# Entropy growth during free expansion of an ideal gas

Subhadip Chakraborti1, Abhishek Dhar1, Sheldon Goldstein2, Anupam Kundu1 and
Joel L. Lebowitz2 1International Centre for Theoretical Sciences, Tata
Institute of Fundamental Research, Bengaluru 560089, India  2Departments of
Mathematics and Physics, Hill Center, Rutgers, The State University of New
Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA

###### Abstract

To illustrate Boltzmann’s construction of an entropy function that is defined
for a microstate of a macroscopic system, we present here the simple example
of the free expansion of a one dimensional gas of non-interacting point
particles. The construction requires one to define macrostates, corresponding
to macroscopic variables. We define a macrostate $M$ by specifying the
fraction of particles in rectangular boxes $\Delta x\Delta v$ of the single
particle position-velocity space $\\{x,v\\}$. We verify that when the number
of particles is large the Boltzmann entropy, $S_{B}(t)$, of a typical
microstate of a nonequilibrium ensemble coincides with the Gibbs entropy of
the coarse-grained time-evolved one-particle distribution associated with this
ensemble. $S_{B}(t)$ approaches its maximum possible value for the dynamical
evolution of the given initial state. The rate of approach depends on the size
of $\Delta v$ in the definition of the macrostate, going to zero at any fixed
time $t$ when $\Delta v\to 0$. Surprisingly the different curves $S_{B}(t)$
collapse when time is scaled with $\Delta v$ as: $t\sim\tau/\Delta v$. We find
an explicit expression for $S_{B}(\tau)$ in the limit $\Delta v\to 0$. We also
consider a different, more hydrodynamical, definition of macrostates for which
$S_{B}(t)$ is monotone increasing, unlike the previous one which has small
decaying oscillations near its maximum value. Our system is non-ergodic, non-
chaotic and non-interacting; our results thus illustrate that these concepts
are not as relevant as sometimes claimed, for observing macroscopic
irreversibility and entropy increase. Rather, the notions of initial
conditions, typicality, large numbers and coarse-graining are the important
factors. We demonstrate these ideas through extensive simulations as well as
analytic results.

Boltzmann’s entropy, Free expansion, Thermalization

##  I Introduction

According to the second law of thermodynamics, any spontaneous change in an
isolated system leads to an increase of the thermodynamic entropy, $S$ (as
defined by Clausius). The second law thus provides in a sense an arrow of time
and quantifies the irreversibility that we observe in everyday physical
phenomena. Understanding how such irreversibility emerges from the microscopic
reversible Newtonian dynamics of a many-particle system was the remarkable
achievement of Boltzmann. He pointed out the key idea that the observed
irreversibility is the typical macroscopic behavior given appropriate initial
conditions, that becomes a certainty when we take the system size truly
macroscopic. Boltzmann also provided a clear prescription for the construction
of an entropy function (which we denote as $S_{B}$) that is defined for a
_single_ microstate of a macroscopic system in a given macrostate. This
entropy function is defined for a system in or out of equilibrium. It is equal
to the thermodynamic entropy for a system in equilibrium.

The deep and somewhat subtle ideas of Boltzmann [1] have been widely discussed
[2, 3, 4, 5] and clarified in recent work [6, 7, 8, 9]. We mention here a
particularly relevant quote from Ref. [10]: Time-asymmetric behavior as
embodied in the second law of thermodynamics is observed in individual
macroscopic systems. It can be understood as arising naturally from time-
symmetric microscopic laws when account is taken of a) the great disparity
between microscopic and macroscopic sizes, b) initial conditions, and c) that
what we observe are “typical” behaviors — not all imaginable ones. Common
alternate explanations, such as those based on equating irreversible
macroscopic behavior with ergodic or mixing properties of ensembles
(probability distributions) already present for chaotic dynamical systems
having only a few degrees of freedom or on the impossibility of having a truly
isolated system, are either unnecessary, misguided or misleading.

The present work is an attempt to provide a numerical demonstration of some of
the above ideas presented in [9] through a simple example.

Our microscopic model is a gas of $N(\gg 1)$ non-interacting point particles
of unit masses confined to move inside a one-dimensional box of length $L$.
Initially the gas is in thermal equilibrium (to be defined more precisely
later) and confined, by a partitioning wall, to the left half of the box. We
consider its subsequent evolution on removal of the partition. In our work we
consider two distinct (families of) macroscopic variables. For the first, we
consider a coarse graining of the single particle phase space
$\\{\mu\equiv(x,v)\\}$ into rectangles $\Delta_{\mu}$ with volumes $\Delta
x\Delta v$ and look at the distribution $f(x,v,t)$, at time $t$, of particles
in this space. This leads to a definition of $S_{B}$, that we refer to as
$S_{B}^{f}$. The second macroscopic description is given by the three locally
conserved fields $U=\\{\rho(x,t),p(x,t),e(x,t)\\}$ corresponding to mass,
momentum and energy — defined using a spatial coarse-graining. The Boltzmann
entropy corresponding to $U$ will be referred to as $S_{B}^{U}$.

We study the time evolution of the two choices of macrovariables, $f$ and $U$,
and the associated entropies, $S_{B}^{f},S_{B}^{U}$. The simplicity of the
model allows us to perform highly accurate simulations with large number of
particles (of order $10^{7}$) and compute both mean distributions (averaged
over initial ensembles) analytically as well as empirical ones (with single
realizations). The results from the empirical distributions allow us to test
the typicality of macroscopic behavior, i.e., that typical microstates—that
is, the overwhelming majority of microstates—corresponding to a given coarse-
grained description defining a macrostate in terms of values of certain
macrovariables yield the same future behavior of those macrovariables, with
some tolerance. We assume, as is usually done in statistical mechanics, that
“overwhelming majority” is defined with respect to the projection of the
microcanonical measure onto the relevant macrostate, i.e. the uniform
distribution on the region of phase space corresponding to the macrostate.

We find that as expected, both $S^{f}_{B}$ and $S^{U}_{B}$ approach for long
times their equilibrium values with the behavior of a typical microstate being
the same as that averaged over the initial ensemble. There are however some
interesting surprises in the time evolution of $S^{f}_{B}(t)$. The rate at
which $S^{f}_{B}(t)$ increases depends strongly on $\Delta v$ with
$dS^{f}_{B}(t)/dt$ apparently going to zero as $\Delta v\to 0$. However, upon
rescaling time, $t\to\tau/\Delta v$, the different curves collapse to a single
curve $S^{f}_{B}(\tau)$. This curve has small decaying oscillations near its
maximum. We obtain an analytic expression for $S^{f}_{B}(\tau)$ which agrees
with the observations. There are no such surprises for $S^{U}_{B}(t)$ which
increase monotonically to the equilibrium value.

We want to point to some of the earlier studies related to this issue. Of
particular relevance are the works of Frisch [11] and of De Bievre and Parris
[12], who studied the time evolution of an initially spatially non uniform
ideal gas, as do we. They do not however consider the entropy change in the
process, which is our focus here. The paper [12] is particularly nice and
highly recommended. It explains in a clear and rigorous way the resolution of
the objections to Boltzmann by Zermelo and Loschmidt — i.e., the “paradoxes”
of Poincare recurrence times in a finite system and of reversibility of the
microscopic dynamics.

The evolution of Boltzmann’s entropy has been earlier investigated numerically
in interacting systems such as fluid models [13, 14, 15, 16] and in systems
evolving via maps [17]. Some subtleties for dense fluids were pointed out by
Jaynes in [18], discussed further in [19] and numerically investigated in
[20]. The one dimensional gas of equal mass hard point particles and hard rods
was extensively studied earlier as one of the tractable models where dynamical
properties can be obtained analytically and where the question of entropy
increase has been investigated. Some of the interesting questions addressed
concern dynamical correlations and the evolution of the single particle
distribution function [11, 21, 22, 23, 24, 25, 26]. The Euler hydrodynamic
equations for the hard rod system were first obtained in [27] and have more
recently been discussed in [28] as an example of an interacting integrable
model, where it is also shown that there are dissipative Navier-Stokes
corrections which vanish when one goes from rods to point particles. The
effect of integrability-breaking on entropy growth was studied in [29] for
hard rods in a harmonic trap. Boltzmann’s ideas also appear in recent
discussions of thermalization in isolated quantum systems [30, 31]. In
contrast to these studies, the present work considers the case of a completely
non-interacting system, namely the ideal gas in one dimension.

The plan of the paper is as follows. In Sec. II we define the Boltzmann
entropy for a general classical macroscopic system. We then describe the
precise model and the different choices of macrostates. In Sec. III we present
our numerical results on the evolution of the macroscopic fields and the
entropy functions for the two different choices of macrovariables. This
section also contains the derivation of the expression for $S^{f}_{B}(\tau)$
in the rescaled time $\tau$. For $S^{U}_{B}$, we present an analysis of the
results based on the “hydrodynamic” equations for the macroscopic fields. In
Sec. IV we study how these macrovariables and the associated (Boltzmann)
entropies evolve with time for atypical initial conditions. A geometric
picture of the dynamics in phase space is provided in Sec. V and we conclude
with a discussion in Sec. VI. Some exact results for the evolution of the
macroscopic fields are presented in App. A.

##  II Boltzmann’s entropy, definition of the microscopic model and choice of
macrostates

###  II.1 Boltzmann’s entropy

The microstate of a classical system of $N$ particles of unit mass confined in
a box, denoted by $X$, is specified by the positions ${\bf x}_{i}$ and
velocities ${\bf v}_{i}={\bf p}_{i}$, with $i=1,2,...,N$, _i.e._ , $X=({\bf
x}_{1},{\bf x}_{2},\ldots{\bf x}_{N},{\bf v}_{1},{\bf v}_{2},\ldots,{\bf
v}_{N})$. The dynamics of the system is given by a Hamiltonian
$H(X)=\sum_{i=1}^{N}v_{i}^{2}/2$.

We now consider a macroscopic or “coarse-grained” description for the case
$N\gg 1$. A simple example of such a description is provided by the
macrovariable $N_{\rm left}$ which gives the total number of particles in the
left half of the box. Clearly, this is a function of the microstate $X$ and we
can write $N_{\rm left}(t)=M(X(t))$, with $M(X)=N_{\rm left}(X)$.

In general we can describe a macrostate by specifying a set of macrovariables
$M(X)=\\{M_{1}(X),M_{2}(X),...,M_{n}(X)\\}$, with resolution $\Delta
M=\\{\Delta M_{j}\\}$ [9]. We identify these macrostates with the elements of
a partition of the full phase space $\Gamma$ into sets $\Gamma_{\hat{M}}$ of
the form

$\displaystyle\Gamma_{\hat{M}}=\\{X\in\Gamma|\hat{M}_{j}\leq
M_{j}(X)\leq\hat{M}_{j}+\Delta M_{j},j=1,...,n\\}.$ (1)

These provide a coarse-grained description in the sense that many different
$X$ correspond to the same range of values of the macrovariable $M(X)$, and
hence to the same set $\Gamma_{\hat{M}}$.

