# On the thermal relaxation of a dense gas described by the modified Enskog
equation in a closed system in contact with a heat bath

<EMAIL_ADDRESS>
###### Abstract.

The thermal relaxation of a dense gas described by the modified Enskog
equation is studied for a closed system in contact with a heat bath. As in the
case of the Boltzmann equation, the Helmholtz free energy $\mathcal{F}$ that
decreases monotonically in time is found under the conventional kinetic
boundary condition that satisfies the Darrozes–Guiraud inequality. The
extension to the modified Enskog–Vlasov equation is also presented.

###### Key words and phrases:

Enskog equation, kinetic theory, dense gas, Helmholtz free energy, H theorem.

###### 1991 Mathematics Subject Classification:

Primary: 82C40, 82B40; Secondary: 76P99.

∗ Corresponding author: Shigeru Takata

Shigeru TAKATA∗

Department of Aeronautics and Astronautics, Kyoto University,

Kyoto daigaku-katsura, Nishikyo-ku, Kyoto 615-8540, Japan

(Communicated by the associate editor name)

##  1\. Introduction

Behavior of ideal gases is well described by the Boltzmann equation for the
entire range of the Knudsen number, the ratio of the mean free path of gas
molecules to a characteristic length of the system. The kinetic theory based
on the Boltzmann equation and its model equations has been applied
successfully to analyses of various gas flows in low pressure circumstances,
micro-scale gas flows, and gas flows caused by the evaporation/condensation at
the gas-liquid interface.

The extension of the kinetic theory to non-ideal gases would go back to the
dates of Enskog [8]. He took account of the displacement effect of molecules
in collision integrals for a hard-sphere gas and proposed a kinetic equation
that is nowadays called the (original) Enskog equation. In the original Enskog
equation, there appears a weight function that represents an equilibrium
correlation function at the contact point of two colliding molecules. On one
hand, satisfactory outcomes of the original Enskog equation, such as the dense
gas effects on the transport properties, led to recent developments of
numerical algorithms [9, 20] and their applications to physical problems,
e.g., [10, 14, 20, 11]. On the other hand, the intuitive choice of the
correlation was recognized to cause some difficulties in recovering the H
theorem, as well as the Onsager reciprocity in the case of mixtures, and
triggered off further intensive studies on the foundation of the equation
around from late 60’s to early 80’s, see, e.g., [19, 16, 15] and references
therein.

Among many efforts in the above-mentioned period, Resibois [16] succeeded to
prove the H theorem, not for the original but for the modified Enskog equation
[19] equipped with another form of correlation function. In most cases,
including the work of Resibois, the H theorem was discussed mainly for
periodic or unbounded spatial domains, or for cases where the influence of a
boundary was not necessary to consider, e.g., [16, 12, 1, 13]. Rather
recently, a proof was given by Maynar et al. [15] for an isolated system,
assuming the specular reflection condition, where special care was directed to
a restriction on the range of collision integral near the boundary. It seems,
however, that the thermal relaxation in contact with a heat bath receives
little attention in the literature, despite the fact that it is one of the
fundamental issues in the thermo-statistical physics. Although the interaction
with the thermostat boundary is considered in a recent monograph of Dorfman et
al. [6], we are not aware of a direct discussion on the thermal relaxation of
a dense gas in a closed system in contact with a heat bath in the context of
the modified Enskog equation.

In the present paper, we would like to fill the gap by a simple argument and
to show that, if the boundary condition satisfies the Darrozes–Guiraud
inequality [7] that is conventionally required in the kinetic theory, the
Helmholtz free energy $\mathcal{F}$ that decreases monotonically in time can
be found for a closed system described by the modified Enskog equation as in
the case of the Boltzmann equation.

##  2\. Problem and formulation

Consider a dense gas in a domain that is surrounded by a simple resting solid
wall kept at a uniform temperature $T_{w}$, i.e., a heat bath with temperature
$T_{w}$. We will study the relaxation of the gas toward a thermal equilibrium
state with the heat bath under the following assumptions:

  1. (1)

The behavior of the gas is described by the modified Enskog equation for a
single species gas;

  2. (2)

The gas molecules are hard spheres with a common diameter $\sigma$ and mass
$m$ and the collisions among themselves are elastic;

  3. (3)

The velocity distribution of gas molecules reflected on the surface of the
heat bath is described by the kinetic boundary condition that is
conventionally used for the Boltzmann equation, the details of which will be
given in (8).

Let $D$ be a fixed spatial domain that the centers of molecules of a gas can
occupy. Let $t$, $\bm{X}$ and $\bm{Y}$, and $\bm{\xi}$ be a time, spatial
positions, and a molecular velocity, respectively. Then, denoting the one-
particle distribution function of gas molecules by $f(t,\bm{X},\bm{\xi}$) and
the correlation function by $g(t,\bm{X},\bm{Y})$, the modified Enskog equation
is written as

$\displaystyle\frac{\partial f}{\partial t}+\xi_{i}\frac{\partial f}{\partial
X_{i}}=J_{ME}(f)\equiv J_{ME}^{G}(f)-J_{ME}^{L}(f),\quad\mathrm{for\
}\bm{X}\in D,$ (1a) $\displaystyle
J_{ME}^{G}(f)\equiv\frac{\sigma^{2}}{m}\int{g(\bm{X}_{\sigma\bm{\alpha}}^{+},\bm{X})f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{+})f^{\prime}(\bm{X})}V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}_{*},$
(1b) $\displaystyle
J_{ME}^{L}(f)\equiv\frac{\sigma^{2}}{m}\int{g(\bm{X}_{\sigma\bm{\alpha}}^{-},\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})}V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}_{*},$
(1c)

where $\bm{X}_{\bm{x}}^{\pm}=\bm{X}\pm\bm{x}$, $\bm{\alpha}$ is a unit vector,

$\theta(x)=\begin{cases}1,&x\geq 0\\\ 0,&x<0\end{cases},$ (2)

$d\Omega(\bm{\alpha})$ is a solid angle element in the direction of
$\bm{\alpha}$, and the following notation convention is used:

$\displaystyle\begin{cases}{f(\bm{X})=f(\bm{X},\bm{\xi}),\
f^{\prime}(\bm{X})=f(\bm{X},\bm{\xi}^{\prime})},\\\
{f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})=f(\bm{X}_{\sigma\bm{\alpha}}^{-},\bm{\xi}_{*}),\
f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})=f(\bm{X}_{\sigma\bm{\alpha}}^{-},\bm{\xi}_{*}^{\prime})},\end{cases}$
(3)
$\displaystyle\bm{\xi}^{\prime}=\bm{\xi}+V_{\alpha}\bm{\alpha},\quad\bm{\xi}_{*}^{\prime}=\bm{\xi}_{*}-V_{\alpha}\bm{\alpha},\quad
V_{\alpha}=\bm{V}\cdot\bm{\alpha},\quad\bm{V}=\bm{\xi_{*}}-\bm{\xi}.$ (4)

Here and in what follows, the argument $t$ is suppressed, unless confusion is
anticipated. Our correlation function $g$ is adjusted to the domain $D$ in
such a way that the usual correlation function $g_{2}(t,\bm{X},\bm{Y})$ is
modified as

$\displaystyle
g(t,\bm{X},\bm{Y})=g_{2}(t,\bm{X},\bm{Y})\chi_{D}(\bm{X})\chi_{D}(\bm{Y}),$
(5a) $\displaystyle\chi_{D}(\bm{X})=\begin{cases}1,&\bm{X}\in D\\\
0,&\mbox{otherwise}\end{cases},$ (5b)

where $\chi_{D}$ plays the same role as the Heaviside function $\theta$.
Consequently, the range of integration in (1b) and (1c) can be treated as the
whole space of $\bm{\xi}_{*}$ and all directions of $\bm{\alpha}$ even near
the surface of the domain $\partial D$. In contrast to the original Enskog
equation, $g_{2}$ takes a complicated form that requires further supplemental
notation. For the moment, it suffices to mention that $g_{2}$ has a symmetric
property $g_{2}(t,\bm{X},\bm{Y})=g_{2}(t,\bm{Y},\bm{X})$ and is a functional
of a gas density

$\rho=\int fd\bm{\xi}.$ (6)

Therefore (1) is closed as the equation for $f$. By (5), $g$ has the same
symmetric property as $g_{2}$:

$g(t,\bm{X},\bm{Y})=g(t,\bm{Y},\bm{X}).$ (7)

Further details of $g_{2}$ can be found in Appendix A.

