Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state

Thermodynamic quantities of the hard-sphere system in the steady state with a small heat flux are calculated within the continuous media approach. Analytical expressions for pressure, internal energy, and entropy are found in the approximation of the fourth order in temperature gradients. It is shown that the gradient contributions to the internal energy depend on the volume, while the entropy satisfies the second law of thermodynamics for nonequilibrium processes. The calculations are performed for dimensions 3D, 2D, and 1D.


Introduction
It is known from statistical mechanics that the interparticle interaction manifests itself in thermodynamic quantities gradually passing from low gas densities to intermediate ones, e.g. [1]. A similar picture concerning the effect of the interaction on thermodynamic behaviour should be expected for nonequilibrium states. They are more complicated and thus are usually investigated for the case of weak deviations from equilibrium.
As concerns the weakly nonequilibrium states with a heat flux, the main attention was paid to the phenomenon of heat conduction [1][2][3][4] as well as to calculations of the linear thermal conductivity coefficient [5][6][7]. Such nonequilibrium macroscopic quantities as pressure, internal energy, and entropy have remained less studied. Interests in the entropy were mainly associated with calculations of its production [1][2][3][4][5][6][7] closely related to the approaching to equilibrium due to relaxation processes.
Theoretical investigations of the thermodynamic properties of systems in the heat-conduction steady state can be divided into two groups in which a) the effect of the heat flux on the pressure, entropy, and other quantities and corresponding densities is studied and b) attempts are made to suggest some general formalism analogous to the equilibrium Gibbs relation (the basic thermodynamic equality). For the hardsphere model as one of the simplest interparticle interactions, the Enskog kinetic equation [5,7,8] is often used in the both cases.
and developed by Bird [37][38][39]. In [40], the spectral method is used for solving the Enskog equation for hard spheres (both elastic and inelastic) in the heat-conduction states. The profiles of the number density, kinetic as well as potential components of the pressure and heat flux are shown to agree well with the direct simulation Monte Carlo data [36].
Morriss with co-workers consider a simplified spatial configurationthe quasi-one-dimensional system of hard disks in a narrow linear channel with model thermal baths on the ends [41]. The disks are coupled to the thermostats by deterministic rules [41][42][43]. The temperature profile, the local entropy density, its production, and the heat flux through the system are obtained for both low [44] and intermediate and high [45] densities. The effect of spatial correlations on the local entropy is examined in [46].
These numerical methods provide results describing the heat-conduction steady states in detail. However, they do not solve the problem of establishing theoretical interrelations between different macroscopic quantities. In recent works by del Pozo et al. [47,48] computer simulation data for the twodimensional hard disks in the heat-conduction steady states are analyzed in terms of the equilibrium-like equation of state and the local Fourier law. Bulk behaviour of the temperature and particle density profiles are shown to obey specific scaling relations valid even for strong nonequilibrium conditions. High accuracy and reliability of these objective laws considerably deepen the understanding of the nature of the steady states.
In [49], there is calculated the pressure, internal energy, entropy, and free energy (not accurately) of the low-density gas in the weakly nonequilibrium heat-conduction steady state by means of the continuous media approach. Simplicity of the method and the fact that the entropy found satisfies the second law of thermodynamics show the usefulness of these results. However, an interaction potential does not enter the thermodynamic quantities with regard to low densities. Here, we attempt to take interparticle interaction into account for the particular case of the hard-sphere system at intermediate densities making use of one of its simplest equations of state. This demonstrates the applicability of the method to the calculation of thermodynamic quantities of gases in the situations where the size of particles becomes important.
In section 2 we describe the heat-conduction steady state. Next, we find the pressure and internal energy, section 3. The entropy calculations and conclusions are given in section 4 and section 5.

Heat-conduction state of the hard spheres
Our aim is to study the effect of the size of molecules on thermodynamic quantities of the intermediate density gas in the heat-conduction steady state using the hard-sphere model. We restrict ourselves to a simple case of weak nonequilibrium. N hard spheres are contained in a vessel of macroscopic size and of a parallelepiped form. The length of the edge and the cross area are denoted by L and Ω (figure 1). Heat is transferred in the direction parallel to the edge, while the local temperature is independent of time and changes slowly along this direction.
Local temperature. Putting the explicit determination of the temperature profile off, we consider the problem of calculation of the thermodynamic quantities from rather general grounds and as before [49] we describe the steady state by the set of temperature value T 0 and values {G 1 , . . . ,G r } of its r successive gradients referred to the geometrical middle-point of the vessel. If axis OZ of the reference system is chosen to be parallel to the heat flux (figure 2), then the quantities can approximately determine the local temperature: The approximation is defined by the number of the gradients in equation (2.1).  The weak nonequilibrium means that any two neighbouring terms in equation (2.1) differ by an order

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For boundary values z = ±L/2, these inequalities read: Such conditions are standard in nonequilibrium statistical mechanics and kinetic theory, e.g., [2,4,5]. It is convenient to distinguish different orders by a formal small parameter δ introduced into expansion (2.1): . . ,G r }. According to this δ-expansion, any macroscopic quantity A will be represented below as a series in powers of δ: where A i contains contributions from the gradient combinations of order i .