Each microstate $X$ belongs to some set $\Gamma_{\hat{M}}$ corresponding to
the coarse-grained value of the macrovariable $\hat{M}=\hat{M}(X)$ (thus for
$X\in\Gamma_{\hat{M}}$ as in Eq. (1), $\hat{M}(X)=\hat{M}$). Boltzmann’s
insight was to associate to each microscopic state $X$ an entropy, through the
set $\Gamma_{\hat{M}}$ to which it belongs [6, 8, 9, 32]:

$S_{B}(X)=S_{B}[\hat{M}(X)]=\ln|\Gamma_{\hat{M}}|,$ (2)

where we have set Boltzmann’s constant $k_{B}=1$. The volume of the set
$\Gamma_{\hat{M}}$ is

$\displaystyle|\Gamma_{\hat{M}}|$
$\displaystyle=\int\prod_{i=1}^{N}\frac{d{\bf x}_{i}d{\bf
p}_{i}}{h}~{}\mathbbm{1}[X\in\Gamma_{\hat{M}}],$ (3)

where $\mathbbm{1}$ represents the indicator function and $h$ is a constant
with the dimension of angular momentum. Here, without loss of generality, we
set $h=1$. As the system evolves under the Hamiltonian dynamics, the
microstate is given by $X(t)$ while the macrovariable evolves as
$M(t)=M(X(t))$. Consequently the corresponding set
$\Gamma_{\hat{M}}(t)=\Gamma_{\hat{M}(X(t))}$ also evolves, thereby specifying
the time evolution of the Boltzmann entropy as
$S_{B}(t)=S_{B}[\Gamma_{\hat{M}}(t)]$. Boltzmann argued that for an isolated
system starting from a microstate corresponding to a low entropy $S_{B}(0)$,
the system evolves in such a way that $S_{B}(t)$ “typically” increases for
macroscopic systems even though the microscopic evolution is completely time-
reversal symmetric. (In what follows we shall drop the hats on the
macrovariables, slightly abusing notation.)

Among all possible macrostates of a system there are two very important ones:
the equilibrium macrostate $M_{eq}$ and the initial macrostate $M_{\rm ini}$.
The entropy of the initial macrostate $S_{B}(t=0)=S_{B}(M_{\rm ini})$ is low
by assumption. On the other hand, the macro-region $\Gamma_{M_{eq}}$ is
overwhelmingly large compared to other macro-regions associated to other
macrostates. It is so large that it contains most of the phase space volume of
$\Gamma_{E}$, an energy shell, assumed to contain all the macrostates $M$.
More precisely, for large $N$, the ratio of their volumes
$|\Gamma_{M_{eq}}|/|\Gamma_{E}|\approx 1-e^{-cN}$ where $c$ is a positive
constant [32, 33, 34]. This property corresponds to equilibrium because the
system should stay in (or near) $\Gamma_{M_{eq}}$ for long times, consistent
with the observed stationarity in thermodynamic equilibrium.

Since $\Gamma_{M_{\rm eq}}$ takes up almost all the volume of $\Gamma_{E}$,
when the system starts from a microstate $X$ belonging to a non-equilibrium
macrostate $M_{\rm ini}$, so that $|\Gamma_{M_{\rm ini}}|\ll|\Gamma_{M_{\rm
eq}}|$, its microscopic dynamics should ‘typically’ take the microstate to
regions $\Gamma_{M}$ of larger phase space volume and thus of larger entropy
$S_{B}$ and eventually to $\Gamma_{M_{\rm eq}}$, unless the dynamics given by
the Hamiltonian has strong constraints, for example additional conservation
laws, or the initial state $X$ is very special [35]. Hence we expect the
quantity $S_{B}(M)$ to increase for the majority (in fact the overwhelming
majority) of microstates in $\Gamma_{M_{\rm ini}}$ except for a few whose
total volume relative to $|\Gamma_{M_{\rm ini}}|$ goes to zero in the
$N\to\infty$ limit. Because of this expectation one can make direct connection
between $S_{B}(M_{\rm eq})$ and $S$ (the thermodynamic entropy) in an
equilibrium state as suggested by Boltzmann. For an isolated system in
equilibrium with energy $E$, in a box $\Lambda$ of volume $V$ and $N$
particles [8]

$\displaystyle S(E,V,N)\approx S_{B}(M_{\rm eq}(E,\Lambda,N)),~{}~{}{\rm
for~{}large}~{}N.$ (4)

Here $M_{\rm eq}(E,\Lambda,N)$ is the equilibrium macrostate of the system.
Its volume $|M_{\rm eq}(E,\Lambda,N)|$ depends effectively only on $V$ for
boxes of reasonable shape. (If, in addition to the energy $E$ and particle
number $N$, there are other conserved quantities, as there are for integrable
systems such as those we are considering here, these could be considered as on
the same footing as $N$ and $E$, with Eq. (4) modified accordingly. However,
we shall not do so here.)

We briefly comment here on why Gibbs’ definition of entropy cannot be used in
the nonequilibrium situation. We recall first that the Gibbs entropy of an
equilibrium canonical ensemble $\varrho_{\rm eq}$ is defined as

$\displaystyle S_{G}[\varrho_{\rm
eq}(X)]=-\int\prod_{i=1}^{N}dx_{i}dp_{i}~{}\varrho_{\rm
eq}(X)~{}\ln\varrho_{\rm eq}(X),$ (5)

and this can be identified for macroscopic systems with the thermodynamic
entropy $S$. Extending this definition to the non-equilibrium situation
described by an evolving ensemble $\varrho_{t}(X)$ one obtains the
corresponding Gibbs-Shannon entropy $S_{G}(t)=S_{G}[\varrho_{t}(X)]$. However,
we note that the volume preserving dynamics is described by the Liouville
equation

$\displaystyle\partial\varrho/\partial t=\\{H,\varrho\\}.$ (6)

This ensures that this entropy does not change with time, i.e.,
$dS_{G}(t)/dt=0$.

A general study of the Boltzmann entropy and the coarse grained Gibbs entropy,
related to the $F_{\alpha}$ in Eq. (11), is given in Chapters IV and V of Ref.
[36]. The discussions there are enlightening. Another book with a clear
analysis of the issues discussed in this article is by Oono [37].

###  II.2 Definition of the model and choices of macrostates

Our model consists of $N$ non-interacting point particles of mass $1$ confined
in a one dimensional box of size $L$. The Hamiltonian of the system
$H=\sum_{j=1}^{N}v_{j}^{2}/2$ consists of only kinetic energy. In between
collisions with the walls (at $x=0,L$), each particle moves at constant
velocity. On collisions with the walls, the velocities are reversed.

We now describe the two families of macrovariables that we will consider in
this study.

Figure 1: Plot of evolution of the empirical particle density
$\tilde{f}_{\alpha}(t)=f_{\alpha}(t)/N$ (black dashed lines) starting from a
single initial microscopic configuration in the two-dimensional $\mu$-space
for grid size $\Delta x=\Delta v=0.5$ and $N=10^{7}$, $L=4$. In the single
initial configuration, the positions of the particles are distributed
uniformly between $(0,L/2)$ and the velocities are drawn from the Maxwell
distribution given by Eq. (17) with canonical temperature $T_{0}=2.5$. We
observe that $\tilde{f}_{\alpha}(t)$ approaches towards its equilibrium form
at large times, however the convergence is oscillatory as can be seen from the
recurrences at times $t=16,~{}32,~{}48,~{}64$ to very close to the equilibrium
form. The evolution is also compared with the analytical result for the
averaged single particle distribution $\tilde{F}_{\alpha}=F_{\alpha}(t)/N$
from Eqs. (11) and (49) which is obtained after averaging over initial
configurations chosen from uniform position distribution over $(0,L/2)$ and
Maxwell velocity distribution at temperature $T_{0}=2.5$ (this is the
equilibrium state in the left half of the box). The good agreement between
$\tilde{f}_{\alpha}(t)$ and $\tilde{F}_{\alpha}(t)$ is a consequence of
typicality.

Choice I — The distribution of particles in the single-particle phase space:
We consider $\mu$-space $\\{(x,v)\\}$ and divide it into cells
$\Delta_{\alpha}$, each of size $|\Delta_{\alpha}|=\Delta x\Delta v$. For a
given microstate $X=\\{x_{i},v_{i}\\}$ we specify the number of particles
$N_{\alpha}$ in each cell. We then obtain the particle number density in each
cell:

$\displaystyle f_{\alpha}=\frac{N_{\alpha}}{|\Delta_{\alpha}|}.$ (7)

This satisfies the normalization $\sum_{\alpha}f_{\alpha}|\Delta_{\alpha}|=N$.
The set $\\{f_{\alpha}\\}$ specifies our first family of macrovariables, with
its corresponding macrostates. The “number” of microstates (volume) for a
given specification of $\\{N_{\alpha}\\}$ is given by
$|\Gamma_{M}|=\prod_{\alpha}[|\Delta_{\alpha}|^{N_{\alpha}}/N_{\alpha}!]$ 111
The phase space volume of $N_{\alpha}$ identical particles in a box of size
$|\Delta_{\alpha}|$ is given by
$\frac{1}{N_{\alpha}!}(|\Delta_{\alpha}|/h)^{N_{\alpha}}$. For the
specification $\\{N_{\alpha}\\}$ over the set of all boxes, we then arrive at
the product form (with $h$ set to $1$). Thus, with
$S^{f}_{B}=\ln|\Gamma_{M}|$, we have using Stirling’s formula for large $N$
the entropy per particle

$\displaystyle
s_{B}^{f}=S_{B}^{f}/N=-\frac{1}{N}\sum_{\alpha}|\Delta_{\alpha}|f_{\alpha}\ln
f_{\alpha},$ (8)

up to an additive constant.

To get a handle on the behavior of $s_{B}^{f}$ we also consider, for a finite
number of particles, an average (indicated below by $\langle...\rangle$) over
initial microscopic configurations chosen from a phase space distribution
$\varrho_{0}(\\{x_{i},v_{i}\\})$:

$\displaystyle
F(x,v,t)=\sum_{i=1}^{N}\left\langle\delta(x_{i}(t)-x)\delta(v_{i}(t)-v)\right\rangle,$
(9)

where $\\{x_{i}(t),v_{i}(t)\\}$ are the positions and velocities of the
particles at time $t$, obtained from the non-interacting dynamics. We note
that $F(x,v,t)$ is the single-particle marginal obtained from the full phase
space density $\varrho_{t}(\\{x_{i},v_{i}\\})$ with initial distribution
$\varrho_{0}$. For our ideal gas, $F(x,v,t)$ obeys the autonomous equation

$\partial_{t}{F}+v\partial_{x}{F}=0,$ (10)

and can be computed analytically as shown in App. A. We can now define a
coarse-grained distribution corresponding to a partition of the $\mu$-space as

$\displaystyle
F_{\alpha}=\frac{1}{|\Delta_{\alpha}|}\int_{x,v\in\Delta_{\alpha}}dx~{}dvF(x,v,t),$
(11)

and a corresponding coarse-grained entropy:

$\displaystyle
s^{F}_{\Delta}=-\frac{1}{N}\sum_{\alpha}|\Delta_{\alpha}|F_{\alpha}\ln
F_{\alpha}.$ (12)

Note that this has a similar form to Eq. (8); however, here we have used mean
distributions instead of the empirical distributions used there. These will in
fact typically be more or less the same,

$s^{f}_{B}\approx s^{F}_{\Delta},$ (13)

a consequence of the law of large numbers crucial for our analysis.

