributed, implying that
their strengths can be related through Dσ/(1 −σ2
0)=Dτ=
D, arising from nonzero σ0. From the definitions of cand C,
we have c/C=b+band cC=2a(1 −σ2
0). Therefore, we
may rewrite S(t)as
S(t)=1−σ2
0
2(2a+b+b)[I1(t)+DI2(t)].(56)
The parameter Ddetermines the relative contributions of the
two integrals to S(t). The integral I1(t) is linear to the lead-
ing order in t, whereas integral I2(t) is quadratic (refer to
Appendix). To second order, the function S(t) is given by
S(t)=1−σ2
0
2(2a+b+b)t−3ν+D
2t2+O(t3).
(57)
The expression of the linear slope from mean field theory
matches our exact calculation in Sec. V. Moreover, the linear
and quadratic terms in Eq. (57) always have opposite signs,
for any a,b,and b. Hence, at early times, S(t) is expected
to show an extremum, as seen in Fig. 8, and in agreement with
simulations.
D. Comparison with simulations
The integrals I1(t) and I2(t)inEqs.(A2) and (A3) can be
evaluated numerically. Using Eq. (56), we derive the analyti-
cal evolution of S(t) from the mean field theory. In Fig. 10 we
have compared the analytical S(t) with the plots from simu-
lations. For the value of parameter D=1, we observe that the
correspondence holds very well at short times across the entire
phase diagram [Fig. 10(a)], and also for varying mean particle
density ρ(•)=(1 +σ0)/2 [Figs. 10(b) and 10(c)]. Therefore,
despite neglecting correlations, the linearized mean field the-
ory succeeds in describing the early-time behavior of S(t).
This is because in our system we have chosen an initial state
without any correlations, which is exactly of the form assumed
by mean field theory.
The expression in Eq. (56) leads to different behaviors of
S(t) in the ordered and disordered regimes since the constant
cwhich enters in the eigenvalues [Eq. (48)] can be real,
imaginary, or zero across the phase boundary. We discuss the
different regimes separately below.
Disordered regime (b+b<0). The constant cis imagi-
nary and its magnitude determines the speed of the density-tilt
kinematic wave. The local cross correlation S(t) approaches
a constant saturation value, as seen in simulations. For the
case of bunchwise balance (2a+b+b=0), the state re-
mains totally disordered (uncorrelated), and satisfies the mean
field condition of absence of correlations at all times. Out-
side the bunchwise balance plane, and beyond the short time
correspondences, the simulations of S(t) depart from their
respective mean field analogs at later times, attaining nonzero
constant values subsequently in the steady state. These depar-
tures result from the buildup of correlations between particles,
which have been neglected in the mean field theory. In
Fig. 10(a) we see that the departure sets in earlier, and to
a greater extent as we move farther away from bunchwise
balance. This trend indicates an increase of the correlation
length as we move away from the bunchwise balance locus,
toward FDPO. This is consistent with the hypothesis [52] that
the correlation length diverges as the transition locus is ap-
proached from the disordered phase, indicating a mixed-order
transition, as discussed at the end of Sec. II.
FDPO (b+b=0). The condition c=0 identifies the
order-disorder phase boundary, where the tilt field evolves
autonomously and governs the evolution of the particle field.
Within the linearized theory, S(t) attains a constant saturation
value in the thermodynamic limit, whereas simulations indi-
cate a slow decay ∼t−φwith φ0.11.