The boundary condition is applied on the surface $\partial D$ of the domain
$D$:

$f(t,\bm{X},\bm{\xi})=\int_{\bm{\xi}_{*}\cdot\bm{n}<0}K(\bm{\xi},\bm{\xi}_{*}|\bm{X})f(t,\bm{X},\bm{\xi}_{*})d\bm{\xi}_{*},\quad(\bm{\xi}\cdot\bm{n}>0,\
\bm{X}\in\partial D),$ (8a) where $K(\bm{\xi},\bm{\xi}_{*}|\bm{X})$ is a
scattering kernel assumed to be time-independent, $\bm{n}$ is the inward unit
normal to the surface $\partial D$ at position $\bm{X}$, and the boundary is
assumed to be at rest. The following properties are conventionally supposed
for a kinetic boundary condition: [17]

  1. (1)

Non-negativeness:

$K(\bm{\xi},\bm{\xi}_{*}|\bm{X})\geq 0,\quad(\bm{\xi}\cdot\bm{n}>0,\
\bm{\xi}_{*}\cdot\bm{n}<0);$ (8b)

  2. (2)

Normalization:

$\int_{\bm{\xi}\cdot\bm{n}>0}\Big{|}\frac{\bm{\xi}\cdot\bm{n}}{\bm{\xi}_{*}\cdot\bm{n}}\Big{|}K(\bm{\xi},\bm{\xi}_{*}|\bm{X})d\bm{\xi}=1,\quad(\bm{\xi}_{*}\cdot\bm{n}<0),$
(8c)

where the integrand in (8c) is the so-called reflection probability density.
Equation (8c) implies that the boundary $\partial D$ is impermeable;

  3. (3)

Preservation of equilibrium: The resting Maxwellian $f_{w}$ characterized by
the surface temperature $T_{w}$, i.e.,

$f_{w}=\frac{a}{(2\pi RT_{w})^{3/2}}\exp(-\frac{\bm{\xi}^{2}}{2RT_{w}}),$ (8d)

with $a(>0)$ being arbitrary, satisfies the boundary condition (8a), and the
other Maxwellians do not satisfy (8a).

The diffuse reflection, the Maxwell, and the Cercignani–Lampis condition [4,
2, 17] that are widely used for the Boltzmann equation are specific examples
of (8). Note that the uniqueness in the third property listed above excludes
the adiabatic boundary such as the specular reflection condition. As to the H
theorem for the specular reflection case, the reader is referred to [15].

The form of the modified Enskog equation (1) is identical to the one for a
confined isolated system discussed in [15]. In our formulation, $\chi_{D}$ is
used to make simpler the integration range near the surface $\partial D$.

##  3\. Collisional contributions to the momentum and the energy transport

Before going into details, we recall three types of operation that are useful
in the transformation of the moments of collision integrals:

(I):

to exchange the letters $\bm{\xi}$ and $\bm{\xi}_{*}$;

(II):

to reverse the direction of $\bm{\alpha}$ (or introduce
$\bm{\beta}=-\bm{\alpha}$);

(III):

to change the integration variables from $(\bm{\xi},\bm{\xi}_{*},\bm{\alpha})$
to $(\bm{\xi}^{\prime},\bm{\xi}_{*}^{\prime},\bm{\alpha})$ and then to change
the letters $(\bm{\xi}^{\prime},\bm{\xi}_{*}^{\prime})$ to
$(\bm{\xi},\bm{\xi}_{*})$.

These operations will be used also in Sec. 4.

We then notice that, by (III) and (II),

$\int{\varphi}J_{ME}^{G}(f)d\bm{\xi}=\int{\varphi}^{\prime}J_{ME}^{L}(f)d\bm{\xi},$
(9)

holds for any ${\varphi}(\bm{\xi})$ and thus

$\displaystyle\frac{m}{\sigma^{2}}\int{\varphi}(\bm{\xi})J_{ME}(f)d\bm{\xi}$
$\displaystyle=$
$\displaystyle\int({\varphi}^{\prime}-{\varphi})g(\bm{X}_{\sigma\bm{\alpha}}^{-},\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}_{*}d\bm{\xi}.$
(10)

First, it is obvious from (10) with ${\varphi}=1$ that $\int
J_{ME}(f)d\bm{\xi}=0$. Hence, the continuity equation is obtained by the
integration of (1a) with respect to $\bm{\xi}$:

$\frac{\partial\rho}{\partial
t}+\frac{\partial}{\partial\bm{X}}\cdot(\rho\bm{v})=0.$ (11)

Here $\bm{v}$ (or $v_{i}$) is a flow velocity defined by

$v_{i}=\frac{1}{\rho}\int\xi_{i}fd\bm{\xi}.$ (12)

Next, consider two kinds of collision invariants $\psi$ as ${\varphi}$ in
(10): (i) $\psi(\bm{\xi})=\xi_{i}$ and (ii) $\psi(\bm{\xi})=\bm{\xi}^{2}/2$.
One of the main qualitative differences from the Boltzmann equation is that
$\psi$-moment of the collision term does not vanish in general. For both (i)
and (ii), (10) with ${\varphi}=\psi$ can be transformed as

$\displaystyle\frac{m}{\sigma^{2}}\int\psi(\bm{\xi})J_{ME}(f)d\bm{\xi}=$
$\displaystyle\int(\psi_{*}^{\prime}-\psi_{*})g(\bm{X}_{\sigma\bm{\beta}}^{+},\bm{X})f(\bm{X}_{\sigma\bm{\beta}}^{+})f_{*}(\bm{X})V_{\beta}\theta(V_{\beta})d\Omega(\bm{\beta})d\bm{\xi}d\bm{\xi}_{*}$
$\displaystyle=$
$\displaystyle\int(\psi-\psi^{\prime})g(\bm{X}_{\sigma\bm{\beta}}^{+},\bm{X})f(\bm{X}_{\sigma\bm{\beta}}^{+})f_{*}(\bm{X})V_{\beta}\theta(V_{\beta})d\Omega(\bm{\beta})d\bm{\xi}d\bm{\xi}_{*},$
(13a) where (I) and (II) are used at the first equality, while
$\psi$+$\psi_{*}$=$\psi^{\prime}$+$\psi_{*}^{\prime}$ is used at the second
equality. Combining (13a) and (10) for ${\varphi}=\psi$ gives
$\displaystyle\frac{m}{\sigma^{2}}\int\psi(\bm{\xi})J_{ME}(f)d\bm{\xi}=$
$\displaystyle\frac{1}{2}\int(\psi^{\prime}-\psi)\\{g(\bm{X}_{\sigma\bm{\alpha}}^{-},\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})$
$\displaystyle-g(\bm{X}_{\sigma\bm{\alpha}}^{+},\bm{X})f(\bm{X}_{\sigma\bm{\alpha}}^{+})f_{*}(\bm{X})\\}V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}_{*}d\bm{\xi}.$
(13b)

Since

$\displaystyle
g(\bm{X}_{\sigma\bm{\alpha}}^{-},\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})-g(\bm{X}_{\sigma\bm{\alpha}}^{+},\bm{X})f(\bm{X}_{\sigma\bm{\alpha}}^{+})f_{*}(\bm{X})$
$\displaystyle=$
$\displaystyle-\int_{0}^{\sigma}\frac{\partial}{\partial\lambda}g(\bm{X}_{\lambda\bm{\alpha}}^{+},\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f_{*}(\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f(\bm{X}_{\lambda\bm{\alpha}}^{+})d\lambda$
$\displaystyle=$
$\displaystyle-\bm{\nabla}\cdot\int_{0}^{\sigma}\bm{\alpha}g(\bm{X}_{\lambda\bm{\alpha}}^{+},\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f_{*}(\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f(\bm{X}_{\lambda\bm{\alpha}}^{+})d\lambda,$
(14)

(13b) gives rise to the notion of collisional contributions to the stress
tensor $p_{ij}^{(c)}$ and the heat flow $q_{i}^{(c)}$ defined as

$\displaystyle p_{ij}^{(c)}=$
$\displaystyle\frac{\sigma^{2}}{2m}\int{\int_{0}^{\sigma}}\alpha_{i}\alpha_{j}V_{\alpha}^{2}\theta(V_{\alpha})$
$\displaystyle\quad
g(\bm{X}_{\lambda\bm{\alpha}}^{+},\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f_{*}(\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f(\bm{X}_{\lambda\bm{\alpha}}^{+}){d\lambda}d\Omega(\bm{\alpha})d\bm{\xi}_{*}d\bm{\xi},$
(15a) $\displaystyle q_{i}^{(c)}=$ $\displaystyle-
p_{ij}^{(c)}v_{j}+\frac{\sigma^{2}}{4m}\int{\int_{0}^{\sigma}}\alpha_{i}[(\bm{\xi}+\bm{\xi}_{*})\cdot\bm{\alpha}]V_{\alpha}^{2}\theta(V_{\alpha})$
$\displaystyle\quad
g(\bm{X}_{\lambda\bm{\alpha}}^{+},\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f_{*}(\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f(\bm{X}_{\lambda\bm{\alpha}}^{+}){d\lambda}d\Omega(\bm{\alpha})d\bm{\xi}_{*}d\bm{\xi}$
$\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{4m}\int{\int_{0}^{\sigma}}\alpha_{i}[(\bm{c}+\bm{c}_{*})\cdot\bm{\alpha}]V_{\alpha}^{2}\theta(V_{\alpha})$
$\displaystyle\quad
g(\bm{X}_{\lambda\bm{\alpha}}^{+},\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f_{*}(\bm{X}_{(\lambda-\sigma)\bm{\alpha}}^{+})f(\bm{X}_{\lambda\bm{\alpha}}^{+}){d\lambda}d\Omega(\bm{\alpha})d\bm{\xi}_{*}d\bm{\xi},$
(15b)

see e.g., [5, 10]. Here $\bm{c}=\bm{\xi}-\bm{v}$,
$\bm{c}_{*}=\bm{\xi}_{*}-\bm{v}$, and

$\psi^{\prime}-\psi=\begin{cases}V_{\alpha}\alpha_{i},&{\displaystyle(\psi=\xi_{i}),}\\\
{\displaystyle\frac{1}{2}V_{\alpha}(\bm{\xi}+\bm{\xi}_{*})\cdot\bm{\alpha},}&({\displaystyle\psi=\frac{1}{2}\bm{\xi}^{2}),}\end{cases}$
(16)

have been used. Note that, thanks to the factor $\chi_{D}$ in $g$, the range
of integration with respect to $\lambda$ is simply from $0$ to $\sigma$,
regardless of the position $\bm{X}$ in $D$.

To summarize, two expressions for the same quantity have been obtained. For
the quantity related to the energy,

$\displaystyle\int\frac{1}{2}\bm{\xi}^{2}J_{ME}(f)d\bm{\xi}$ $\displaystyle=$
$\displaystyle-\frac{\sigma^{2}}{2m}\int[(\bm{\xi}+\bm{\xi}_{*})\cdot\bm{\alpha}]V_{\alpha}^{2}\theta(V_{\alpha})g(\bm{X}_{\sigma\bm{\alpha}}^{+},\bm{X})f(\bm{X}_{\sigma\bm{\alpha}}^{+})f_{*}(\bm{X})d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*},$
(17a) and
$\int\frac{1}{2}\bm{\xi}^{2}J_{ME}(f)d\bm{\xi}=-\frac{\partial}{\partial
X_{i}}(p_{ij}^{(c)}v_{j}+q_{i}^{(c)}),$ (17b)

see (13a) with (16) and (15); for the quantity related to the momentum,

$\displaystyle\int\xi_{i}J_{ME}(f)d\bm{\xi}$ $\displaystyle=$
$\displaystyle-\frac{\sigma^{2}}{m}\int\alpha_{i}V_{\alpha}^{2}\theta(V_{\alpha})g(\bm{X}_{\sigma\bm{\alpha}}^{+},\bm{X})f(\bm{X}_{\sigma\bm{\alpha}}^{+})f_{*}(\bm{X})d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*},$
(18a) and $\int\xi_{i}J_{ME}(f)d\bm{\xi}=-\frac{\partial}{\partial
X_{j}}p_{ij}^{(c)},$ (18b)

see (13a) with (16) and (15a).

Finally by integrating (17a) over the domain $D$ and recalling (5a), it is
seen that

$\displaystyle\int_{D}\int\frac{1}{2}\bm{\xi}^{2}J_{ME}(f)d\bm{\xi}d\bm{X}$
$\displaystyle=$
$\displaystyle-\frac{\sigma^{2}}{2m}\int[(\bm{\xi}+\bm{\xi}_{*})\cdot\bm{\alpha}]V_{\alpha}^{2}\theta(V_{\alpha})g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})d\bm{\xi}d\bm{\xi}_{*}d\Omega(\bm{\alpha})d\bm{X}$
$\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2m}\int[(\bm{\xi}+\bm{\xi}_{*})\cdot\bm{\beta}]V_{\beta}^{2}\theta(-V_{\beta})g(\bm{X},\bm{X}_{\sigma\bm{\beta}}^{+})f(\bm{X})f_{*}(\bm{X}_{\sigma\bm{\beta}}^{+})d\bm{\xi}d\bm{\xi}_{*}d\Omega(\bm{\beta})d\bm{X}$
$\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2m}\int[(\bm{\xi}+\bm{\xi}_{*})\cdot\bm{\beta}]V_{\beta}^{2}\theta(V_{\beta})g(\bm{X},\bm{X}_{\sigma\bm{\beta}}^{+})f_{*}(\bm{X})f(\bm{X}_{\sigma\bm{\beta}}^{+})d\bm{\xi}_{*}d\bm{\xi}d\Omega(\bm{\beta})d\bm{X}$
$\displaystyle=$
$\displaystyle-\int_{D}\int\frac{1}{2}\bm{\xi}^{2}J_{ME}(f)d\bm{\xi}d\bm{X},$
(19)

where the position is shifted by $-\sigma\bm{\alpha}$ at the first equality,
(II) and (I) are applied respectively at the second and the third equality,
and (17a) is used at the last equality. Hence

$\int_{D}\int\frac{1}{2}\bm{\xi}^{2}J_{ME}(f)d\bm{\xi}d\bm{X}=0,$ (20)

and by (17b)

$-\int_{D}\frac{\partial}{\partial
X_{i}}(p_{ij}^{(c)}v_{j}+q_{i}^{(c)})d\bm{X}=\int_{\partial
D}(p_{ij}^{(c)}v_{j}+q_{i}^{(c)})n_{i}dS=0.$ (21)

Here the divergence theorem has been used and $\bm{n}$ is the inward unit
normal to the surface $\partial D$. In the same way, it can be shown that

$\int_{D}\int\xi_{i}J_{ME}(f)d\bm{\xi}d\bm{X}=0,$ (22)

and by (18b)

$-\int_{D}\frac{\partial}{\partial X_{j}}p_{ij}^{(c)}d\bm{X}=\int_{\partial
D}p_{ij}^{(c)}n_{j}dS=0.$ (23)

######  Lemma 3.1.

In total, there are no collisional contributions to the momentum and energy
transport:

$\int_{D}\int\xi_{i}J_{ME}(f)d\bm{\xi}d\bm{X}=0,\quad\int_{D}\int\frac{1}{2}\bm{\xi}^{2}J_{ME}(f)d\bm{\xi}d\bm{X}=0.$
(24)

Accordingly, there are no collisional contributions to the net momentum and
energy transport to the surface $\partial D$:

$\int_{\partial D}p_{ij}^{(c)}n_{j}dS=0,\quad\int_{\partial
D}(p_{ij}^{(c)}v_{j}+q_{i}^{(c)})n_{i}dS=0.$ (25)

In particular, if $D$ is convex, $p_{ij}^{(c)}\equiv 0$ and $q_{i}^{(c)}\equiv
0$ on the surface $\partial D$.