We note that if we let $|\Delta_{\alpha}|\to 0$ in Eqs. (11) and (12) then for
any fixed $t$,

$\displaystyle s^{F}_{\Delta}\to s^{F}=-\frac{1}{N}\int dxdvF\ln F,$ (14)

which, since the evolution of $F$ satisfies Eq. (10), makes $s^{F}$
independent of $t$ 11footnotetext: It should be noted however that in the
limit as $t\to\infty$, $F(t)$ weakly approaches a uniform spatial distribution
(Prosser mixing [prosser1969]). This would also be the case for $s^{f}_{B}$
when $\Delta_{\alpha}\to 0$, $N\to\infty$, and $f_{\alpha}$ is suitably
normalized. This is due to the fact that we are dealing with a non-interacting
system so that $s^{F}$ is just, up to normalization, the Gibbs entropy of the
entire $N$-particle system (whose distribution can be taken to be the evolving
product of $F$’s), which does not change under the time evolution. We shall
see later that even for the ideal gas if we look on a time scale proportional
to $1/\Delta v$ we will see $s^{F}_{\Delta}\approx s^{f}_{B}$ increase with
time albeit non-monotonically.

Choice II - The spatial distribution of mass, momentum and energy: We divide
the box $(0,L)$ into $K$ cells $\delta_{a}$, $a=1,2...K$, each of size
$\ell=L/K$. For a given microscopic configuration $X$, let $N_{a}$ be the
number of particles in cell $\delta_{a}$ and let $P_{a}$ and $E_{a}$ be the
total momentum and total energy of these particles. In this case the
macrostate is defined by these set of locally conserved quantities
$U=\\{N_{a},P_{a},E_{a}\\}$ and we obtain the Boltzmann entropy
$S_{B}^{U}=\ln|\Gamma_{U}|$ where $|\Gamma_{U}|$ is the volume of the phase
space region $\Gamma_{U}$ corresponding to the macrostate $U$. For large $N$
this entropy per particle attains the form

$\displaystyle
s_{B}^{U}=\frac{S_{B}^{U}}{N}=\frac{1}{N}\sum_{a}\rho_{a}\ell~{}s(\rho_{a},\epsilon_{a}),$
(15)

where $s(\rho_{a},\epsilon_{a})$ is the equilibrium ideal gas entropy per
particle for density $\rho_{a}=N_{a}/\ell$ and internal energy density
$\epsilon_{a}=[E_{a}-P_{a}^{2}/(2N_{a})]/\ell=e_{a}-p_{a}^{2}/(2\rho_{a})$,
with $p_{a}=P_{a}/\ell,e_{a}=E_{a}/\ell$ being the momentum density and total
energy density respectively. This is given explicitly (up to additive constant
terms) by:

$\displaystyle
s(\rho,\epsilon)=-\ln\rho+\frac{1}{2}\ln\left(\frac{\epsilon}{\rho}\right)~{}.$
(16)

##  III Results for the time evolution of macrostates and entropy increase

###  III.1 Choice I of the macrovariables

####  III.1.1 Numerical results

We consider $N=10^{7}$ particles initially uniformly distributed in the left
half $(0,L/2)$ of the box with box size $L=4$. For our non-interacting point
particle system, the choice of system size $L$ is inconsequential and hence we
arbitrarily set $L=4$. Since we keep the system length fixed, changing $N$
corresponds to changing the density in our system. There is no upper bound to
the density since there is no interaction between the particles. In real
systems the number of particles would scale with the volume. We consider first
the case where the initial velocities of our microstate are drawn from the
Maxwell distribution given by [19]

$g_{\rm eq}(v,T_{0})=\left(\frac{1}{2\pi
T_{0}}\right)^{1/2}\exp\left(-\frac{v^{2}}{2T_{0}}\right),$ (17)

with temperature $T_{0}=2.5$. This is the canonical ensemble corresponding to
the equilibrium macrostate with particles in the left half of the box. We
choose a _single random realization_ from this canonical ensemble as our
initial microstate. Equivalently we can choose the initial configuration from
a microcanonical ensemble with total energy $E=NT_{0}/2$. The region
$\Gamma_{M_{\rm ini}}$ then consists of all $X\in\Gamma_{E}$ such that $N_{\rm
left}(X)=N$.

We divide the $\mu$-space ($x$-$v$ space) of the system into grids of size
$\Delta_{\alpha}=\Delta x\Delta v$ and calculate the evolution of the
empirical single particle density $f_{\alpha}$ given by Eq. (7) by performing
microscopic simulations of the evolution of the given microstate. In Fig. (1)
we plot $\tilde{f}_{\alpha}=f_{\alpha}(t)/N$ at different points
$x_{\alpha},v_{\alpha}$ in $\mu$-space and at different times, for $\Delta
x=\Delta v=0.5$. We observe that $\tilde{f}_{\alpha}(t)$ approaches its
equilibrium form non-monotonically in time with near-recurrences to the
equilibrium distribution. At large times, the $\tilde{f}_{\alpha}(t)$ finally
reaches the equilibrium form where particles are uniformly distributed between
$(0,L)$ and velocities are Maxwellian with temperature $T_{0}=2.5$. We also
compare the empirical $\tilde{f}_{\alpha}(t)$ (black dashed lines) and mean
distribution $\tilde{F}_{\alpha}(t)=F_{\alpha}(t)/N$ (red solid lines),
calculated for the same grid size, at different times. The mean distribution
${F}_{\alpha}(t)$ is computed analytically from Eqs. (11) and (49). We find
good agreement between the empirical density $\tilde{f}_{\alpha}(t)$ and the
mean distribution $\tilde{F}_{\alpha}(t)$ — a consequence of the typicality
implied by the law of large numbers for this non-interacting model. We have
verified that this agreement is also valid when we choose the initial random
configuration from a microcanonical distribution with energy per particle
given by $T_{0}/2$.

Figure 2: A comparison between $s^{f}_{B}(t)$ obtained from simulation of a
single realization, and $s^{F}_{\Delta}(t)$ obtained after an ensemble
average, plotted as a function of time during free expansion. For evaluating
$s^{f}_{B}(t)$ we use the same single initial condition used in Fig. (1) for
$N=10^{7}$ in a box of size $L=4$. We compute $s^{f}_{B}(t)$ for different
grid sizes $|\Delta_{\alpha}|=\Delta x\Delta v$ by keeping $\Delta x=0.5$
fixed and varying $\Delta v=0.5$ (red empty squares), 0.25 (blue empty
inverted triangles), 0.1 (yellow empty circles), and 0.05 (green empty
triangles). The solid lines are $s^{F}_{\Delta}(t)$ obtained analytically from
Eq. (12) for $T_{0}=2.5$. We again observe excellent agreement between
$s^{f}_{B}(t)$ and $s^{F}_{\Delta}(t)$ and notice that they both eventually
increase and saturate to the equilibrium value. However the approach to this
equilibrium value is oscillatory with decaying amplitude and period $2L/\Delta
v$. Also note that the growth rate at any given time decreases with decreasing
$\Delta v$.  Figure 3: Plot showing the dependence of growth of the entropy,
$s^{f}_{B}(t)$, for different spatial resolutions $\Delta x$. For any given
$\Delta v$ we find that there is a weak dependence on the size of $\Delta x$
at early times. The parameter values and initial conditions are the same as in
Fig. (1). Figure 4: Plot showing the distribution of $N=10^{6}$ particles in
the $(x,v)$ plane at different instants of time. These results correspond to a
single realization of the time-evolution of the system, for the same
parameters and initial conditions as used in Fig. (1).

In Fig. (2) we show the evolution of the corresponding entropy $s^{f}_{B}(t)$
[given by Eq. (8), where we fix the additive constant so that at $t=0$, this
agrees with Eqs. (15,16)] during free expansion, for the same random single
realization and parameters as in Fig. (1). We plot $s^{f}_{B}(t)$ for
different grid sizes by keeping $\Delta x$ fixed and varying $\Delta v$. The
solid lines correspond to the entropy $s^{F}_{\Delta}(t)$ [given by Eq. (12)]
calculated from the exact expression for the mean distribution $F(x,v,t)$. We
observe that there is very good agreement between $s_{B}^{f}$ and
$s^{F}_{\Delta}$, as expected. Both the entropies grow, initially
monotonically with time, touching a value slightly above the final equilibrium
value and then exhibit small oscillations in time with a period
$\tau_{p}=2L/\Delta v$; these eventually die and the entropy saturates to its
equilibrium value. Note that these oscillations were also seen in the
recurrences in Fig. (1) and we will discuss their origin in the next
subsection.

Though the final increase of entropy appears to be always equal to $\ln(2)$,
we observe in Fig. (2) that the entropy growth rate decreases with decreasing
$\Delta v$. In other words, at any fixed time, with decreasing $\Delta v$ one
observes a correspondingly lower entropy. On the other hand we see in Fig. (3)
that the entropy growth rate shows convergence on decreasing $\Delta x$. This
can be understood from the plot of the $\mu$-space distribution shown in Fig.
(4). We see that with time, the system keeps developing more and more
structure in the velocity direction, while, in the spatial direction it
becomes more or less homogeneous after some initial time. Thus, decreasing the
grid size $\Delta x$ does not give us more information about the system, while
decreasing $\Delta v$ does.

To understand the dependence of $s_{B}^{f}$ on $\Delta v$ consider the limit
of vanishing grid size. For large $N$, corresponding to $f_{\alpha}$ defined
for a given microstate $X=\\{x_{i},v_{i}\\}$, one can define a smooth function
$f(x,v,t)$ such that $N_{\alpha}=\int_{x,v\in\Delta_{\alpha}}dxdvf(x,v,t)$ and
$\int dxdvf(x,v,t)=N$. Eq. (8) then becomes

$\displaystyle{s}_{B}^{f}(t)\approx-\frac{1}{N}\int dx~{}dv~{}f(x,v,t)\ln
f(x,v,t),$ (18)

up to an additive constant. In the large $N$, small grid size limit, the
function $\tilde{f}=\lim_{\Delta x\to 0,\Delta v\to 0}\lim_{N\to\infty}f/N$
satisfies the equation

$\partial_{t}\tilde{f}+v\partial_{x}\tilde{f}=0,$ (19)

using which it follows that its associated “entropy”

$\displaystyle\mathcal{s}_{B}^{f}(t)=-\int
dx~{}dv~{}\tilde{f}(x,v,t)\ln\tilde{f}(x,v,t),$ (20)

obeys $d\mathcal{s}_{B}^{f}(t)/dt=0$. Thus it would seem that there is no
entropy increase in the large $N$, $\Delta_{\alpha}\to 0$ limit. However, as
is apparent from the numerical findings in Sec. III, for any fixed grid size
$|\Delta_{\alpha}|$ the exact $s_{B}^{f}$, or its approximation on the right
hand side of Eq. (8), will typically increase (if initially its value is not
at its maximum) over time. For $N$ large and $|\Delta_{\alpha}|$ small,
significant increase may not begin for a very long time (the time at which
$\tilde{f}(x,v,t)$ develops structure on the scale $|\Delta_{\alpha}|$), a
reflection of the fact that the entropy in Eq. (20) does not change with time.