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MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
101 102
0.1
-0.1
0
0.1
0.2
0.3
101 102
0.10.01
0.04
0
0.08
0.12
0.1
0
0.2
0.3
101 102
0.10.01
(a)
(b)
(c)
FDPO
Ordered
Disordered
Totally
Disordered
Simulations
Ordered (SPS)
Simulations
Disordered
Simulations
tt
t
S(t)
S(t)
S(t)
1
2
3
4
5
6
7
8
9
10
1
2
3
4
3
4
2
1
FIG. 10. Comparison of the early-time evolution of S(t) derived from mean field theory (colored plots) with simulations (gray, dashed
plots). Linear-log plots are shown for (a) varying parameters a,b,bacross the phase diagram with mean particle density ρ(•)=1+σ0
2=1
2,for
different ordered and disordered steady-state phases, (b) varying particle densities ρ(•)=1
2,1
4,1
8,and 1
16 in plots 1–4, with parameters a=0.4,
and b=b=0.2 fixed in the ordered regime (SPS), and (c) varying particle densities ρ(•)=1
2,1
4,1
8,and 1
16 in plots 1–4, with parameters a=
0.4,b=0.2, and b=−0.3 fixed in the disordered regime. In all three cases (a), (b), and (c), the analytical evolution in Eq. (57) corresponds
very well with simulations at short times. The deviation at later times occurs due to the buildup of correlations in the system, neglected in the
mean field theory. The system size used here is Nsys =2048. The parameter values used in (a) for (a,b,b) are, respectively, 1(0.4,0.3,0.3),
2(0.4,0.4,0), 3(0.4,0.4,−0.2), 4(0.4,0.2,−0.2), 5(0.4,−0.1,−0.1), 6(0.4,−0.2,−0.2), 7(0.4,−0.3,−0.3), 8(0.4,−0.4,−0.4), 9
(0.2,−0.3,−0.3), and 10 (0.2,−0.4,−0.4).
Ordered regime (b+b>0). cis real, giving rise to an
instability which leads to a divergence of S(t). In reality,
nonlinearities curb the runaway behavior predicted by the
linearized theory, and in fact S(t) approaches zero in the
thermodynamic limit, corresponding to coarsening toward or-
dered states as seen in simulations.
VII. LATE-TIME BEHAVIOR OF S
In this section, we discuss the evolution of S(t)at
late times, particularly during coarsening toward the phase-
separated steady states in the ordered regime. Unlike quanti-
ties such as two-point correlation functions, which have been
employed routinely to study out-of-equilibrium systems ap-
proaching steady state, S(t)isalocal quantity which also
captures the extent of coarsening toward ordered phases. We
show that the average length of irreducible sequences provides
a good estimation of the coarsening length scale at late times.
We also discuss the occurrence of a stretch of time where
S(t) decays with a diffusive power law preceding the onset of
coarsening, as predicted by the linearized mean field theory.
A. Coarsening with two-point correlation functions
Earlier studies of coarsening pertaining to the ordered
phases in the LH model have shown that the two-point corre-
lation functions exhibit scaling, as in phase-ordering kinetics
[56]. The particle density correlation G(r,t)=σj(t)σj+r(t)
has the following scaling form:
G(r,t)=gr
L(t).(58)
Here, L(t) represents a coarsening length scale which grows
in time typically, but not always, as L(t)∼t1/z, where zis
the dynamic exponent. The arrangements of particles and tilts
situated within a stretch of length L(t) resemble those in
the steady state of a finite system of size Nsys =L(t). The
manner in which L(t) grows with tdepends on the ordered
phase toward which the system coarsens. For instance, while
heading toward the SPS phase, the system undergoes very
slow coarsening proceeding through an activation process
[15], leading to L(t)∼log t, verified numerically [45]. In the
case of IPS and FPS, however, the system coarsens faster, as
apowerlaw∼t1/zwith z2[17]. Further, the system also
undergoes coarsening with z1.5 as it approaches steady
state on the transition line of order and disorder, i.e., FDPO
[14,40].
A particularly significant feature of the scaling function
g(y)inEq.(58) is its behavior at small argument y:
g(y)=m2
0[1 −g1|y|α], |y|1.(59)
In any ordered phase, the intercept of g(y)asy→0 is equal to
the long-range order m2
0[56]. In the case of the three ordered
phases SPS, IPS, and FPS, we have m2
0=1[18], whereas in
FDPO, m2
00.71 [40]. For the ordered states SPS, IPS, and
FPS, the clusters of Hand Lparticles are separated by sharp
interfaces. This implies a linear fall of g(y) for small y, i.e.,
the exponent α=1inEq.(59). This is consistent with the
Porod law [57], observed normally in phase ordering with a
scalar order parameter [56]. However, in FDPO, the clusters
are separated by broad interfacial regions, smaller than but
of the order of L(t). Consequently, the scaling function g(y)
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LIGHT AND HEAVY PARTICLES ON A FLUCTUATING … PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
FIG. 11. Sas a counter of coarsening “structures” in the LH
lattice. The Hparticles are represented by colored circles, while the L
particles and tilts are represented together by linespoints in grayscale.