###### Proof.

Equations (24) and (25) are simply a summary of the present section. When $D$
is convex,
$\chi_{D}(\bm{X}_{+(\lambda-\sigma)\bm{\alpha}})\chi_{D}(\bm{X}_{+\lambda\bm{\alpha}})=0$
for $\bm{X}\in\partial D$, except for the special case that $\partial D$ is
flat at $\bm{X}$. However, the exception occurs only for $\bm{\alpha}$ in the
directions tangential to $\partial D$ and thus has no contribution to the
integration of the angle in (15a) and (15b). ∎

######  Remark 1.

Equation (24) in Lemma 3.1 is physically a natural consequence, since the
collisional transport of momentum and energy comes from interactions within
gas molecules. The collisional stress tensor $p_{ij}^{(c)}$ and heat flow
$q_{i}^{(c)}$ are, however, not likely to vanish pointwisely on the surface
$\partial D$ if the domain $D$ is not convex.

##  4\. H function

In this section, we shall recall the discussions on the H theorem in the
literature [16, 15, 6]. Consider first the so-called kinetic part of the H
function111To be precise, it is necessary to make the argument of the
logarithmic function dimensionless, like $\ln(f/c_{0})$ with a constant
$c_{0}$ having the same dimension as $f$. We, however, leave the argument
dimensional to avoid additional calculations that do not affect the results.

$\mathcal{H}^{(k)}\equiv\int_{D}\int f{\ln f}d\bm{\xi}d\bm{X},$ (26)

Then, multiplying $1+{\ln f}$ with the modified Enskog equation (1a) gives

$\frac{\partial}{\partial t}\langle f{\ln f}\rangle+\frac{\partial}{\partial
X_{i}}\langle\xi_{i}f{\ln f}\rangle=\langle J_{ME}(f){\ln f}\rangle,$ (27)

after the integration with respect to $\bm{\xi}$, where
$\langle\bullet\rangle=\int\bullet\ d\bm{\xi}$. The first step toward the H
theorem is to apply (10) with ${\varphi=\ln f}$ to the right-hand side:

$\displaystyle\frac{m}{\sigma^{2}}\langle J_{ME}(f){\ln f}\rangle$
$\displaystyle=$
$\displaystyle\int\ln[f^{\prime}(\bm{X})/f(\bm{X})]g(\bm{X}_{\sigma\bm{\alpha}}^{-},\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}_{*}d\bm{\xi}.$
(28)

Then, the integration of (28) over the domain $D$ is again a relevant step for
the position shift by $+\sigma\bm{\alpha}$ and gives

$\displaystyle\frac{m}{\sigma^{2}}\int_{D}\langle J_{ME}(f){\ln f}\rangle
d\bm{X}$ $\displaystyle=$
$\displaystyle\int\ln[f^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{+})/f(\bm{X}_{\sigma\bm{\alpha}}^{+})]g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{+})f_{*}(\bm{X})f(\bm{X}_{\sigma\bm{\alpha}}^{+})V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}_{*}d\bm{\xi}d\bm{X}$
$\displaystyle=$
$\displaystyle\int\ln[f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})/f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})]g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*}d\bm{X}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int\ln\Big{(}\frac{f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})f^{\prime}(\bm{X})}{f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})}\Big{)}g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*}d\bm{X},$
(29)

where (II) and (I) are applied at the second equality, while the third line
and (28) are combined at the last equality. Since for any $x,y>0$

$x\ln(y/x)\leq y-x,$ (30)

where equality holds if and only if $y=x$,

$\int_{D}\langle J_{ME}(f){\ln f}\rangle d\bm{X}\leq I(t),$ (31)

holds, where

$I(t)=\frac{\sigma^{2}}{2m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})[f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})f^{\prime}(\bm{X})-f(\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})]V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*}d\bm{X}.$
(32)

Equation (32) can be transformed as

$\displaystyle I(t)=$ $\displaystyle-\frac{\sigma^{2}}{2m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})f^{\prime}(\bm{X})V_{\alpha}^{\prime}\theta(-V_{\alpha}^{\prime})d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*}d\bm{X}$
$\displaystyle-\frac{\sigma^{2}}{2m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})V_{\alpha}\theta(V_{\alpha})d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*}d\bm{X}$
$\displaystyle=$ $\displaystyle\frac{\sigma^{2}}{2m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})[(\bm{\xi}-\bm{\xi}_{*})\cdot\bm{\alpha}]d\Omega(\bm{\alpha})d\bm{\xi}d\bm{\xi}_{*}d\bm{X}$
$\displaystyle=$ $\displaystyle\frac{\sigma^{2}}{2m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})\rho(\bm{X})\rho(\bm{X}_{\sigma\bm{\alpha}}^{-})\bm{v}(\bm{X})\cdot\bm{\alpha}d\Omega(\bm{\alpha})d\bm{X}$
$\displaystyle-\frac{\sigma^{2}}{2m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{-})\rho(\bm{X})\rho(\bm{X}_{\sigma\bm{\alpha}}^{-})\bm{v}(\bm{X}_{\sigma\bm{\alpha}}^{-})\cdot\bm{\alpha}d\Omega(\bm{\alpha})d\bm{X}$
$\displaystyle=$ $\displaystyle\frac{\sigma^{2}}{2m}\int
g(\bm{X}_{\sigma\bm{\alpha}}^{+},\bm{X})\rho(\bm{X}_{\sigma\bm{\alpha}}^{+})\rho(\bm{X})\bm{v}(\bm{X}_{\sigma\bm{\alpha}}^{+})\cdot\bm{\alpha}d\Omega(\bm{\alpha})d\bm{X}$
$\displaystyle+\frac{\sigma^{2}}{2m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{+})\rho(\bm{X})\rho(\bm{X}_{\sigma\bm{\alpha}}^{+})\bm{v}(\bm{X}_{\sigma\bm{\alpha}}^{+})\cdot\bm{\alpha}d\Omega(\bm{\alpha})d\bm{X}$
$\displaystyle=$ $\displaystyle\frac{\sigma^{2}}{m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{+})\rho(\bm{X})\rho(\bm{X}_{\sigma\bm{\alpha}}^{+})\bm{v}(\bm{X}_{\sigma\bm{\alpha}}^{+})\cdot\bm{\alpha}d\Omega(\bm{\alpha})d\bm{X},$
(33)

where
$V_{\alpha}^{\prime}\equiv(\bm{\xi}_{*}^{\prime}-\bm{\xi}^{\prime})\cdot\bm{\alpha}=-V_{\alpha}$
is used at the first equality, (III) is used at the second equality, the
integration with respect to $\bm{\xi}$ and $\bm{\xi}_{*}$ is performed at the
third equality, and the shift operation by $+\sigma\bm{\alpha}$ and (II) are
used at the fourth equality. The last line of (33) is further transformed as

$\displaystyle I(t)=$ $\displaystyle\frac{\sigma^{2}}{m}\int
g(\bm{X},\bm{X}_{\sigma\bm{\alpha}}^{+})\rho(\bm{X})\rho(\bm{X}_{\sigma\bm{\alpha}}^{+})\bm{v}(\bm{X}_{\sigma\bm{\alpha}}^{+})\cdot\bm{\alpha}d\Omega(\bm{\alpha})d\bm{X}$
$\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{m}\int\delta(|\bm{X}-\bm{Y}|-\sigma)g(\bm{X},\bm{Y})\rho(\bm{X})\rho(\bm{Y})\bm{v}(\bm{Y})\cdot\frac{\bm{Y}-\bm{X}}{\sigma^{2}|\bm{Y}-\bm{X}|}d\bm{Y}d\bm{X}$
$\displaystyle=$ $\displaystyle\frac{1}{m}\int
g(\bm{X},\bm{Y})\rho(\bm{X})\rho(\bm{Y})\bm{v}(\bm{Y})\cdot\frac{\partial}{\partial\bm{Y}}\theta(|\bm{X}-\bm{Y}|-\sigma)d\bm{Y}d\bm{X}$
$\displaystyle=$ $\displaystyle\frac{1}{m}\int_{D\times
D}g_{2}(\bm{X},\bm{Y})\rho(\bm{X})\rho(\bm{Y})\bm{v}(\bm{X})\cdot\frac{\partial}{\partial\bm{X}}\theta(|\bm{X}-\bm{Y}|-\sigma)d\bm{X}d\bm{Y},$
(34)

and the last line is reduced by (60) in Appendix A to

$\displaystyle I(t)$
$\displaystyle=\int_{D}\rho\bm{v}\cdot\frac{\partial}{\partial\bm{X}}{\ln\frac{\rho}{w}}d\bm{X}$
$\displaystyle=-\int_{\partial
D}\rho\bm{v}\cdot\bm{n}{\ln\frac{\rho}{w}}dS-\int_{D}({\ln\frac{\rho}{w}})\frac{\partial}{\partial\bm{X}}\cdot(\rho\bm{v})d\bm{X}$
$\displaystyle=\int_{D}\frac{\partial\rho}{\partial
t}{\ln\frac{\rho}{w}}d\bm{X}$
$\displaystyle=\frac{d}{dt}\int_{D}\rho({\ln\frac{\rho}{w}}-1)d\bm{X}+\int_{D}\frac{\rho}{w}\frac{\partial
w}{\partial t}d\bm{X}$
$\displaystyle=\frac{d}{dt}\Big{(}\int_{D}\rho{\ln\frac{\rho}{w}}d\bm{X}+m\ln\phi\Big{)}.$
(35)