The situation is different for a gas of hard spheres of diameter $a$ in 3D
where in the Boltzmann-Grad limit, $a\to 0,N\to\infty$ with $Na^{2}\to b>0$,
one can define the macrostate by a smooth one-particle empirical density which
satisfies the Boltzmann equation, given by Eq. (19) modified by collision
terms on the right [3, 39]. As shown by Boltzmann’s $H$-theorem, this leads to
increase of the entropy $\mathcal{s}^{f}_{B}$.

Figure 5:  This figure shows a collapse of the data presented in Fig. (2) for
different values of $\Delta v$, on plotting the entropy as a function of the
scaled time $\tau=t\Delta v/(2L)$. The dashed curve is the analytic prediction
from Eq. (21) Inset: A zoom up of the plot. The horizontal dotted line
corresponds to the entropy given by the right hand side of Eq. (18) with
$f=\rho_{0}g_{\rm eq}$, where $\rho_{0}=N/L$ and $g_{\rm eq}$ is in Eq. (17).
The small overshoot (of $s^{f}_{B}$) that we see for larger $\Delta v$ arises
since a coarse-grained velocity distribution is effectively broader than the
Maxwellian from which it comes, leading to a larger effective temperature.

####  III.1.2 Scaling analysis for the evolution of entropy

We now return to the question of the observed oscillation period
$\tau_{p}=2L/\Delta v$ in Fig. (2). This is easiest to understand once we
consider a mapping of the dynamics of particles in a box of length $L$ to the
dynamics on a circle of length $2L$. This mapping corresponds to the
$\mu$-space map $\phi$ taking $[0,L]\times\mathbb{R}$ to
$[0,2L]\times[0,\infty)$ and given by $(x,v)\to(x,v)$ for $v>0$ and
$(x,v)\to(2L-x,-v)$ for $v<0$ . With this mapping we see that two particles,
initially at the same spatial point but with velocity difference $\Delta v$
(size of the velocity grid), will meet again (possibly in different locations)
at times $n\tau_{p}$, where $n$ is an integer. Furthermore, at the times
$t=n\tau_{p}$, all points that are initially within a cell $\Delta x\Delta v$
will lie on a narrow strip that winds around precisely $n$ times around the
circle and still within $\Delta v$ [see Fig. (6)]. Thus, the spatial
distribution of points becomes exactly uniform within the region $v$ to
$(v+\Delta v)$ at the times $t=n\tau_{p}$ and this explains the fact that
$s^{F}_{\Delta}$ (in Eq. (12)) reaches its maximum value at these times. At
intermediate times, the winding on the circle is incomplete and we get a lower
entropy.

In Fig. (5) we show plots of the entropy time-evolution data for different
values of $\Delta v$ [from Fig. (2)] as a function of the scaled time
$\tau=t\Delta v/(2L)\equiv t/\tau_{p}$. We find a remarkable collapse of the
data to a single curve. The physical picture in the preceding paragraph in
fact leads to an analytical understanding of this and we can obtain an
explicit expression for the evolution of the entropy in the scaled time
variable — this is given by the function:

$\displaystyle
s^{\bar{F}}(\tau)=-\ln\rho_{0}+\frac{1}{2}\ln\frac{T}{2}-\frac{1}{2}\int_{0}^{2}dzR(z,\tau)\ln
R(z,\tau),$ (21)

where $\rho_{0}=N/L$ and $R(z,\tau)$ is known explicitly (see below); the
integral above can be numerically computed. In Fig. (5) we find excellent
agreement between the collapsed data and the analytic result.

We now present the details of our analytic understanding of the observed
scaling and of Eq. (21). For this we use the mapping between the dynamics with
reflecting boundary conditions and the dynamics in a periodic box. The
dynamics on the circle simply consists of rotations at constant positive
velocities which implies $F(x,v,t)=F(x-vt,v,0)$ [with $F$ periodic in $x$ with
period $2L$]. We claim that the following space averaged distribution function
will in fact capture the evolution of the entropy of the system at the
rescaled time $\tau=t\Delta v/(2L)$:

$\displaystyle\bar{F}(x,v,\tau)$
$\displaystyle=\frac{1}{2L\tau}\int_{0}^{2L\tau}dx^{\prime}F(x-x^{\prime},v,0).$
(22)

We define a corresponding Gibbs entropy per particle:

$s^{\bar{F}}(\tau)=-\frac{1}{N}\int_{0}^{2L}dx\int_{0}^{\infty}dv~{}\bar{F}\ln\bar{F}.$
(23)

More precisely we now show that in the limit $\Delta x\to 0,\Delta v\to 0$, we
get

$\displaystyle\lim_{\Delta\to 0}s^{F}_{\Delta}(2L\tau/\Delta
v)=s^{\bar{F}}(\tau),$ (24)

which explains the observed scaling. With the system defined on the circle let
us consider the averaged distributions:

$\displaystyle F_{\alpha}(\Delta x,\Delta v)$ $\displaystyle=\frac{1}{\Delta
x\Delta v}\int_{x_{\alpha}}^{x_{\alpha}+\Delta
x}dx^{\prime}\int_{v_{\alpha}}^{v_{\alpha}+\Delta
v}dv^{\prime}F(x^{\prime},v^{\prime},t)$ (25) $\displaystyle=\frac{1}{\Delta
x\Delta v}\int_{x_{\alpha}}^{x_{\alpha}+\Delta
x}dx^{\prime}\int_{v_{\alpha}}^{v_{\alpha}+\Delta
v}dv^{\prime}F(x^{\prime}-v^{\prime}t,v^{\prime},0)$
$\displaystyle\approx\frac{1}{\Delta v}\int_{v_{\alpha}}^{v_{\alpha}+\Delta
v}dv^{\prime}F(x_{\alpha}-v^{\prime}t,v_{\alpha},0)$
$\displaystyle=\frac{1}{t\Delta v}\int_{0}^{t\Delta
v}dx^{\prime}F(x_{\alpha}-v_{\alpha}t-x^{\prime},v_{\alpha},0)$ (26)
$\displaystyle=\bar{F}(x_{\alpha}-v_{\alpha}t,v_{\alpha},t;\tau).$ (27)

Since $x$ is on the circle, we have for small $\Delta x$ and small $\Delta v$,

$\displaystyle
s^{F}_{\Delta}(t)=-\frac{1}{N}\sum_{\alpha}|\Delta_{\alpha}|\bar{F}(x_{\alpha}-v_{\alpha}t,v_{\alpha},\tau)\ln\bar{F}(x_{\alpha}-v_{\alpha}t,v_{\alpha},\tau)$
$\displaystyle\approx-\frac{1}{N}\sum_{\alpha}\int_{0}^{2L}dx\Delta
v\bar{F}(x-v_{\alpha}t,v_{\alpha},\tau)\ln\bar{F}(x-v_{\alpha}t,v_{\alpha},\tau)$
$\displaystyle=-\frac{1}{N}\sum_{\alpha}\int_{0}^{2L}dx\Delta
v\bar{F}(x,v_{\alpha},\tau)\ln\bar{F}(x,v_{\alpha},\tau)$
$\displaystyle\approx-\frac{1}{N}\int_{0}^{2L}dx\int_{0}^{\infty}dv\bar{F}(x,v,\tau)\ln\bar{F}(x,v,\tau)$
$\displaystyle=s^{\bar{F}}(\tau),$ (28)

where we used the translational invariance in going from the second to the
third step. We note that this result explains the main observations in Fig.
(5), namely slowly decaying oscillatory approach to the final value, with
precise returns at integer values of $\tau$. Since $F$ has period $2L$, we can
write it in the form $F(x,v,0)=\rho_{0}\bar{g}(v)+\phi(x,v,0)$ where
$\rho_{0}$ is the average density in the original box,
$\bar{g}(v)=\rho_{0}^{-1}\int_{0}^{2L}dxF(x,v,0)/(2L)$ is the averaged global
velocity distribution [$\bar{g}$ is normalized to $1/2$ on $v>0$] while
$\phi(x,v,0)$ has period $2L$ and mean zero (i.e.
$\int_{0}^{2L}dx\phi(x,v,0)=0$). Hence clearly
$\psi(x,v,\tau)=\int_{0}^{2L\tau}dx^{\prime}\phi(x-x^{\prime},v,0)/(2L)$ is
periodic in $\tau$ with period $1$. Then we have from Eq. (22) that

$\displaystyle\bar{F}(x,v,\tau)=\rho_{0}\bar{g}(v)+\frac{\psi(x,v,\tau)}{\tau}.$
(29)

We see that for integer values of $\tau=1,2,3,\ldots$, $\bar{F}$ attains the
value $\rho_{0}\bar{g}(v)$, which yields the time-maximum of the entropy
$s^{\bar{F}}$ given by:

$\displaystyle s^{\bar{F}}_{\rm max}$
$\displaystyle=-\frac{1}{N}\int_{0}^{2L}dx\int_{0}^{\infty}dv\rho_{0}\bar{g}(v)\ln[\rho_{0}\bar{g}(v)]$
$\displaystyle=-\ln\rho_{0}-2\int_{0}^{\infty}dv\bar{g}(v)\ln[\bar{g}(v)].$
(30)

We also see that the deviations of $\bar{F}$, from the value
$\rho_{0}\bar{g}(v)$, that occur at values of $\tau$ between these integers
are at most of order $1/\tau$, implying that the same thing is true for the
entropy.

For the special initial condition $F(x,v,0)=\rho_{0}[1+a\cos(\pi
x/L)]h(v)\Theta(v)$, with $h(v)$ any even velocity distribution, we get
$\bar{F}(x,v,\tau)=\rho_{0}\\{1-a[\sin[\pi(x/L-2\tau)]-\sin[\pi
x/L]]/(2\pi\tau)\\}h(v)$. We thus explicitly find here that
$\bar{F}=\rho_{0}h(v)$ for $\tau=1,2,3,...$, with $1/\tau$ deviations for
intermediate values as described above. In particular $\bar{F}$ approaches
$\rho_{0}h(v)$ as $\tau\to\infty$.

Figure 6: Plot showing the evolution of $N=10^{5}$ particles, in the $(x,v)$
plane, where the particles move on a circle of length $2L=8$. The particles
were initially distributed uniformly in a small box with $\Delta x=0.25,\Delta
v=0.5$. With time the box gets continually stretched and, at times that are
multiples of $2L/\Delta v=16$, the stretched pieces wind completely around the
box. Comparing with Fig. (2) we see that the dips in $s^{f}_{B}(t)$ occur at
times $\approx 24,40$, at which the winding around the length $2L$ is
complete.