Local equation of state.
If we select (figure 2) the macroscopically small layer [z − 1 2 dz; z + 1 2 dz] (but sufficiently large in comparison with the hard-sphere diameter), then with regard to the weak nonequilibrium of the state, the pressure in this layer can be approximated by the equilibrium equation of state. We choose the latter to be the van der Waals equation for hard spheres, e.g., [1,50]: where V is the volume of an equilibrium system and b means the volume referred to a particle in the close-packing state. The corresponding internal energy and entropy read: with D being the dimensionality and ξ (D) where m is the mass of the particle and h is Planck's constant, see e.g., [1].
We substitute the real local values of the temperature T (z) and the number density n(z) into equation (2.5) to get the local pressure assumption for the weakly nonequilibrium heat-conduction steady state: (2.8)

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Hard spheres in the heat-conduction steady state The local densities of the internal energy and entropy can be obtained from the equilibrium counterparts E eq /V and S eq /V in the same way: (2.10)

Baric and caloric equations of state
Next, we turn to calculation of the pressure. The fact that the hard spheres are maintained in a mechanical equilibrium means that the pressure has the same value all over the vessel: This statement is a natural condition for the heat-conduction steady state. It is involved in the statisticalmechanical description of light scattering [51][52][53] and the BGK-model kinetic calculations [54,55], while in computer simulations checking of this condition ensures additionally the validity of results for the steady state, e.g., [44]. Consequently, we have a constant quantity in the left-hand side of equation (2.8) for the case of the steady state, henceforth denoted as P .
Number density n(z) obeys the normalization condition: where integrations with respect to transverse coordinates x and y have been performed 1 in the integral over the volume Ω×L; N is the total number of particles in the system. The density n(z) can be expressed through T (z) and P using equation here and below, the middle-point temperature value is denoted by T (in place of T 0 ). It follows from the weak nonequilibrium conditions (2.2) that δ k γ k z k ≪ 1 and the fraction in equation (3.3) can be expanded (up to the r -th order): n(z) = n 0 ν 0 + δν 1 z + . . . + δ r ν r z r + . . . , (3.5) with coefficients dependent on C through the parameters {γ}, equation (3.4): In what follows, we restrict ourselves to the fourth order though it is not so hard to derive higher-order contributions.
Perturbations for the pressure. Equation (3.5) inserted into the normalization condition (3.2) can be integrated explicitly n(T + bC ) = C 1 + δ 2 κ 2 (C ) + δ 4 κ 4 (C ) + . . . , (3.6) where equation (3.4) has been used for n 0 ; here, n ≡ N /ΩL is the number density in the state of thermal equilibrium, while the coefficients introduced read: Expression (3.6) is an equation to determine the constant C . Since ν 2 , ν 4 , . . . depend on C too, equation (3.6) is highly nonlinear. Its solution for the weak nonequilibrium can be sought by perturbations: Even orders in δ are absent here because they are absent in equation (3.6).
We note that the coefficients κ 2 , κ 4 , . . . are also to be expanded in δ: . . . , (3.9) where κ (k) i is caused by those contributions to C whose order is not higher than k; in particular, κ (0) i is determined by the term C (0) , κ (2) i is determined by the terms C (0) and C (2) , etc. After substitution of expansion (3.9) into equation (3.6), the series in the square brackets is rearranged: n T + ηC (0) = C (0) , ηC (2) = C (0) κ (2) + C (2) , ηC (4) = C (0) κ (4) + C (2) κ (2) + C (4) , with η ≡ nb being the reduced partial volume. Finally, we obtain the solutions: (2) ], Here,η ≡ 1−η denotes the reduced accessible volume. The result for C (4) has been obtained by the use of the formula for C (2) . The contributions C (k) can be expressed through the gradients {g } and we deduce the baric equation of the weakly nonequilibrium heat-conduction steady state for hard spheres in the van der Waals approximation [56]: where powers of δ are omitted, and p 0 ≡ 1, p 2 ≡ 1 12 g 2 − g 2 1η L 2 , and The quantities p 2 and p 4 describe the corrections to the pressure from the gradients in corresponding orders. The effect of particle's size is involved in the reduced volume η referred to the particles, mainly in combinationη ≡ 1 − η. Tending b → 0 causes η → 0,η → 1 which results in the expressions transforming to those for the low-density case [49]. The gradient expansion for the middle-point value of the number density, equation (3.4), can be also found as n 0 = n (0) 0 + n (2) 0 + n (4) 0 + . . . , (3.13) in which the coefficients are defined as follows [56]: n (0) 0 = n, n (2) 0 = n 1 12 g 2η − g 2 1η 2 L 2 , and n (4) 0 = n The internal energy is calculated by integration of its density ε(z), equation (2.9): with coefficients e k = p k dependent on {g } and η = bN /(ΩL). We conclude that the internal energy of hard spheres in the heat-conduction steady state depends on the volume ΩL and differs from the lowdensity result [49], while the equilibrium energies of these systems are known to be identical and independent on volume.

Entropy
Expression ( to be the entropy of the weakly nonequilibrium heat-conduction steady state. After its calculation, S is shown to satisfy the second law of thermodynamics for nonequilibrium processes [2,3,57].

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the σ α 's multiplying the gradients in s 4 are written as follows: We notice that the coefficients s i for the entropy (an additive quantity) depend on the dimensionality D, contrary to the pressure ones, p i . In the limit b → 0 (and η → 0), the low-density gas results are recovered, which coincide for value D = 3 with those found earlier [49].
Any nonequilibrium state undergoes a relaxation at the conditions of the free evolution, which is accompanied by the entropy increase [2,3,57]. We show that the entropy calculated (4.10) possesses this feature. To this end, let us imagine that the system is made isolated on the boundaries z = ∓ 1 2 L and afterwards it is allowed to relax during a macroscopically large time interval. The entropy S fin of the final equilibrium can be compared to that of the initial steady state, equation (4.10).
The internal energy of the hard spheres does not change after the isolation, thus E = D 2 N k B T fin , where T fin is the temperature ascribed to the final state. We derive from equation where we have used that e k = p k , while the third term in the curly brackets is an expansion of the logarithm.
The entropy difference ∆S ≡ S − S fin takes the form ∆S = ∆S (2) + ∆S (4) + . . . with It is obvious that ