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

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.

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

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

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

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

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

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

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

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

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 $\partial D$.

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),

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

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 $\bm{\xi}$:

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

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

Since

(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

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

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,

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

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

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

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

and by (17b)

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

and by (18b)

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.

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

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:

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

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 $x,y>0$

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

holds, where

Equation (32) can be transformed as

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

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

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

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

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

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

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