Here $\bm{v}\cdot\bm{n}=0$ on $\partial D$, the continuity equation (11), and
the relation

$\displaystyle\frac{1}{\phi}\frac{d\phi}{dt}=$
$\displaystyle\frac{N}{\phi}\int_{D^{N}}\frac{\partial w(\bm{X}_{1})}{\partial
t}w(\bm{X}_{2})\cdots
w(\bm{X}_{N})\Theta(\bm{X}_{1},\cdots,\bm{X}_{N})d\bm{X}_{1}\cdots
d\bm{X}_{N}$ $\displaystyle=$ $\displaystyle\frac{1}{m}\int_{D}\frac{\partial
w(\bm{X}_{1})}{\partial t}\frac{\rho(\bm{X}_{1})}{w(\bm{X}_{1})}d\bm{X}_{1},$
(36)

have been used; see (54a), (54b), and (56) in Appendix A, as for (36). Hence,
we finally arrive at

$I(t)=-\frac{d\mathcal{H}^{(c)}}{dt},$ (37)

where $\mathcal{H}^{(c)}$ is a so-called collisional part of the H function
defined by

$\mathcal{H}^{(c)}(t)\equiv-\int_{D}\rho(\bm{X}){\ln\frac{\rho(\bm{X})}{w(\bm{X})}}d\bm{X}-m\ln\phi.$
(38)

The total H function $\mathcal{H}\equiv\mathcal{H}^{(k)}+\mathcal{H}^{(c)}$
thus satisfies the following inequality:

$\frac{d\mathcal{H}}{dt}+\int_{D}\frac{\partial}{\partial
X_{i}}\langle\xi_{i}f{\ln f}\rangle d\bm{X}\leq 0,$ (39)

where the equality holds if and only if
$f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})f^{\prime}(\bm{X})=f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})$.

######  Remark 2.

The above $\mathcal{H}$ is bounded. See Appendix B.

######  Remark 3.

If the system is isolated, the second term on the left-hand side of (39)
vanishes, and $\mathcal{H}$ monotonically decreases in time as shown in [15].
Therefore, $-R\mathcal{H}$ is identified as a natural extension of the
thermodynamic entropy to the case of non-equilibrium state. Equation (39)
combined with the following lemma, i.e., Lemma 4.1, can be found in [6, p.
270].

######  Lemma 4.1.

(Darrozes–Guiraud [7, 2, 17]) If the velocity distribution function $f$
satisfies the boundary condition (8), then it holds that

$\int_{\partial D}\langle(\bm{\xi}\cdot\bm{n})f\ln\frac{f}{f_{w}}\rangle
dS\leq 0,$ (40)

where $\bm{n}$ is the inward unit normal to the surface $\partial D$ and the
equality holds if and only if $f=f_{w}$.

##  5\. Main results: Free energy and its monotonicity

After the presentation of the known results [16, 15, 1] in Sec. 4, we now
discuss the thermal relaxation of a dense gas in a closed system with the aid
of Lemma 3.1. Consider the multiplication of $1+\ln(f/f_{w})$ with the
modified Enskog equation (1a) and integrate it with respect to $\bm{\xi}$.
Since $f_{w}$ depends on neither $t$ nor $\bm{X}$, we have

$\frac{\partial}{\partial t}\langle
f\ln(f/f_{w})\rangle+\frac{\partial}{\partial
X_{i}}\langle\xi_{i}f\ln(f/f_{w})\rangle=\langle\ln(f/f_{w})J_{ME}(f)\rangle.$
(41)

Since ${\ln f_{w}}=a_{w}-\bm{\xi}^{2}/(2RT_{w})$ with $a_{w}$ being a
constant, the right-hand side of (41) is reduced to

$\langle\ln(f/f_{w})J_{ME}(f)\rangle=\langle J_{ME}(f){\ln
f}\rangle+\frac{1}{2RT_{w}}\langle\bm{\xi}^{2}J_{ME}(f)\rangle.$ (42)

Once we integrate (41) with respect to $\bm{X}$ over the domain $D$, the
contribution from $\langle\bm{\xi}^{2}J_{ME}(f)\rangle$ vanishes by Lemma 3.1
and we arrive at

$\displaystyle\frac{d}{dt}\int_{D}\langle f\ln(f/f_{w})\rangle d\bm{X}=$
$\displaystyle\int_{\partial D}\langle\xi_{i}n_{i}f\ln(f/f_{w})\rangle
dS+\int_{D}\langle J_{ME}(f){\ln f}\rangle d\bm{X}$ $\displaystyle\leq$
$\displaystyle\int_{\partial D}\langle\xi_{i}n_{i}f\ln(f/f_{w})\rangle
dS-\frac{d\mathcal{H}^{(c)}}{dt}\leq-\frac{d\mathcal{H}^{(c)}}{dt},$ (43)

where Lemma 4.1 has been used at the last inequality. By transposing the most
right-hand side to the left-hand side, it is seen that $\mathcal{F}$ defined
by

$\mathcal{F}\equiv RT_{w}(\int_{D}\langle f\ln(f/f_{w})\rangle
d\bm{X}+\mathcal{H}^{(c)}),$ (44)

decreases monotonically in time:

$\frac{d\mathcal{F}}{dt}\leq 0,$ (45)

where the equality holds if and only if
$f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})f^{\prime}(\bm{X})=f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})f(\bm{X})$
for $\bm{X},\ \bm{X}_{\sigma\bm{\alpha}}^{-}\in D$ and $f=f_{w}$ on $\partial
D$; see the equality condition for (39) and in Lemma 4.1. Since $\mathcal{F}$
is bounded from below (see Appendix B), $\mathcal{F}$ approaches a stationary
value as $t\to\infty$. The extension to the case of the modified Enskog–Vlasov
equation is discussed in Appendix C.

######  Theorem 5.1.

(thermal relaxation in a closed system surrounded by a heat bath) Suppose that
the behavior of a dense gas in a closed system surrounded by a heat bath with
a constant temperature $T_{w}$ is described by the modified Enskog equation
(1) and the boundary condition (8). Then a quantity $\mathcal{F}$ defined by

$\mathcal{F}=RT_{w}(\int_{D}\langle f\ln(f/f_{w})\rangle
d\bm{X}+\mathcal{H}^{(c)}),$ (46)

monotonically decreases in time and approaches a stationary value as
$t\to\infty$, where $f_{w}$ and $\mathcal{H}^{(c)}$ are respectively defined
by (8d) and (38).