Every macroscopic valley filled by a cluster of Hparticles contains
only one triad valley •in excess of the number of residing
triad hills •, which is counted as +1. All the other triads •
(S=+1) and •(S=−1) inside the macroscopic valley can be
grouped into noncontributing pairs. Similarly, every macroscopic hill
overlapping with an Lcluster contributes +1toS, by means of its one
excess triad ◦(S=+1).
displays a cusp singularity as y→0 with exponent α<1(
0.15) [43], indicating the breakdown of the Porod law.
The discussion above is consistent with the following pic-
ture: a coarsening landscape predominantly comprises several
large, slowly evolving structures, whose typical size at time
tcorresponds to the coarsening length scale L(t). Two such
adjacent structures c1and c2undergo sequential mergers over
a timescale t∗∼Lz(t), thus forming a larger structure c∗
1.As-
suming that a local steady state is reached within L(t) by time
t, and referring to the ordered steady-state profiles depicted
in Fig. 2(c), we infer that the coarsening structures typically
consist of H-particle clusters overlapping with macroscopic
valleys of tilts, along with their neighboring L-particle clus-
ters overlapping with macroscopic hills. Moreover, these
structures also include the interfacial regions between the H-
particle valleys and L-particle hills, which may be quite broad
in the case of FDPO.
B. S(t) as a local indicator of coarsening
Aside from its “microscopic” interpretation in Sec. IV as
a local measure of cross correlation, S(t) also counts the
number of coarsening structures in the lattice at any time t.
As discussed earlier, the sizes of these structures L(t) may be
quite large.
On the basis of our understanding that a stretch of length
L(t) in an ordered phase is mainly composed of H-particle
valleys and L-particle hills, every coarsening structure is ex-
pected to contribute +2toS(t). This is illustrated in Fig. 11,
which shows how compact structures with sharp interfaces in
a typical coarsening landscape are counted by S(t). Within
every macroscopic valley filled by a cluster of Hparticles at
any given time, all local triad valleys •and hills •
can be grouped into pairs whose contributions to S(t) can-
cel, except for a single remaining triad valley •which
contributes +1. Likewise, every macroscopic hill overlapping
with a cluster of Lparticles has a single local hill ◦in
excess, which also contributes +1. In Sec. VII D we show that
in the ordered phases, the coarsening length scale L(t) can be
extracted from the late-time behavior of the disorder-averaged
correlation S(t) through the relation
S(t)∼1
L(t).(60)
During coarsening toward FDPO, however, the coarsening
structures have broad interfacial regions which also contribute
substantially to S(t). Thus, the magnitude of S(t) does not
directly reflect the number of coarsening structures. Never-
theless, S(t) shows a slow decay at late times.
C. Irreducible sequences and coarsening length scale
Next, we show that the coarsening length scale discussed
above corresponds to the length scale of the irreducible se-
quences involving interfaces and bends described in Sec. III.
The dynamics of the system proceeds through local updates
of the particles and tilts or, equivalently, the interfaces and
bends. Since we have established a direct relation between the
numbers of these sequences and the local cross correlation
in Eq. (32), this naturally leads to a length scale describing
S, namely, the lengths of the irreducible sequences. This can
be established as follows. At any given time, the sites of
the system can be grouped into sequences that are reducible,
irreducible, as well as sites not belonging to any sequence. Fo-
cusing specifically on the irreducible sequences which govern
S, we may then assign spin variables to all the sites of the
system with +for sequences of type (),−for sequences of
type (), and 0 for sites belonging to reducible sequences as
well as sites not belonging to any sequence. These ±1 indices
are assigned to all sites from the start to end of an irreducible
sequence. At any time t, there are a finite number of sites in
each of these states given by n+(t), n−(t), and n0(t), respec-
tively, with n+(t)+n−(t)+n0(t)=2Nsys. Additionally, we
assume that the sizes of the irreducible sequences are well de-
scribed by their average length L+and L−, such that the total
number of sequences are given by N+=n+/L+and N−=
n−/L−. We can then compute Sfor each configuration, using
Eq. (32), as S(t)=(2n+/L+)−(2n−/L−). As discussed in
Sec. III, one type of sequence is “favored” whereas the other
is “unfavored,” i.e., the system preferentially evolves toward
favored structures. In our simulations we study the regime
of the parameter space a>0,b+b0, and therefore R=
2a+b+b>0. Thus, irreducible sequences of type +dom-
inate at late times. This behavior is illustrated in Fig. 12,
where we plot N±, the number of ±sequences averaged over
different evolutions, showing the asymmetry in the number of
+and −sequences at late times.