Next, we consider the case discussed in Sec. (III.1) where the gas is
initially confined on the left half of the box. The initial distribution
considered is of the product form $F(x,v,0)=\rho(x)g_{\rm eq}(v)\Theta(v)$ and
we then get:

$\displaystyle\bar{F}(x,v,\tau)$ $\displaystyle=g_{\rm
eq}(v)\Theta(v)\frac{1}{2L\tau}\int_{0}^{2L\tau}dx^{\prime}\rho_{c}(x-x^{\prime}),$
(31)

where $\rho_{c}(x)$ is the initial density profile on the circle $[0,2L]$,
given by $\rho_{c}(x)=2\rho_{0}$ for $x\in[0,L/2]\cup[3L/2,2L]$ and zero
elsewhere. To perform the above integral, we Fourier-decompose the density
profile as $\rho_{c}(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{in\pi x/L}$, where
$a_{n}=(2L)^{-1}\int_{0}^{2L}\rho_{c}(x)e^{-in\pi x/L}$. Hence we obtain:

$\displaystyle\bar{F}(x,v,\tau)$ $\displaystyle=g_{\rm
eq}(v)\Theta(v)\left[{\rho_{0}}+\sum_{n\neq
0}\frac{a_{n}e^{in\pi(x/L-\tau)}\sin(n\pi\tau)}{n\pi\tau}\right].$ (32)

For our initial condition with a half-filled box one finds
$a_{n}=2\rho_{0}\sin(n\pi/2)/(n\pi)$ for $n\neq 0$. Hence we get

$\displaystyle\bar{F}(x,v,\tau)=g_{\rm
eq}(v)\Theta(v)\rho_{0}R(x/L,\tau),~{}~{}{\rm where}$ (33) $\displaystyle
R(z,\tau)$
$\displaystyle=\left[1+\frac{4}{\pi^{2}\tau}\sum_{n=1}^{\infty}\frac{\cos[n\pi(z-\tau)]\sin(n\pi/2)\sin(n\pi\tau)}{n^{2}}\right].$
(34)

The product form of $\bar{F}$ leads to simplifications for the entropy given
by Eq. (23) and, we finally obtain Eq. (21) [after fixing additive constants
so that at $t=0$, it agrees with Eqs. (15,16)].

Figure 7: Plot of the spatial profiles of the three conserved fields density
$\rho(x,t)$, velocity $u(x,t)$, and temperature $T(x,t)$ at different times
$t=0$ (magenta crosses), 0.1 (green empty squares), 0.4 (black empty
triangles), 1 (yellow empty diamonds), 2 (red inverted empty triangles), and 4
(blue empty circles) obtained from simulation of a single typical microstate
of $N=10^{7}$ particles. The density is normalized by the mean value
$\rho_{0}=N/L$. Initial configuration is one realization of the canonical
ensemble for particles in the left half $(0,L/2)$ with $L=4$. The initial
positions of the particles are distributed uniformly between $(0,L/2)$ and the
initial velocities are drawn from Maxwell distribution given by Eq. (17) with
canonical temperature $T_{0}=2.5$. The solid lines are analytically obtained
fields $\bar{\rho},\bar{u}=\bar{p}/\bar{\rho},$ and $\bar{T}$ given by Eqs.
(50), (52), (54) (see App. A for details). The excellent agreement between the
empirical densities and the mean densities once again establish typicality.
Figure 8: Testing the dependence of typicality on the number of particles $N$.
We plot the empirical density $\rho(x,t)$ (normalized by $\rho_{0}=N/L$) for
different $N$ and at different times, along with the mean density. The
agreement of the empirical profiles with the averaged ones (dashed-dotted
lines) becomes better as $N$ is increased. Figure 9: Plot of entropy per
particle, $s^{U}_{B}$, as a function of time during free expansion. Parameter
values are: $L=4,N=10^{5},T_{0}=2.5$. We consider the same single initial
configuration as in Fig. (1). The different plots correspond to partitions of
size $\ell=0.2,0.4,1$. Unlike in Fig. (2), here we see a monotonic increase
and a convergence of the growth rate on decreasing $\ell$. The solid lines
correspond to the mean field analytic profiles and we find very good agreement
with the entropy computed from the empirical fields.

###  III.2 Choice II of the macrovariables

####  III.2.1 Numerical results

We again start from a typical single realization with $N=10^{7},~{}L=4$ and
$T_{0}=2.5$ (the same as that used in Fig. (1)). In this case we partition the
box into $K=40$ cells each of size $\ell=L/40=0.1$ and calculate the
corresponding empirical density $\rho(x,t)$, velocity
$u(x,t)=p(x,t)/\rho(x,t)$ and energy $e(x,t)$ fields. Suppressing the time
dependence, we have that $\rho_{a}=\rho(x_{a}),p_{a}=p(x_{a})$, and
$e_{a}=e(x_{a})$, $x_{a}\in\delta_{a}$, with corresponding temperature field
$T(x)=2e(x)/\rho(x)-u^{2}(x)$. In Fig. (7), we plot these fields at different
times. The solid lines are the analytically obtained averaged fields
$\bar{\rho},~{}\bar{u},$ and $\bar{T}$ given by Eqs. (50), (52), (54). The
details of the analytical calculation of mean fields are provided in App. A.
We find excellent agreement between the empirical and mean densities, as
expected. We also find that at long times these fields converge to their
equilibrium values given by the uniform fields $\rho(x)=\rho_{0}$, $u(x)=0$
and $T(x)=T_{0}$. Unlike for the case of the $f$-macrovariable, here we do not
see an oscillatory approach to the equilibrium state. In fact from the
analytic results (see App. A) one can see that the approach to equilibrium at
long times takes the form $A(x,t)-A_{\rm eq}(x)\sim B(x)~{}e^{-at^{2}}$ with
$a=T_{0}\pi^{2}/(2L^{2})$, where $A(x,t)$ can be any of the three fields
${\rho},u,{T}$ discussed above, $A_{\rm eq}(x)$ represents its equilibrium
value and $B(x)$ is some real known function. Next, we compute the empirical
density field for different values of $N$. In Fig. (8) we plot the evolution
of $\rho(x,t)$ for the different values of $N$ and compare them with the
respective mean profiles $\bar{\rho}(x,t)$ at different times (black dot-
dashed lines). We notice that the empirical density shows fluctuations for
small $N$ which decrease for increasing $N$, leading to better agreement of
the empirical profiles with the averaged ones.

We next insert these three fields into Eqs. (15) and (16) to obtain the
intensive empirical entropy $s^{U}_{B}(t)$. In Fig. (9) we plot $s^{U}_{B}(t)$
with time $t$ for different cell sizes $\ell$. The solid lines correspond to
theoretical computation of $s^{U}_{B}(t)$ using the analytical expressions of
the mean fields $\bar{\rho}(x,t),~{}\bar{p}(x,t)$ and $\bar{e}(x,t)$ given in
App. A. In this case we see that the increase of $s^{U}_{B}(t)$ is monotonic
and the entropy growth rate converges as we decrease the cell size $\ell$. The
final increase of entropy is again equal to $\ln(2)$, as expected.

####  III.2.2 Entropy increase for $S^{U}_{B}$ and hydrodynamics

We now explore the connection between the increase of the entropy $S^{U}_{B}$
and the behavior of the $U$-macrovariables in the hydrodynamic limit. It is
believed that the Euler equations for the three conserved fields describe, in
a suitable regime, the hydrodynamics of a one-dimensional fluid of interacting
particles. At the level of the Euler equations there is no entropy increase.
While this is well known, we provide an argument for it here, since we will
need to refer to the argument later. So consider the one-dimensional Euler
equations:

$\displaystyle\partial_{t}\rho+\partial_{x}(\rho u)=0,$ (35a)
$\displaystyle\partial_{t}(\rho u)+\partial_{x}(\rho u^{2}+P)=0,$ (35b)
$\displaystyle\partial_{t}\left(\rho\tilde{e}+\frac{1}{2}\rho
u^{2}\right)+\partial_{x}\left[u\left(\rho\tilde{e}+\frac{1}{2}\rho
u^{2}+P\right)\right]=0,$ (35c)

where $\tilde{e}(x,t)=e/\rho-u^{2}/2$ is the internal energy per particle and
the pressure, for an ideal gas system is given by $P=\rho T$. These equations
can be written in the form

$\displaystyle\frac{D\rho}{Dt}+\rho\partial_{x}u=0,$ (36a)
$\displaystyle\frac{Du}{Dt}+\frac{1}{\rho}\partial_{x}P=0,$ (36b)
$\displaystyle\frac{D\tilde{e}}{Dt}+\frac{1}{\rho}P{\partial_{x}u}=0,$ (36c)

where $D/Dt=\partial_{t}+{u\partial_{x}}$ denotes the advective derivative.

Figure 10: Verification of entropy production rate as in Fig. (9) using Eq.
(41). The red line is the LHS of Eq. (41) where $S^{U}_{B}(t)$ is calculated
from the definition given by Eq. (15) and the green points are the RHS of Eq.
(41) calculated from hydrodynamics. Inset: Plot of the integrand in Eq. (41)
with space $x$ at different times.

Now we use the Euler hydrodynamic equations along with Clausius’ laws of
thermodynamics to determine the entropy production rate in the slowly evolving
local equilibrium state. Clausius’ laws of thermodynamics provide a well-known
thermodynamic relation for the ideal gas, given by $TdS=d\tilde{E}+PdL$, where
$S$ is the Clausius entropy, $\tilde{E}$ is total internal energy and $L$ is
the volume. Applying this relation to a small volume $\ell$ with a fixed
number of particles $n_{\ell}$ we find, after some manipulations,

$\displaystyle Tds=d\tilde{e}+Pd(\ell/n_{\ell}),$ $\displaystyle{\rm
hence,}~{}~{}\frac{Ds}{Dt}=\frac{1}{T}\left[\frac{D\tilde{e}}{Dt}-\frac{P}{\rho^{2}}\frac{D\rho}{Dt}\right],$

with $s(x,t)$ being the entropy per particle. From Eqs. (36a) and (36c) we
then immediately obtain that $\frac{Ds}{Dt}=0$. The total entropy
$S(t)=\int_{0}^{L}\rho s(x,t)~{}dx$ also remains constant, since
$dS/dt=-\int_{0}^{L}dx~{}\partial_{x}({\rho us})=0$, using the boundary
conditions ${u(0,t)=u(L,t)=0}.$ The standard mechanisms of entropy growth in
the hydrodynamic description are either additional dissipative (Navier-Stokes-
Fourier) terms or the formation of shocks.

Figure 11: Plot of the spatial profiles of the three conserved fields density
$\rho(x,t)$, velocity ${u(x,t)}$, and temperature $T(x,t)$ at different times
obtained from simulation of a single “atypical” configuration with a binary
choice of velocities. The initial positions of the particles ($N=10^{7}$) are
distributed uniformly between $(0,L/2)$ with $L=4$, the initial velocities of
odd particles are set $v_{0}$ and of even particles are set to $-v_{0}$. For
comparison with previous cases we chose $v_{0}=\sqrt{T_{0}}$ with $T_{0}=2.5$.
The profiles repeat themselves after a time period $8\tau$ with
$\tau=L/(4v_{0})$ and thus the system does not reach an equilibrium at large
time in this ‘atypical’ case. We have used grid size $\ell=0.1$.