######  Remark 4.

From (44) and (26), $\mathcal{F}$ can be rewritten as

$\displaystyle\mathcal{F}=$ $\displaystyle RT_{w}(\int_{D}\langle f{\ln
f}\rangle d\bm{X}-\int_{D}\langle f{\ln f_{w}}\rangle
d\bm{X}+\mathcal{H}^{(c)})$ $\displaystyle=$
$\displaystyle(\mathcal{H}^{(k)}+\mathcal{H}^{(c)})RT_{w}+\int_{D}\langle\frac{1}{2}\bm{\xi}^{2}f\rangle
d\bm{X}+\mathrm{const.}$ (47)

Since $\int_{D}\langle\frac{1}{2}\bm{\xi}^{2}f\rangle d\bm{X}$ and
$-\mathcal{H}R$ are respectively the internal energy $E$ and the entropy $S$
of the closed system (see Remark 3), $\mathcal{F}$ is identified as $E-T_{w}S$
up to an additive constant, i.e., an extension of the Helmholtz free energy in
thermodynamics to a non-equilibrium system. The present result shows that the
same statement for the Boltzmann equation mentioned in [6, p. 270] holds for
the modified Enskog equation, thanks to Lemma 3.1. In the case of the
Boltzmann equation, the consideration of Lemma 3.1 was not required.

When $d\mathcal{F}/dt=0$, two conditions

$\displaystyle\ln f_{*}^{\prime}(\bm{X}_{\sigma\bm{\alpha}}^{-})+\ln
f^{\prime}(\bm{X})=\ln f_{*}(\bm{X}_{\sigma\bm{\alpha}}^{-})+\ln
f(\bm{X}),\quad\textrm{for\ }\bm{X},\bm{X}^{-}_{\sigma\bm{\alpha}}\in D,$
(48a) $\displaystyle f(t,\bm{X},\bm{\xi})=\frac{\rho(t,\bm{X})}{(2\pi
RT_{w})^{3/2}}\exp(-\frac{\bm{\xi}^{2}}{2RT_{w}}),\quad\textrm{for\
}\bm{X}\in\partial D,$ (48b)

hold. On condition that (48a) is identical to

$\ln
f(t,\bm{X},\bm{\xi})=b_{0}(t,\bm{X})+b_{i}(t)\xi_{i}+b_{4}(t)\bm{\xi}^{2}+c_{i}(t)\epsilon_{ijk}X_{j}\xi_{k},$
(49)

or equivalently to

$f(t,\bm{X},\bm{\xi})=\frac{\rho(t,\bm{X})}{(2\pi
RT(t))^{3/2}}\exp(-\frac{(\bm{\xi}-\bm{v}(t,\bm{X}))^{2}}{2RT(t)}),$ (50)

with $\bm{v}(t,\bm{X})=\bm{V}(t)+\bm{X}\times\bm{W}(t)$ [15], (48b) leads to
$T(t)=T_{w}$ and $\bm{v}(t,\bm{X})=0$. Furthermore, $\rho$ is independent of
$t$ because of the continuity equation (11) with $\bm{v}=\bm{0}$. Therefore,
when $d\mathcal{F}/dt=0$, $f$ is a time-independent resting Maxwellian

$\frac{\rho(\bm{X})}{(2\pi RT_{w})^{3/2}}\exp(-\frac{\bm{\xi}^{2}}{2RT_{w}}),$
(51)

which represents the thermal equilibrium state with the heat bath
characterized by the uniform temperature $T_{w}$.

##  6\. Conclusion

In the present work, the thermal relaxation of a dense gas in a closed system
surrounded by a heat bath has been studied on the basis of the modified Enskog
equation. The H theorem established by Resibois [16] for the infinite domain
and for a periodic domain and then later by Maynar et al. [15] for a bounded
domain surrounded by the specular-reflection wall has been arranged in a form
suitable for a closed system surrounded by a heat bath. The case of the
modified Enskog–Vlasov equation has also been considered in Appendix C.
Different from the case of the Boltzmann equation, it is required to pay
attention to collisional contributions to the momentum and the energy
transport. We have confirmed, however, that their net contributions on the
boundary vanish. It is physically natural in view of the origin of those
transports. As the result, the Darrozes–Guiraud inequality plays the same role
as in the case of the Boltzmann equation to find a quantity $\mathcal{F}$ that
corresponds to the Helmholtz free energy in the thermodynamics. This quantity
has been shown to be bounded and to decrease monotonically in time.

##  Appendix A N-particle distribution and correlation function $g_{2}$

In the case of the modified Enskog equation, the $N$-particle (factorized)
distribution function $\rho_{N}$ is introduced:

$\rho_{N}=\frac{1}{\phi(t)}\Theta(\bm{X}_{1},\cdots,\bm{X}_{N})W(t,\bm{X}_{1},\bm{\xi}_{1})\cdots
W(t,\bm{X}_{N},\bm{\xi}_{N}),$ (52)

and the velocity distribution function $f$ is expressed in terms of
$\rho_{N}$:

$\displaystyle f(t,\bm{X}_{1},\bm{\xi}_{1})=$ $\displaystyle
mN\int_{(D\times\mathbb{R}^{3})^{(N-1)}}\rho_{N}(t,\bm{Z}_{1},\dots,\bm{Z}_{N})d\bm{Z}_{2}\cdots
d\bm{Z}_{N}$ $\displaystyle=$
$\displaystyle{\frac{mN}{\phi(t)}W(t,\bm{X}_{1},\bm{\xi}_{1})Y(t,\bm{X}_{1}),}$
(53)

where and in what follows $\bm{Z}_{i}=(\bm{X}_{i},\bm{\xi}_{i})$,
$(D\times\mathbb{R}^{3})^{N}$ (or $D^{N}$) is the $N$-times direct multiple of
$D\times\mathbb{R}^{3}$ (or $D$), $N$ is the number of molecules in $D$, and

$\displaystyle{Y(t,\bm{X}_{1})=\int_{D^{N-1}}w(t,\bm{X}_{2})\cdots
w(t,\bm{X}_{N})\Theta(\bm{X}_{1},\cdots,\bm{X}_{N})d\bm{X}_{2}\cdots
d\bm{X}_{N},}$ (54a) $\displaystyle\phi(t)=\int_{D^{N}}w(t,\bm{X}_{1})\cdots
w(t,\bm{X}_{N})\Theta(\bm{X}_{1},\cdots,\bm{X}_{N})d\bm{X}_{1}\cdots
d\bm{X}_{N},$ (54b) $\displaystyle w(t,\bm{X})=\int
W(t,\bm{X},\bm{\xi})d\bm{\xi},$ (54c)
$\displaystyle\Theta(\bm{X}_{1},\cdots,\bm{X}_{N})=\prod_{i=1}^{N}\prod_{j>i}^{N}\theta(|\bm{X}_{ij}|-\sigma),\quad\bm{X}_{ij}=\bm{X}_{i}-\bm{X}_{j}.$
(54d)

Note that $\rho_{N}$ is normalized as

$\int_{(D\times\mathbb{R}^{3})^{N}}\rho_{N}d\bm{Z}_{1}\cdots d\bm{Z}_{N}=1,$
(55)

and the density $\rho$ is also expressed as

$\rho(t,\bm{X})=\frac{mN}{\phi(t)}w(t,\bm{X})Y(t,\bm{X}),$ (56)

by a simple integration of (53) with respect to $\bm{\xi}_{1}$.

The correlation function $g_{2}$ in (5a) is then defined in terms of the
quantities in (54) as222In the literature, $\Theta$ is often used in place of
$\Theta_{(1,2)}$ in the definition of $g_{2}$. The definition (57a) is adopted
in order to avoid any ambiguity occurring in the derivation of (37).