Next, we make the assumption that the numbers of sites n+
in the favored sequences remain constant, or near constant,
over the coarsening dynamics, indicating that the favored
043279-17
MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
FIG. 12. A log-log plot of N±(t), the average number of irre-
ducible sequences of types +and −[defined in Eq. (26)] with time
tduring coarsening toward the ordered phases SPS, IPS, FPS, and
FDPO. The parameter values chosen here are the same as in plots
1–4 in Fig. 7. In the regime of the parameter space that we study
(a>0,b+b0), the +sequences are favored, whereas the −
sequences are unfavored and their number tends to zero at late times.
Inset: Log-log plot of the time evolution of the average number of
lattice sites n±(t) within the +and −sequences, for different ordered
phases. The system size used here is Nsys =2048.
sequences primarily lengthen through mergers. This behavior
is illustrated in the inset of Fig. 12. This leads us to an estimate
of the scaling of the local cross correlation S∼1/L+.In
Fig. 13 we show the evolution of the average length of the +
sequences for the various phases in the LH model. We find
SPS
IPS
FPS
FDPO
60
0
20
40
80
100
L+(t)
L+(t)
t
102
10
10
102
10
1
103
103104
102103104105
t
∼t0.5
∼t
0.11
∼log t
FIG. 13. Growth of L+(t), the average length of irreducible se-
quences of type +with time tas the system coarsens toward the
ordered phases. A log-log plot of L+(t) in the different ordered
phases. IPS, FPS, and FDPO display a power-law growth in the
length of sequences. Inset: Linear-log plot of L+(t) in the SPS phase,
displaying a logarithmic increase. The parameter values chosen here
are the same as in plots 1–4in Fig. 7. The system size used here is
Nsys =2048.
that indeed the length scale associated with the irreducible
sequences is governed by the same coarsening exponents as
the local cross-correlation function S. We note that the length
scale of irreducible sequences in fact provides a coarsening
length scale that can be measured in every configuration and
not only in the average over evolutions.
Finally, we note the direct relationship between the coars-
ening structures introduced in the previous subsection and the
irreducible sequences. A coarsening structure in the ordered
regime consists of a cluster of heavy particles within a valley
adjacent to a cluster of light particles on a hill, as can be seen
in Fig. 11. This is naturally described in terms of interfaces
and bends defined in Sec. IV as (......)..., which is an ir-
reducible sequence of type +,asgiveninEq.(26). Therefore,
the length of irreducible sequences provides a direct measure
of the length of the coarsening structures, through which the
coarsening length scale can be probed.
D. Coarsening results from S(t)
We discuss below the decay characteristics of S(t) during
coarsening, its scaling behavior, and finite-size effects.
Coarsening toward ordered states (SPS, IPS, and FPS).
The decay profile of S(t) at late times reflects the dynamics
of the coarsening regime. We argue below that S(t) accu-
rately counts the diminishing number of coarsening structures,
which is inversely proportional to their growing sizes L(t).
Identifying the coarsening length scale of the system as the
average length of the favored irreducible sequences L(t)≡
L+(t), leads to Eq. (60). Based on this and the discussion on
L(t) in the previous subsection, we expect S(t) to decay as
∼1/log tfor SPS, and as a power law ∼t−φwith φ0.5
for IPS and FPS. These behaviors are verified by numerical
simulation, as shown in Fig. 14. For a finite-sized system,
the steady state is reached when L(t) becomes as large as the
system size Nsys, i.e., when tis of the order ts(Nsys ). Beyond
this time S(t) saturates to a constant value Sss ≈A/Nsys ,as
discussed earlier in Sec. IV. For the SPS phase, ts(Nsys)∼
eλNsys , where λis a constant, whereas for IPS and FPS phases,
we have ts(Nsys)∼Nz
sys. The average number of structures in
the steady state is proportional to the number of active triads
present, i.e., A∼O(1), independent of Nsys .