Figure 12: Panel (a): Plot of $s_{B}^{f}(t)$ and $s_{B}^{U}(t)$ for the
‘atypical’ initial configuration with binary velocities, and other parameters
considered in Fig. (11). The periodic oscillation in both cases imply that the
system never reaches equilibrium. For the left figure we have used grid size
$\Delta x=\Delta v=0.5$ and for the right figure we have used $\ell=0.1$.
Panel (b): In this case we consider a small perturbation of the binary
velocity initial conditon in (a). We choose $v_{i}=v_{0}+r_{i}$ for odd
particles and $v_{i}=-v_{0}+r_{i}$ for even particles, where $r_{i}$ are iid
random variables chosen from the uniform distribution over
$(-\epsilon,\epsilon)$ with $\epsilon=0.1v_{0}$. We find that, unlike the
recurrent behaviour observed in (a), both the entropies increase and saturate
to a final value with a final change of $\ln 2$. The insets in (b) show the
early time evolution, where we see decaying oscillations, reminiscent of the
recurrent ones seen in (a).

We now discuss entropy production in our non-interacting gas using a similar
description, keeping in mind that we now do not expect a closed set of
hydrodynamic equations with the three fields. In fact from Eq. (19) we see
that the conserved densities
$\rho(x,t)=N\int_{-\infty}^{\infty}dv\tilde{f}(x,v,t)$,
$\rho(x,t)u(x,t)=N\int_{-\infty}^{\infty}dv~{}v\tilde{f}(x,v,t)$ and
$e(x,t)=N\int_{-\infty}^{\infty}dv(v^{2}/2)\tilde{f}(x,v,t)$ satisfy
continuity equations. The first two of these are precisely the Euler equations
in Eq. (35a) and (35b) whereas the third equation for the energy field
$e(x,t)$ is different from Eq. (35c) [11]. Formally the energy field satisfies
the exact conservation equation

$\displaystyle\partial_{t}e+\partial_{x}J=0,$ (37) $\displaystyle{\rm
where}~{}~{}J(x,t)=\frac{1}{2}\int_{-\infty}^{\infty}dv~{}v^{3}N\tilde{f}(x,v,t)$
(38)

is the energy current density. It is instructive to rewrite this equation in
the following form:

$\displaystyle\partial_{t}e+\partial_{x}[{u(e+P)}]=-\partial_{x}J_{s},$ (39)
$\displaystyle{\rm where}~{}J_{s}(x,t)=J-{u(e+P)}$ (40)

is the current after subtracting the reversible Euler part. This current,
$J_{s}$, can be interpreted as a “heat” current. Then, repeating the steps as
before we find that the entropy production rate is finite and given by

$\displaystyle\frac{dS(t)}{dt}=-\int_{0}^{L}dx~{}\frac{\partial_{x}J_{s}}{T}=-\int_{0}^{L}dx\frac{J_{s}\partial_{x}T}{T^{2}},$
(41)

where in the last step we used the fact that the current vanishes at the
boundaries. For our system we can compute the fields $J_{s}(x,t)$ and $T(x,t)$
directly from the exact solution of the microscopic dynamics and thereby
compute the entropy production rate from the above equation. In Fig. (10) we
compare this with the entropy production rate obtained from the definition
given by Eq. (15) and find perfect agreement between the two. In addition, as
shown in the inset of Fig. (10), we find that the integrand is non-negative
everywhere (although we are not able to prove it explicitly) which leads to a
non-negative entropy growth after integration. Note that for generic
interacting non-integrable systems, the term $J_{s}$ should be expressible in
terms of the three basic fields and in fact given by the Fourier’s law
$J_{s}=-\kappa\partial_{x}T$. This form would then guarantee non-negativity of
the entropy production rate.

We briefly comment on the growth of $S_{B}^{f}$. For the case of our non-
interacting gas, Eq. (19) or Eq. (10) are analogous to the Euler equations and
the growth of $S_{B}^{f}$ was purely a result of the discretization of
$\mu$-space. Another well known trivially integrable system is the harmonic
chain. The Euler equations for this system were written in [40] where it was
also noted that a finite space-time scaling parameter led to a Navier-Stokes
type correction term [41]. Interestingly, for the disordered harmonic chain,
closed form Euler equations can be written for just the stretch and momentum
variables, even though the system has a macroscopic number of conserved
quantities [42].

Figure 13: Plot of evolution of the empirical particle density
$\tilde{f}_{\alpha}(x,v,t)=f_{\alpha}(x,v,t)/N$, starting from a single two-
temperature initial microstate, at two spatial locations $x=L/4$ and $x=3L/4$.
The initial position of $N=10^{7}$ particles are distributed uniformly within
$(0,L)$ with $L=4$, while the initial velocities of the particles in the left
and right halves are drawn from Maxwell distributions at temperatures
$T_{L}=1$ and $T_{R}=10$, respectively. The grid size was taken as $\Delta
x=\Delta v=0.5$. At $t=0$ the empirical density $\tilde{f}_{\alpha}(x,v,t)$ is
Maxwellian with $T_{0}=1$ and $T_{0}=10$ at $x=L/4$ and $x=3L/4$,
respectively. As time evolves, the empirical density at any position gets
contribution from particles originating initially from both the Maxwell
distributions. After a large time, the distribution
$\tilde{f}_{\alpha}(x,v,t)$ is seen to approach the form $g(v)/L$ (shown by
the black dash-dot line in the left-most panel), where $g(v)=[{g_{\rm
eq}(v,1)+g_{\rm eq}(v,10)}]/{2}$ and $g_{\rm eq}(v,T_{0})$ is given in Eq.
(17).

Figure 14: Plot of the spatial profiles of the three conserved fields density
$\rho(x,t)$, velocity ${u(x,t)}$, and temperature $T(x,t)$ for the ‘two-
temperature case’ at different times $t=0$ (magenta circles), 0.1 (green
dashed lines), 0.4 (blue dotted lines), 1 (yellow dash-dot lines), 2 (red
dash-dot-dot lines), and 4 (black solid lines) obtained from simulation of a
single initial microscopic configuration with $N=10^{7}$ particles. The
initial condition is the same as that used in Fig. (13). Notice that after a
highly nontrivial evolution, all three profiles become flat and thus the
system reaches the equilibrium state at long times. Figure 15: Plot of
$s_{B}^{f}(t)$ and $s_{B}^{U}(t)$ obtained from simulation of a single
realization starting from the two-temperature initial configuration. The
initial condition is the same as that used in Fig. (13). We have used several
grid sizes for computing $s^{f}_{B}$ and $s^{U}_{B}$. While both entropies
saturate at large times, only the final value of $s^{U}_{B}$ corresponds to
the thermodynamic equilibrium value. Since the final velocity distribution at
long times is not Maxwellian, the saturation value of $s^{f}_{B}$ does not
correspond to the equilibrium value. Observe that the growth is non-monotonic
in both cases. We also see convergence of $s^{U}_{B}$ with coarse-graining
scale $\ell$ while for $s^{f}_{B}$ we again see a decreasing growth rate with
decreasing $\Delta v$.

##  IV Other initial conditions

So far we have considered a single typical initial condition for a macrostate
in which the initial positions are uniformly distributed in $(0,L/2)$ and the
initial velocities are chosen from a (uniform) Maxwell distribution. We found
that at large times the system goes to equilibrium, with the profiles of the
conserved fields becoming flat and the corresponding entropy $s^{U}_{B}(t)$
reaching a steady value. It is also interesting to study the evolution for a
single initial condition, atypical for all the particles being on the left
side.

To do that we first consider a single configuration of $N=10^{7}$ particles
initially in the left half $(0,L/2)$ distributed uniformly. The initial
velocities of odd particles are set to $v_{0}=\sqrt{T_{0}}$ and that of even
particles are set to $v_{0}=-\sqrt{T_{0}}$, with $T_{0}=2.5$. Interestingly,
in this case, each particle comes back to its original position with its
original velocity periodically after a time period $2L/\sqrt{T_{0}}$. This
recurrence is observed in Fig. (11), where we plot the profiles of the three
conserved fields density $\rho(x,t)$, velocity $v(x,t)$, and temperature
$T(x,t)$ at different times. We note that the profiles repeat themselves after
a time period $8\tau$ with $\tau=L/(4\sqrt{T_{0}})$. Thus, unlike for the
typical initial configuration in Fig. (7), for this atypical initial condition
the system never settles down into an equilibrium state for either of our two
choices of macrovariables. We have also looked at the evolution of the
entropies $s_{B}^{f}(t)$ and $s_{B}^{U}(t)$ for this atypical initial
configuration in Fig. (12a), where we find, of course, that the entropy in
both cases keeps oscillating for all time.

We now demonstrate that even a small perturbation of this atypical initial
condition changes the entropy evolution drastically — from recurrent to
irreversible growth (for times that are not extremely large). For this we add
a small random perturbation to the previously discussed binary velocity
setting. We choose $v_{i}=v_{0}+r_{i}$ for odd particles and
$v_{i}=-v_{0}+r_{i}$ for even particles, where $r_{i}$ are iid random
variables chosen from the uniform distributions over $(-\epsilon,\epsilon)$
with $\epsilon\ll v_{0}$. The outcome is remarkable – as seen in Fig. (12b),
both the entropies $s_{B}^{f}(t)$ and $s_{B}^{U}(t)$ now again increase by the
amount $\ln(2)$ and finally reach equilibrium (though not their thermal
equilibrium values). This clearly shows that the atypical initial microstate
is very special. A slight perturbation makes it a more typical one for which
the entropy increases and we observe macroscopic irreversibility. In the
insets of Fig. (12b), we see oscillations of the entropy at early times,
corresponding to some memory of the atypical initial condition.

The perturbed initial microstate just described is in fact still atypical,
even for the $f$-macrostate to which it belongs—a consequence of the
alternation of the sign of the velocity of successive particles. Nonetheless,
the analysis of Sec. (III.1.2) continues to apply, providing an analytical
demonstration of the irreversible behavior just described for the perturbed
microstate.

We also consider another initial microstate corresponding to an $f$-macrostate
(choice-I) where the particles are uniformly distributed over the full box
with zero momentum and with two different temperatures on the left and right
half of the box, i.e., with the velocities chosen from the corresponding
Maxwellians. In Fig. (13) we show the time-evolution of $f(x,v,t)$ while Fig.
(14) shows the evolution of the three fields – density, velocity and
temperature. In the former case we see that the long-time form of the single
particle distribution is non-thermal, i.e. non-Maxwellian, as demonstrated and
explained in Fig. (13). On the other hand, the fields $\rho(x,t),~{}{u(x,t)}$
and $T(x,t)$ are converging to their expected thermal equilibrium values, as
shown in Fig. (14). In Fig. (15) we compare the evolution of $s^{f}_{B}$ and
$s_{B}^{U}$ for the case where the left and right halves are initially at
temperatures $T_{L}=1$ and $T_{R}=10$ respectively. We note that both
$s^{U}_{B}$ and $s^{f}_{B}$ saturate at long times but the increase in entropy
is less for $s^{f}_{B}$. This is because the conserved fields evolve at long
times to their thermodynamic equilibrium values, with uniform density, zero
momentum and temperature $T=(T_{L}+T_{R})/2$, with $s^{U}_{B}$ thus attaining
the corresponding equilibrium value. On the other hand, the total velocity
distribution does not evolve with time and hence remains non-Maxwellian at all
times. Thus $s^{f}_{B}$ saturates to a value lower than the equilibrium one.
From Eq. (29) and the discussion in that section we find that the saturation
value of $s^{f}_{B}$ is given by Eq. (30) with $\bar{g}(v)=[g_{\rm
eq}(v,T_{L})+g_{\rm eq}(v,T_{R})]/2$. This agrees with the measured saturation
value in Fig. (15). The invariant velocity distribution $\bar{g}(v)$ which is
just the mean of two Maxwellians in fact defines a corresponding generalized
Gibbs ensemble (GGE) that describes the long-time equilibrium state of the
system. Note that the entropy growth for both definitions of entropy is non-
monotonic, unlike what is seen for $s^{U}_{B}$ for free expansion.