$\displaystyle g_{2}(t,\bm{X}_{1},\bm{X}_{2})$ $\displaystyle=$
$\displaystyle\frac{m^{2}N(N-1)}{\phi(t)}\frac{w(t,\bm{X}_{1})w(t,\bm{X}_{2})}{\rho(t,\bm{X}_{1})\rho(t,\bm{X}_{2})}$
$\displaystyle\quad\times\int_{D^{N-2}}w(t,\bm{X}_{3})\cdots
w(t,\bm{X}_{N})\Theta_{(1,2)}(\bm{X}_{1},\cdots,\bm{X}_{N})d\bm{X}_{3}\cdots
d\bm{X}_{N},$ (57a) where
$\Theta_{(1,2)}(\bm{X}_{1},\cdots,\bm{X}_{N})=\prod_{i=1}^{N}\prod_{j>\max(i,2)}^{N}\theta(|\bm{X}_{ij}|-\sigma).$
(57b) Note that
$\Theta(\bm{X}_{1},\cdots,\bm{X}_{N})=\theta(|\bm{X}_{12}|-\sigma)\Theta_{(1,2)}(\bm{X}_{1},\cdots,\bm{X}_{N}),$
(57c)

by (54d) and (57b). By (56) with (54a), $\rho$ can be regarded as a functional
of $w$ and, if invertible, vice versa. Hence, $\phi$ and $g_{2}$ can also be
regarded as functionals of $\rho$. It is seen from (57c) that

$\displaystyle\int_{D^{N-1}}\Theta_{(1,2)}(\bm{X}_{1},\dots,\bm{X}_{N})\frac{\partial}{\partial\bm{X}_{1}}\theta(|\bm{X}_{12}|-\sigma)F(\bm{X}_{2},\dots,\bm{X}_{N})d\bm{X}_{2}\dots
d\bm{X}_{N}$ $\displaystyle=$
$\displaystyle\frac{1}{N-1}\frac{\partial}{\partial\bm{X}_{1}}\int_{D^{N-1}}\Theta_{(1,2)}(\bm{X}_{1},\dots,\bm{X}_{N})\theta(|\bm{X}_{12}|-\sigma)F(\bm{X}_{2},\dots,\bm{X}_{N})d\bm{X}_{2}\dots
d\bm{X}_{N}$ $\displaystyle=$
$\displaystyle\frac{1}{N-1}\frac{\partial}{\partial\bm{X}_{1}}\int_{D^{N-1}}\Theta(\bm{X}_{1},\dots,\bm{X}_{N})F(\bm{X}_{2},\dots,\bm{X}_{N})d\bm{X}_{2}\dots
d\bm{X}_{N},$ (58)

if $F(\bm{X}_{2},\dots,\bm{X}_{N})$ is a function such that

$F(\bm{X}_{2},\dots,\bm{X}_{i}\dots,\bm{X}_{j}\dots,\bm{X}_{N})=F(\bm{X}_{2},\dots,\bm{X}_{j}\dots,\bm{X}_{i}\dots,\bm{X}_{N}),$
(59)

for $\forall i,j\in\\{2,\dots,N\\}$.

Now, thanks to (58), the reduction used in Sec. 4 is possible as follows:

$\displaystyle\frac{1}{m}\int_{D}\rho(\bm{X}_{2})\rho(\bm{X}_{1})g_{2}(\bm{X}_{1},\bm{X}_{2})\frac{\partial}{\partial\bm{X}_{1}}\theta(|\bm{X}_{12}|-\sigma)d\bm{X}_{2}$
$\displaystyle=$
$\displaystyle\frac{mN(N-1)}{\phi(t)}w(\bm{X}_{1})\int_{D^{N-1}}w(\bm{X}_{2})\cdots
w(\bm{X}_{N})$
$\displaystyle\quad\times\Theta_{(1,2)}(\bm{X}_{1},\cdots,\bm{X}_{N})\frac{\partial}{\partial\bm{X}_{1}}\theta(|\bm{X}_{12}|-\sigma)d\bm{X}_{2}\cdots
d\bm{X}_{N}$ $\displaystyle=$ $\displaystyle
w(\bm{X}_{1})\frac{\partial}{\partial\bm{X}_{1}}\\{\frac{mN}{\phi(t)}\int_{D^{N-1}}w(\bm{X}_{2})\cdots
w(\bm{X}_{N})\Theta(\bm{X}_{1},\cdots,\bm{X}_{N})d\bm{X}_{2}\cdots
d\bm{X}_{N}\\}$ $\displaystyle=$ $\displaystyle
w(\bm{X}_{1})\frac{\partial}{\partial\bm{X}_{1}}\frac{\rho(\bm{X}_{1})}{w(\bm{X}_{1})}=\rho(\bm{X}_{1})\frac{\partial}{\partial\bm{X}_{1}}\ln\frac{\rho(\bm{X}_{1})}{w(\bm{X}_{1})},$
(60)

where (57a), (54a), and (56) have been used and the argument $t$ is omitted
from $\rho$ and $w$.

##  Appendix B Boundedness of $\mathcal{F}$

In this Appendix, we will show that $\mathcal{F}$ is bounded.

With the preparations in Appendix A, we first show that $\mathcal{H}$
occurring in (39) is identical to the following $H$: [16, 15]

$H(t)=m\int_{(D\times\mathbb{R}^{3})^{N}}\rho_{N}\ln\rho_{N}d\bm{Z}_{1}\cdots
d\bm{Z}_{N}.$ (61)

Indeed, since $\Theta\ln\Theta\equiv 0$, the integrations with respect to
$\bm{Z}_{2},\cdots,\bm{Z}_{N}$ are simplified to yield

$\displaystyle H(t)=$ $\displaystyle
m\int_{D^{N}\times\mathbb{R}^{3N}}\rho_{N}(\sum_{i=1}^{N}\ln
W(t,\bm{X}_{i},\bm{\xi}_{i})-\ln\phi)d\bm{X}_{1}\cdots
d\bm{X}_{N}d\bm{\xi}_{1}\cdots d\bm{\xi}_{N}$ $\displaystyle=$
$\displaystyle\int_{D\times\mathbb{R}^{3}}f(t,\bm{X}_{1},\bm{\xi}_{1})\ln
W(t,\bm{X}_{1},\bm{\xi}_{1})d\bm{X}_{1}d\bm{\xi}_{1}-m\ln\phi.$ (62)

Because of (53) and (56),

$\ln W=\ln f-\ln\frac{\rho}{w},$ (63)

and substitution to (62) leads to

$\displaystyle H(t)=$
$\displaystyle\mathcal{H}^{(k)}-\int_{D\times\mathbb{R}^{3}}f(t,\bm{X}_{1},\bm{\xi}_{1})\ln\frac{\rho(t,\bm{X}_{1})}{w(t,\bm{X}_{1})}d\bm{X}_{1}d\bm{\xi}_{1}-m\ln\phi$
$\displaystyle=$
$\displaystyle\mathcal{H}^{(k)}-\int_{D}\rho(t,\bm{X}_{1})\ln\frac{\rho(t,\bm{X}_{1})}{w(t,\bm{X}_{1})}d\bm{X}_{1}-m\ln\phi$
$\displaystyle=$
$\displaystyle\mathcal{H}^{(k)}+\mathcal{H}^{(c)}=\mathcal{H}.$ (64)

Now, thanks to the form (61), the same method as the case of the Boltzmann
equation (see, e.g., [3, Sec. 9.4]) is available to show that $\mathcal{F}$ is
bounded from below, which is as follows. As $x$ increases from $x=0$, $x\ln x$
first monotonically decreases and reaches the minimum at $x=e^{-1}$, and then
increases monotonically for $x>e^{-1}$. Hence, if $\rho_{N}\geq e^{-1}$,
$\rho_{N}\ln\rho_{N}\geq-\rho_{N}$. If $\rho_{N}<e^{-1}$, we split this case
into (i) $\rho_{N}\geq(4\pi
RT_{w})^{-3N/2}V_{D}^{-N}\exp(-\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}})$
and (ii) $\rho_{N}<(4\pi
RT_{w})^{-3N/2}V_{D}^{-N}\exp(-\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}})$,
where $V_{D}$ is the volume of $D$. In case (i),
$\rho_{N}\ln\rho_{N}\geq\rho_{N}[-(3N/2)\ln(4\pi RT_{w})-N\ln
V_{D}-\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}}]$; in case (ii),
$\rho_{N}\ln\rho_{N}>(4\pi
RT_{w})^{-3N/2}V_{D}^{-N}\exp(-\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}})[-(3N/2)\ln(4\pi
RT_{w})-N\ln V_{D}-\sum_{j=1}^{N}\frac{\bm{\xi}_{j}^{2}}{4RT_{w}}]$.
Consequently, it holds that