The late-time behavior of S(t) from numerical simula-
tions is shown for the ordered phases SPS, IPS, and FPS
in Figs. 14(a)–14(c). In SPS [Fig. 14(a)], we observe that
S(t)−1grows as ∼log tfor t<ts(Nsys), and saturates when
t>ts(Nsys). The inset gives evidence that ts∼eλNsys and Sss ∼
1/Nsys . The late-time evolution in the IPS and FPS phases can
be collapsed by suitable rescaling [insets of Figs. 14(b) and
14(c)]. A generalized form of S(t), that applies to IPS, FPS,
as well as FDPO, is given by the scaling ansatz
S(t)=1
Nμ
sys
FL(t)
Nsys ,(61)
where L(t)∼t1/z. In the limit tts(Nsys ), as seen in sim-
ulations we have S(t)∼t−φ, independent of Nsys. Thus, for
small ywe have F(y)∼y−μand φ=μ/z.Fortts(Nsys),
the system approaches steady state, and as seen in simulations,
we have S(t)∼N−μ
sys . Consequently, the scaling function F(y)
approaches an O(1) constant as y→∞. For the IPS and FPS
043279-18
LIGHT AND HEAVY PARTICLES ON A FLUCTUATING … PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
Nsys
1
2
3
4
5
6
7
8
9
32
48
64
80
128
256
512
1024
2048
1
2
3
4
3
5
6
7
8
10 104
1102103105
0.1 106
t
10
102
103
10
30
50
70
∼t0.5
10 104
1102103105
0.1 106
1
10-1
10-2
1
10-4 10-2
t/N2
sys
(a) SPS (b) IPS
3
5
6
7
8
10 104
1102103105
0.1 106
t
10
102
103
∼t0.43
1
10-1
10-2
1
10-4 10-2
t/N2.3
sys
(c) FPS
t
5
6
7
8
9
10 104
1102103105
t
10
20
8
6
30
∼t0.11
10
1
1
10-4 10-2
t/N
1.5
sys
(d) FDPO
0.1
0.3
0.5
0.7
1102
10-2
S(t)−1
S(t)−1/Nsys
S(t)−1/Nsys
t/e0.18Nsys
S(t)−1/Nsys
S(t)−1/N 0.1
7
sys
7
S(t)−1
S(t)−1
S(t)−1
∼log t
FIG. 14. Coarsening and finite-size effects of S(t) in the ordered phases (a) SPS, (b) IPS, (c) FPS, and (d) the order-disorder separatrix
(FDPO). The parameter values chosen here are the same as in plots 1–4in Fig. 7. (a) A linear-log plot of 1/S(t) in the SPS phase displays
logarithmic coarsening. Inset: The saturation value in the steady-state scales with Nsys and occurs at a characteristic timescale that diverges
exponentially with Nsys. (b)–(d) The log-log plots of 1/S(t) in the other phases exhibit power-law coarsening, with a scaling form consistent
with Eq. (61)(shownintheinsets).
phases we have μ=1. The data presented in Figs. 14(b) and
14(c) are consistent with the scaling form in Eq. (61).
Fluctuation-dominated phase ordering (FDPO).Inthis
case, Eq. (60) does not hold as the coarsening structures have
broad interfacial regions which also contribute substantially
to S(t). Nevertheless, the scaling form in Eq. (61) continues
to hold. The numerical results in Fig. 14(d) show that S(t)
decays as a slow power law ∼t−φwith φ0.11 during coars-
ening, as the system evolves toward a steady state with large
fluctuations in the extent of ordering. As discussed in Sec. IV,
the number of structures in steady state for a finite system
scales as A∼N1−μ
sys , consistent with Sss ∼N−μ
sys . The scaling
relation φ=μ/zis satisfied with μ0.17, φ0.11, and
z1.5.