Figure 16: A schematic showing the partition of the full phase space into
subspaces defined by the equilibrium macrostates $f_{\rm eq}$ (boundary shown
by dotted-line), $U_{\rm eq}$ (boundary with long-dashed line) and $g_{\rm
eq}$ (boundary with dashed line). The points $X_{1}(0),X_{2}(0),X_{3}(0)$
correspond to the three initial microstates considered in our study. The point
$X_{1}(0)$ corresponds to the free expansion from thermal equilibrium,
$X_{2}(0)$ corresponds to free expansion from the alternate velocity
microstate and $X_{3}(0)$ corresponds to the uniform density non-Maxwellian
(two-temperature) initial global velocity distribution. After a long time,
$\tau$, these microstates, $X_{1}(\tau),X_{2}(\tau),X_{3}(\tau)$, would remain
in regions of phase space as shown in the figure ( making brief excursions out
of these regions on Poincare recurrence time-scales). We note that
$X_{1}(\tau)$ will be contained in $\Gamma_{f_{\rm eq}}$, $X_{2}(\tau)$ is
outside $\Gamma_{f_{\rm eq}}\cup\Gamma_{U_{\rm eq}}\cup\Gamma_{g_{\rm eq}}$,
and $X_{3}(\tau)$ will end up in $\Gamma_{U_{\rm eq}}\setminus\Gamma_{f_{\rm
eq}}$ (all points in $\Gamma_{U_{\rm eq}}$ that are not in $\Gamma_{f_{\rm
eq}}$).

##  V Geometrical overview

Apart from the macrovariables, $U,f$, let us also define another one,
corresponding to the global velocity distribution: $g(v)=\int dxf(x,v)/N$. For
the equal mass gas, this is a constant of the motion. Each of $U$, $f$ and $g$
define partitions of $\Gamma_{E}$ :
$\Gamma_{E}=\\{\Gamma_{U}\\}=\\{\Gamma_{f}\\}=\\{\Gamma_{g}\\}$, where
$\\{...\\}$ represents the collection of all possible macrostate values. The
last, $\\{\Gamma_{g}\\}$, is a partition of $\Gamma_{E}$ into sets invariant
under the dynamics. Each of the three partitions has a dominant set,
$\Gamma_{U_{\rm eq}}$, $\Gamma_{f_{\rm eq}}$, and $\Gamma_{g_{\rm eq}}$,
respectively. These are shown schematically in Fig. (16). The macrostate
$U_{\rm eq}$ corresponds to uniform profiles of the conserved fields, $f_{\rm
eq}$ corresponds to a macrostate with uniform density profile and Maxwellian
velocity distribution, and $g_{\rm eq}$ corresponds to a global Maxwellian
velocity distribution. Note that the $f$ partition of $\Gamma_{E}$ is a
refinement of the $U$ partition and also a refinement of the $g$ partition.

As shown in the figure, $\Gamma_{f_{\rm eq}}$ is the dominant set in
$\Gamma_{U_{\rm eq}}$ and in $\Gamma_{g_{\rm eq}}$ , while $\Gamma_{g_{\rm
eq}}$ has tiny regions that are outside $\Gamma_{U_{\rm eq}}$ and vice-versa.
Any initial microstate, $X_{1}(0)$, inside $\Gamma_{g_{\rm eq}}$, such as the
one chosen from thermal equilibrium in the left half of the box, will
eventually be in the region $\Gamma_{f_{\rm eq}}\cap\Gamma_{U_{\rm
eq}}\cap\Gamma_{g_{\rm eq}}=\Gamma_{f_{\rm eq}}$ which corresponds to
“complete” thermal equilibrium. On the other hand typical microstates such as
$X_{3}(0)$, chosen from outside of $\Gamma_{g_{\rm eq}}$ will end in
$\Gamma_{U_{\rm eq}}$ but outside $\Gamma_{f_{\rm eq}}$ and so in this case we
have restricted thermalization. This is seen in Fig. (15) where $s^{U}_{B}$ is
seen to reach its equilibrium value while $s^{f}_{B}$ does not. Finally one
has very special atypical microstates, $X_{2}(0)$, such as in the alternate
velocity case considered in Fig. (12), which remains outside $\Gamma_{f_{\rm
eq}}\cup\Gamma_{U_{\rm eq}}\cup\Gamma_{g_{\rm eq}}$ and there is no
thermalization at all. The above features are specific to our system, a non-
interacting integrable model. For non-integrable models it is expected that
almost any initial microstate would end in $\Gamma_{f_{\rm eq}}$ and the
system would thermalize completely.

##  VI Conclusion

We summarize here our main findings. In this paper, we have studied entropy
increase during the free expansion of an ideal gas. In the microscopic
description we start from an initial condition where the molecules are
uniformly distributed in the left half of a box and the velocities are chosen
from a thermal distribution. For this system we study the evolution of the
Boltzmann entropy defined for a single microstate with two choices of
macrovariables: the empirical single particle distribution $f_{\alpha}(t)$
[defined in Eq. (7)] and the profiles of density, momentum and energy
$U=[\rho(x),p(x),e(x)]$ (or equivalently $\rho,v,T$). The corresponding
entropies are $s^{f}_{B}$ and $s^{U}_{B}$, respectively. In equilibrium, both
these choices correspond to the thermodynamic entropy $S$.

  * •

The time evolution of the empirical density $f(x,v,t)$ and the fields
$\rho(x,t),{u(x,t)},T(x,t)$ were obtained for a single realization, starting
from a single typical initial microstate chosen from the equilibrium
macrostate where all particles are in equilibrium in the left half of a box.
We found that these agree, in the large $N$ limit, with the corresponding
fields $F_{\alpha},\bar{\rho},{\bar{u}},\bar{T}$, obtained by _averaging_ over
initial microstates taken from the relevant initial Gibbs distribution or,
more or less equivalently, from the same initial macrostate. For our model,
the averaged fields can easily be computed exactly. This demonstrates that the
evolution of the macrostates and the corresponding entropies, for single
typical microstates (belonging to a macrostate $M_{\rm ini}$), agrees with the
evolution obtained after averaging over the ensemble of initial conditions
(corresponding to the same initial macrostate $M_{\rm ini}$).

  * •

Both $s^{f}_{B}$ and $s^{U}_{B}$ increase with time and eventually reach the
expected equilibrium value with an increase of $\ln(2)$ per particle. However,
while $s^{U}_{B}$ increases monotonically with time, $s^{f}_{B}$ shows
oscillations which decay with time.

  * •

The entropies are defined in terms of coarse graining scales
$|\Delta_{\alpha}|$ for $s^{f}_{B}$ and $\ell$ for $s^{U}_{B}$. We find that
the entropy production rate for $s^{U}_{B}$ does not depend much on decreasing
grid-size $\ell$. On the other hand, $s^{f}_{B}$ decreases with decreasing
$\Delta v$. However, we find a remarkable scaling collapse of the data for
$s^{f}_{B}(t)$ for different $\Delta v$ on plotting them as a function of the
scaled time $\tau=t\Delta v/(2L)$. We provide an analytic understanding of
this collapse and compute the scaling function. This also explains the
oscillations and in particular the observation that the system periodically
goes to the maximum entropy state.

  * •

The fact that the entropy production rate for $s^{U}_{B}$ does not depend much
on grid size in the “continuum limit” of small grid size reflects a general
feature of entropy production: it often is insensitive to the choice of
macrovariables, which tend naturally to be chosen so as to be near the
continuum limit of small grid size. This in part explains why the fact that
the Boltzmann entropy depends upon, and is defined relative to, a choice of
macrovariables and macrostates can often be ignored. However when the entropy
production in the continuum limit vanishes, as it does in an ideal gas for the
$f$-macrovariables, the dependence on grid size can no longer be ignored.

  * •

We showed that the entropy increase for $S^{U}_{B}$ can be related to a
microscopic “heat” current, $J_{s}$, and formally the local entropy production
rate can be written in the form $J_{s}\partial_{x}(1/T)$. This quantity is
positive everywhere and its integral over all space gives a positive entropy
production.

  * •

Other initial conditions: The results above are for the specific case of free
expansion. We have also studied other initial conditions. We considered one
example (with an atypical initial microstate with alternate particles having
different velocities, $\pm v_{0}$) where the system never reaches a steady
state and the entropy shows persistent oscillations. Our second example
involves particles initially distributed uniformly in the box but with a non-
Maxwellian global velocity distribution. For this case we find that
$s^{U}_{B}$ saturates to its equilibrium value at long times, while
$s^{f}_{B}$ does not, corresponding to the observed fact that the
macrovariables $f$ and $U$ evolve in this case to limiting values: $U$ evolves
to its equilibrium value $U_{\rm eq}$ while $f$ evolves to $g/L$,
corresponding to the dominant $f$-macrostate given the total velocity
distribution arising from $f_{\rm initial}$.

Thus our study illustrates the crucial role of typicality, large numbers, and
coarse-graining that lead to entropy increase (starting from a low entropy
state), irreversibility and approach to thermal equilibrium, even in a non-
interacting integrable system. The ideas of ergodicity, interaction and chaos
do not seem to be so relevant.

That there is no change in the continuum limit $\mathcal{s}^{f}_{B}$, the
entropy $s_{B}^{f}$ in the limit $\Delta x,\Delta v\to 0$, agrees
computationally with what would be obtained from generalized hydrodynamics
(GHD) of integrable systems [43, 44, 45, 46, 47]. According to GHD, the
entropy is defined as the integral of a Gibbs entropy density for local
generalized canonical ensembles involving other conserved quantities in
addition to the energy. This is regarded as the fundamental entropy, and not,
as we do, as an idealization of the more fundamental entropy $s_{B}^{f}$ that
depends on the choice of macrovariables and coarse graining. Classical
integrable systems contain a macroscopic number of conserved quantities and,
according to GHD, the correct hydrodynamic description of an integrable system
is given by a specification of all the conserved fields. One then finds that
for non-interacting systems such as the ideal gas or a harmonic chain, the
hydrodynamic equations are given by the (generalized) Euler equations, which
yield no entropy production for the corresponding continuum entropy.