$\displaystyle\rho_{N}\ln\rho_{N}\geq$
$\displaystyle-\rho_{N}-\rho_{N}N\ln[(4\pi
RT_{w})^{3/2}V_{D}]-\rho_{N}\sum_{j=1}^{N}\frac{\bm{\xi}_{j}^{2}}{4RT_{w}}$
$\displaystyle-\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}}\frac{1}{(4\pi
RT_{w})^{3N/2}V_{D}^{N}}\exp(-\sum_{j=1}^{N}\frac{\bm{\xi}_{j}^{2}}{4RT_{w}})$
$\displaystyle-\frac{N\ln[(4\pi RT_{w})^{3/2}V_{D}]}{(4\pi
RT_{w})^{3N/2}V_{D}^{N}}\exp(-\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}}),$
(65)

by which $H$ is evaluated as

$\displaystyle H(t)=$ $\displaystyle
m\int_{(D\times\mathbb{R}^{3})^{N}}\rho_{N}\ln\rho_{N}d\bm{Z}_{1}\cdots
d\bm{Z}_{N}$ $\displaystyle\geq$
$\displaystyle-m\int_{(D\times\mathbb{R}^{3})^{N}}\\{\rho_{N}+\rho_{N}N\ln[(4\pi
RT_{w})^{3/2}V_{D}]+\rho_{N}\sum_{j=1}^{N}\frac{\bm{\xi}_{j}^{2}}{4RT_{w}}$
$\displaystyle+\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}}\frac{1}{(4\pi
RT_{w})^{3N/2}V_{D}^{N}}\exp(-\sum_{j=1}^{N}\frac{\bm{\xi}_{j}^{2}}{4RT_{w}})$
$\displaystyle+\frac{N\ln[(4\pi RT_{w})^{3/2}V_{D}]}{(4\pi
RT_{w})^{3N/2}V_{D}^{N}}\exp(-\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}})\\}d\bm{Z}_{1}\cdots
d\bm{Z}_{N}$ $\displaystyle=$ $\displaystyle-m-2mN\ln[(4\pi
RT_{w})^{3/2}V_{D}]-m\\{\int_{(D\times\mathbb{R}^{3})^{N}}\rho_{N}\sum_{j=1}^{N}\frac{\bm{\xi}_{j}^{2}}{4RT_{w}}$
$\displaystyle+\sum_{i=1}^{N}\frac{\bm{\xi}_{i}^{2}}{4RT_{w}}\frac{1}{(4\pi
RT_{w})^{3N/2}V_{D}^{N}}\exp(-\sum_{j=1}^{N}\frac{\bm{\xi}_{j}^{2}}{4RT_{w}})\\}d\bm{Z}_{1}\cdots
d\bm{Z}_{N}$ $\displaystyle=$ $\displaystyle-\\{m+2mN\ln[(4\pi
RT_{w})^{3/2}V_{D}]+\int_{D\times\mathbb{R}^{3}}f(t,\bm{X},\bm{\xi})\frac{\bm{\xi}^{2}}{4RT_{w}}d\bm{X}d\bm{\xi}$
$\displaystyle+mN\int_{\mathbb{R}^{3}}\frac{\bm{\xi}^{2}}{4RT_{w}}\frac{1}{(4\pi
RT_{w})^{3/2}}\exp(-\frac{\bm{\xi}^{2}}{4RT_{w}})d\bm{\xi}\\}$
$\displaystyle\geq$ $\displaystyle-\\{mN(\frac{5}{2}+\ln[(4\pi
RT_{w})^{3}V_{D}^{2}])+\int_{D\times\mathbb{R}^{3}}f(t,\bm{X},\bm{\xi})\frac{\bm{\xi}^{2}}{4RT_{w}}d\bm{X}d\bm{\xi}\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{2RT_{w}}\int_{D}\langle\frac{1}{2}\bm{\xi}^{2}f\rangle
d\bm{X}+\mathrm{const.}$ (66)

Remind that $mN$ is the total mass in $D$ and thus is finite. Hence (66) means
that $\mathcal{F}\geq\frac{1}{2}\int_{D}\langle\frac{1}{2}\bm{\xi}^{2}f\rangle
d\bm{X}+\mathrm{const.}$ by (47). Moreover, if $\mathcal{F}$ is initially
finite, then $\mathcal{F}$, $\mathcal{H}$, and
$\int_{D}\langle\frac{1}{2}\bm{\xi}^{2}f\rangle d\bm{X}$ are bounded
individually from both below and above for $t\geq 0$.

##  Appendix C The case of modified Enskog–Vlasov equation

In the case of Enskog–Vlasov equation, an external force term $F_{i}{\partial
f}/{\partial\xi_{i}}$ is added on the left-hand side of (1), where

$F_{i}=-\int_{D}\frac{\partial}{\partial
X_{i}}\Phi(|\bm{Y}-\bm{X}|)\rho(t,\bm{Y})d\bm{Y},$ (67)

and $\Phi$ is the attractive isotropic force potential between molecules.

By taking the $(1+{\ln{f}})$-moment of the external force term:

$\langle(1+{\ln f})F_{i}\frac{\partial f}{\partial\xi_{i}}\rangle=\langle
F_{i}\frac{\partial}{\partial\xi_{i}}(f{\ln f})\rangle=0,$ (68)

and thus the external term is found to give no contribution to (27). Hence,
(39) remains unchanged.

Next consider the $(1+\ln({f}/f_{w}))$-moment:

$\langle(1+\ln\frac{f}{f_{w}})F_{i}\frac{\partial
f}{\partial\xi_{i}}\rangle=-\langle({\ln{f_{w}}})F_{i}\frac{\partial
f}{\partial\xi_{i}}\rangle=F_{i}\langle\frac{\bm{\xi}^{2}}{2RT_{w}}\frac{\partial
f}{\partial\xi_{i}}\rangle=-\frac{\rho v_{i}F_{i}}{RT_{w}}.$ (69)

Since $F_{i}$ is given by (67),

$\displaystyle-\int_{D}\frac{\rho v_{i}F_{i}}{RT_{w}}d\bm{X}=$
$\displaystyle\int_{D}\frac{\rho v_{i}}{RT_{w}}\frac{\partial}{\partial
X_{i}}\int_{D}\Phi(|\bm{Y}-\bm{X}|)\rho(t,\bm{Y})d\bm{Y}d\bm{X}$
$\displaystyle=$ $\displaystyle-\int_{\partial D}\frac{\rho
v_{i}}{RT_{w}}n_{i}\int_{D}\Phi(|\bm{Y}-\bm{X}|)\rho(t,\bm{Y})d\bm{Y}dS(\bm{X})$
$\displaystyle-\int_{D}\frac{1}{RT_{w}}\frac{\partial(\rho v_{i})}{\partial
X_{i}}\int_{D}\Phi(|\bm{Y}-\bm{X}|)\rho(t,\bm{Y})d\bm{Y}d\bm{X}$
$\displaystyle=$
$\displaystyle\int_{D}\frac{1}{RT_{w}}\frac{\partial\rho(t,\bm{X})}{\partial
t}\int_{D}\Phi(|\bm{Y}-\bm{X}|)\rho(t,\bm{Y})d\bm{Y}d\bm{X}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{D\times
D}\frac{\Phi(|\bm{Y}-\bm{X}|)}{RT_{w}}\rho(t,\bm{X})\rho(t,\bm{Y})d\bm{X}d\bm{Y},$
(70)

where $v_{i}n_{i}=0$ on $\partial D$ and the continuity equation (11) have
been used. Therefore, in the case of the modified Enskog–Vlasov equation,

$\displaystyle\mathcal{F}^{\prime}\equiv$ $\displaystyle
RT_{w}(\int_{D}\langle f\ln(f/f_{w})\rangle d\bm{X}+\mathcal{H}^{(c)})$
$\displaystyle+\frac{1}{2}\int_{D\times
D}\Phi(|\bm{Y}-\bm{X}|)\rho(t,\bm{X})\rho(t,\bm{Y})d\bm{X}d\bm{Y},$ (71)

decreases monotonically in time:

$\frac{d\mathcal{F}^{\prime}}{dt}\leq 0.$ (72)

This corresponds to the result in Appendix B of [18] for a simple kinetic
model. If $\Phi\geq C$ holds for some constant $C$, $\mathcal{F}^{\prime}$ is
bounded from below and approaches a stationary value as $t\to\infty$.

## Acknowledgements

The present work has been supported in part by the JSPS KAKENHI Grant (No.
22K03923) and the Kyoto University Foundation. The author thanks Masanari
Hattori for his helpful comments to the draft of this paper.

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Received xxxx 20xx; revised xxxx 20xx.