In simulations toward the SPS phase, we also observe a
power-law stretch over time where S(t) decays as ∼t−0.5,
before the onset of the ∼1/log tcoarsening. The time span
of this “precoarsening” stretch grows as we lower the particle
density ρ(•)=(1 +σ0)/2 in our simulations, and can extend
across a decade as shown in Fig. 15. From the expressions of
integrals I1(t) and I2(t)[Eqs.(A5) and (A6)], we observe that
the mean field evolution of S(t) is governed by the interplay
of three timescales (1) tν∼1/ν,(2)tc∼1/c, and (3) tins ∼
ν/c2. The effect of diffusion dominates between timescales
tνand tc, resulting in a diffusive ∼t−0.5decay of S(t). The
timescale tins represents the characteristic time beyond which
the linear instability prevails over diffusion. For strong diffu-
sion (ν1) we have tν<tc<tins. Therefore, the power-law
∼t−0.5stretch observed in simulations can also exist within
the mean field theory. Distinguishing the precoarsening effect
is difficult in the IPS and FPS phases, as S(t) decays during
coarsening as ∼t−φ, with φclose to the precoarsening expo-
nent 0.5.
VIII. CONCLUSIONS AND DISCUSSION
In this paper we have studied the LH model, which de-
scribes light and heavy particles advecting and interacting
with a fluctuating surface. We introduced three theoretical
ideas in this work: bunchwise balance, irreducible sequences,
and a local cross-correlation function. We established a con-
dition 2a+b+b=0 in the parameter space of this model
under which the steady state of the system is characterized
by an equiprobable measure over all configurations. Fur-
thermore, we showed that this condition is necessary and
sufficient for a product-measure steady state. This occurs via a
mechanism which we termed “bunchwise balance,” in which
the incoming probability current into every configuration from
a group of configurations is exactly balanced by the outgoing
current to another uniquely specified group of configurations.
043279-19
MAHAPATRA, RAMOLA, AND BARMA PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
FIG. 15. Precoarsening stretches in the SPS phase for varying
mean particle densities ρ(•)=1+σ0
2=1
2,1
4,and 1
8, with fixed pa-
rameters a=0.4andb=b=0.2. A plot of t0.5S(t)vstshows
that the time span of the ∼t−0.5precoarsening decay increases with
decreasing density ρ(•), and can extend to about a decade in time for
ρ(•)=1/8. Inset: S(t)vst. The system size used here is Nsys =512.
Next, we identified a local cross-correlation function S,
involving the particle density at a site, and its adjacent tilts.
We showed that Sis able to capture and distinguish between
the properties of different phases that occur in this model.
We showed using an exact argument that the initial evolution
of S(t) starting from a totally disordered configuration is
linear, with a slope proportional to R=2a+b+b. We then
used a set of linearized equations derived from a mean-field
expression for the current to describe the early-time dynamics
in the LH model, up to quadratic order in time. We provided
evidence that the point at which the early-time evolution satu-
rates is related to the discreteness of the underlying lattice. We
also provided numerical evidence that this mean field theory
is able to capture several nontrivial aspects of the evolution
of S. Finally, we studied the late-time coarsening behavior
of the system through the local cross-correlation function,
and showed that, surprisingly, this local quantity is able to
characterize several nontrivial coarsening properties of the
system. We also provided numerical evidence that the length
of irreducible sequences, which have a direct relation to S,
provide an accurate estimation of the coarsening length scale
associated with the LH model.
Several interesting directions remain open. As we have
shown, the LH model displays an equiprobable steady state
through a bunchwise balance mechanism, where the bunches
in this model consist of two incoming and two outgoing tran-
sitions. It would be interesting to find models that display such
a condition with larger bunches, or even unequal numbers of
transitions in each bunch. The local cross-correlation function
studied in this paper is able to capture the nontrivial coarsen-
ing properties in the system which are usually probed through
nonlocal quantities. Indeed, we have established a relationship
between the local correlations and the nonlocal “irreducible”
sequences. However, the exact dynamics of these sequences
and how these lead to the various nontrivial exponents asso-
ciated with the coarsening dynamics of the LH model remain
to be established. It would be interesting to extend our study
of the local cross-correlation Sand the concept of irreducible
sequences to other multispecies models with a larger number
of species. As many of the concepts introduced in this work
rely on the one-dimensional nature of the system, it would also
be useful to search for generalizations in higher dimensions.