In fact, for the ideal gas, it is seen that the empirical single particle
density, $\tilde{f}(x,v)$, has complete information about the conserved
quantities and the hydrodynamic equations are equivalently given by Eq. (19),
implying constancy of the continuum entropy $\mathcal{s}_{B}^{f}(t)$ [see Eq.
(20)]. On the other hand, if instead we consider the contiuum entropy for the
$U$-macrostates, involving only three of the conserved quantities, we find
entropy increase. For “interacting” integrable models, such as the hard rod
system and the Toda chain, it turns out that Navier-Stokes dissipative terms
appear in the hydrodynamic equations, and hence there is continuum entropy
generation even when the macrostate description includes all conserved
quantities. We hasten to add, however, that unlike for GHD the choice of
macrovariables required for a Boltzmann entropy need not be limited to those
that are conserved. After all, the one-particle empirical distribution
$f(x,v)$ involved in the Boltzmann equation for a hard sphere gas is certainly
not conserved when collisions occur.

Thermalization in non-chaotic systems, such as the Toda chain, has been
discussed in other recent papers [48, 49]. The one-dimensional hard rod gas,
an integrable model which however has a Navier-Stokes term in the hydrodynamic
equations [28], would be another interesting case to study from the present
perspective. It will also be interesting to extend the study of the present
paper to interacting systems such as the alternate mass hard particle gas
where the masses of alternate particles take different values. This non-
integrable model has been studied extensively in the context of the breakdown
of Fourier’s law of heat conduction in one dimension [50, 51, 52, 53] and
verification of the hydrodynamic description [54, 55, 56, 57]. For this case
we expect important differences from the present integrable model even though
the equilibrium thermodynamics of both are identical. For example we expect
that the entropy production rate for $s_{B}^{f}$ at a fixed time should
converge to a finite positive value even in the limit of vanishing grid-size,
in contrast to what we see in Fig. (2).

## Acknowledgments

SC, AD and AK acknowledge support of the Department of Atomic Energy,
Government of India, under Project No. RTI4001 and would also like to
acknowledge the ICTS program on “Thermalization, Many body localization and
Hydrodynamics (Code:ICTS/hydrodynamics2019/11)” for enabling crucial
discussions related to this work. SC, AD and AK thank Cedric Bernardin, Deepak
Dhar, Francois Huveneers, Christian Maes, Stefano Olla, and Herbert Spohn for
illuminating discussions. We thank Cedric Bernardin, Vir Bulchandani, Benjamin
Doyon, David Huse and Joel Moore for helpful comments on the manuscript. The
work of JLL was supported by the AFOSR. The numerical simulations were
performed on a Mario HPC at ICTS-TIFR.

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##  Appendix A Exact solution for the evolution of macroscopic fields

Here we present exact results for the evolution of average fields
corresponding to the empirical densities $f(x,p,t)$ and
$[\rho(x,t),p(x,t),e(x,t)]$. A similar study on a ring was done earlier in
[11, 58].

The mean density $\bar{\rho}(x,t)$, momentum $\bar{p}(x,t)$ and energy
$\bar{e}(x,t)$ fields are defined as

$\displaystyle\bar{\rho}(x,t)=\left\langle\sum_{j=1}^{N}\delta(x_{j}-x)\right\rangle=\int
dv~{}F(x,v,t),$ (42)
$\displaystyle\bar{p}(x,t)=\left\langle\sum_{j=1}^{N}\delta(x_{j}-x)v_{j}\right\rangle=\int
dv~{}v~{}F(x,v,t),$ (43)
$\displaystyle\bar{e}(x,t)=\left\langle\sum_{j=1}^{N}\delta(x_{j}-x)\frac{v_{j}^{2}}{2}\right\rangle=\int
dv~{}\frac{v^{2}}{2}~{}F(x,v,t).$

Here $\langle...\rangle$ denotes an average over the initial positions and
velocities, $\\{x_{i}(0),v_{i}(0)\\}$, of all the particles, which are chosen
from the canonical distribution at temperature $T$ with all particles in the
left half $(0,L/2)$. In the following, we study the evolution of these fields
in time and space for this system with two reflecting boundaries at $x=0$ and
$x=L$.

The canonical ensemble implies that we distribute the particles uniformly in
$(0,L/2$) and draw their velocities from the Maxwell’s velocity distribution
given by

$g_{\text{eq}}(v_{0},T)=\left(\frac{1}{2\pi
T}\right)^{1/2}\exp\left[\frac{-v_{0}^{2}}{2T}\right].$ (45)

As discussed earlier, we can effectively treat the particles as non-
interacting and the problem reduces to a single particle problem [25]. We then
consider a single particle starting from an initial position $x_{0}$ with
initial velocity $v_{0}$. The final position, $x_{t}$, of the particle, taking
all possible collisions into account, can take the following forms:

$\displaystyle
x_{t}=\begin{cases}x_{0}+v_{0}t-2nL,&{\text{~{}~{}if~{}~{}}}v_{0}>0,v_{t}>0,\\\
x_{0}+v_{0}t+2nL,&{\text{~{}~{}if~{}~{}}}v_{0}<0,v_{t}<0,\\\
-x_{0}-v_{0}t+2nL,&{\text{~{}~{}if~{}~{}}}v_{0}>0,v_{t}<0,\\\
-x_{0}-v_{0}t-2nL,&{\text{~{}~{}if~{}~{}}}v_{0}<0,v_{t}>0,\end{cases}$ (46)

where $v_{t}$, the velocity at time $t$, can take either values $\pm v_{0}$
and $n=0,1,2,3...,$ is the number of collision(s) that the particle has with
both the boundaries. The distribution $F(x,v,t)$ is then obtained by averaging
over the initial position [uniform in $(0,L/2)$] and velocity [drawn from
$g_{\rm eq}(v_{0},T)$]:

$F(x,v,t)=N\langle\delta(x-x_{t})\delta(v-v_{t})\rangle.$ (47)

Using Eq. (46) we get:

$\displaystyle
F(x,v,t)=\frac{2N}{L}\int_{0}^{L/2}dx_{0}\int_{-\infty}^{\infty}dv_{0}g_{\rm
eq}(v_{0})\sum_{n=-\infty}^{\infty}\left[\delta(x-x_{0}-v_{0}t+2nL)\delta(v-v_{0})+\delta(x+x_{0}+v_{0}t-2nL)\delta(v+v_{0})\right],$
(48) $\displaystyle={2\rho_{0}}\frac{\exp(-v^{2}/2T)}{\sqrt{2\pi
T}}\sum_{n=-\infty}^{\infty}[\Theta(x-vt-2nL+L/2)-\Theta(x-vt+2nL-L/2)],$ (49)

where $\rho_{0}=N/L$. Now we calculate the three mean fields by performing the
integrals in Eqs. (A), (A), and (A) to get

$\displaystyle\bar{\rho}(x,t)$
$\displaystyle=\int_{-\infty}^{\infty}dvF(x,v,t),$
$\displaystyle={2\rho_{0}}\frac{1}{\sqrt{2\pi
T}}\sum_{n=-\infty}^{\infty}\int_{0}^{L/2}dx_{0}\int_{-\infty}^{\infty}dv~{}e^{-v^{2}/2T}\left[\delta(x-x_{0}-vt+2nL)+\delta(x+x_{0}-vt-2nL)\right],$
$\displaystyle=2\rho_{0}\frac{1}{\sqrt{2\pi
T}}\sum_{n=-\infty}^{\infty}\int_{0}^{L/2}dx_{0}\frac{1}{t}\left[\exp\left\\{\frac{-(2nL+x-x_{0})^{2}}{2Tt^{2}}\right\\}+\exp\left\\{\frac{-(2nL-
x-x_{0})^{2}}{2Tt^{2}}\right\\}\right],$
$\displaystyle=\rho_{0}\sum_{n=-\infty}^{\infty}\left[\text{erf}\left(\frac{L/2-2nL-x}{\sqrt{2T}t}\right)+\text{erf}\left(\frac{L/2-2nL+x}{\sqrt{2T}t}\right)\right].$
(50)

Using the Poisson resummation formula, this can be rewritten in the
alternative series form:

$\displaystyle\bar{\rho}(x,t)=\rho_{0}+4\rho_{0}\sum_{k=1}^{\infty}\frac{1}{k\pi}\sin\left(\frac{k\pi}{2}\right)\cos\left(\frac{k\pi
x}{L}\right)\exp\left(\frac{-k^{2}\pi^{2}Tt^{2}}{2L^{2}}\right).$ (51)

Following a similar approach we obtain the following expressions for the mean
momentum and energy:

$\displaystyle\bar{p}(x,t)$
$\displaystyle=\int_{-\infty}^{\infty}dv~{}vF(x,v,t),$
$\displaystyle=\rho_{0}\sqrt{\frac{2T}{\pi}}\sum_{n=-\infty}^{\infty}\left[\exp\left\\{-\frac{(2nL-L/2+x)^{2}}{2Tt^{2}}\right\\}-\exp\left\\{-\frac{(2nL-L/2-x)^{2}}{2Tt^{2}}\right\\}\right],$
(52)
$\displaystyle=\frac{4\rho_{0}Tt}{L}\sum_{k=1}^{\infty}\sin\left(\frac{k\pi}{2}\right)\sin\left(\frac{k\pi
x}{L}\right)\exp\left(\frac{-k^{2}\pi^{2}Tt^{2}}{2L^{2}}\right),$ (53)
$\displaystyle\bar{e}(x,t)$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}dv~{}v^{2}F(x,v,t),$
$\displaystyle=\rho_{0}\frac{2T}{\sqrt{\pi}}\sum_{n=-\infty}^{\infty}\left[-\frac{\sqrt{\pi}}{4}\text{erf}\left(\frac{2nL-L/2+x}{\sqrt{2T}t}\right)+\frac{2nL-L/2+x}{2\sqrt{2T}t}\exp\left\\{\frac{-(2nL-L/2+x)^{2}}{2Tt^{2}}\right\\}\right.$
$\displaystyle\left.-\frac{\sqrt{\pi}}{4}\text{erf}\left(\frac{2nL-L/2-x}{\sqrt{2T}t}\right)+\frac{2nL-L/2-x}{2\sqrt{2T}t}\exp\left\\{\frac{-(2nL-L/2-x)^{2}}{2Tt^{2}}\right\\}\right].$
(54)
$\displaystyle=\frac{\rho_{0}T}{2}+2\rho_{0}T\sum_{k=1}^{\infty}\frac{1}{k\pi}\left(1-\frac{k^{2}\pi^{2}Tt^{2}}{L^{2}}\right)\sin\left(\frac{k\pi}{2}\right)\cos\left(\frac{k\pi
x}{L}\right)\exp\left(\frac{-k^{2}\pi^{2}Tt^{2}}{2L^{2}}\right).$ (55)

From Eqs. (51, 53, 55) one can easily see that the approach to equilibrium has
the long-time form $e^{-at^{2}}$ with $a=T\pi^{2}/(2L^{2})$.

##  Appendix B Numerical methods

Here we describe the details of the simulation procedure. We consider a box
$(0,L)$ with particles initially distributed uniformly in the left half
$(0,L/2)$ of the box. The velocity of each particle is chosen independently
from a specified distribution. Since the particles are non-interacting, their
velocities change only during collisions with the walls at $x=0$ and $x=L$.
The position of each particle is updated according to Eqs. (46) which
incorporates the effect of reflecting boundaries. With the system evolving in
time with this dynamics, we compute the required observables.