Finally, it would be intriguing to use the irreducible sequences
introduced in this work to provide a quantitative insight into
the nature of typical FDPO configurations, and in particular
the structure of the interfacial regions in this regime.
ACKNOWLEDGMENTS
We acknowledge useful discussions with S. Chakraborty,
S. Chatterjee, A. S. Rajput, V. V. Krishnan, and R. Dandekar.
S.M. would like to thank UM-DAE CEBS Mumbai and TIFR
Hyderabad for hospitality and academic support. This project
was funded by intramural funds at TIFR Hyderabad from the
Department of Atomic Energy (DAE), Government of India.
M.B. acknowledges support under the DAE Homi Bhabha
Chair Professorship of the Department of Atomic Energy.
APPENDIX: DERIVATION OF INTEGRALS I1(t)
AND I2(t)
Expressing
hP,Q
k(t)inEq.(55) in terms of the initial con-
dition and noise evolutions through Eq. (51), we see that S(t)
can be written as the sum of two terms, each with a distinct
physical origin:
S(t)=S1(t)+S2(t),(A1)
where
S1(t)=C
N2
sys
π
k=−π
sin3k
2
hP
k(0)
hQ
−k(0){e−(λ+
k+λ−
−k)t−e−(λ+
−k+λ−
k)t},(A2)
S2(t)=C
N2
sys
π
k=−π
sin3k
2e−(λ+
k+λ−
−k)tt
0t
0
dtdt e(λ+
kt+λ−
−kt)ηP
k(tηQ
−k(t)
−e−(λ+
−k+λ−
k)tt
0t
0
dtdt e(λ+
−kt+λ−
kt)ηP
−k(t)ηQ
k(t).(A3)
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LIGHT AND HEAVY PARTICLES ON A FLUCTUATING … PHYSICAL REVIEW RESEARCH 2, 043279 (2020)
FIG. 16. A typical plot of S(t) from the linearized mean field
theory, shown in purple. The system shown here evolves from a
totally disordered initial state toward an ordered phase (SPS). The
two constitutent integrals I1(t) (in green) and I2(t) (in blue) encode
the effects of the initial state and noise, respectively. At short times,
the mean field evolution corresponds well with the simulation of S(t)
(in dashed gray). The system size used here is Nsys =2048. The
values of the parameters are a=0.4andb=b=0.2, with mean
particle density ρ(•)=1+σ0
2=1
2.
We next use Eqs. (52) and (53)inEqs.(A2) and (A3), and
take the continuum limit π
k=−π→Nsys
2ππ
−πdk. We find that
S(t) is a linear combination of two integrals I1(t) and I2(t)
derived from S1and S2.Wehave
S(t)=c
CMσ+cCMτI1(t)
!
S1
+c
CDσ+cCDτI2(t)
!
S2
,
(A4)
where the integrals I1(t) and I2(t)aregivenby
I1(t)=1
4ππ
−π
dk sin k
2e−4νsin2(k
2)t
×1
csinh 2csin k
2t,(A5)
I2(t)=1
ππ
−π
dk t
0
dtsin3k
2e−4νsin2(k
2)(t−t)
×1
csinh 2csin k
2(t−t).(A6)
In Eq. (A4), the correlation strengths for the random initial
state and noises are related as Mσ=1−σ2
0,Mτ=1, and
Dσ/(1 −σ2
0)=Dτ=D. The integral I1(t) can be expanded
up to second order in tas
I1(t)∼t−3νt2+O(t3).(A7)
Above, we have expanded the expression in Eq. (A5)
up to second order in tand used the definite integral
π
−πdk sin2k
2=πin the O(t) term and π
−πdk sin4k
2=
3π/4intheO(t2) term. Similarly, I2(t) can be expanded up
to second order in tas
I2(t)∼−3
2Dt2+O(t3).(A8)
Therefore, the linear term in S(t) arises only from I1(t),
while the quadratic term has contributions from both integrals
I1(t) and I2(t). This behavior is illustrated in Fig. 16.From
the definitions of cand C,wehavec/C=b+band cC=
2a(1 −σ2
0). We may therefore rewrite S(t) using the simpli-
fied expressions in Eqs. (A7) and (A8), leading to Eq. (57)in
the main text.
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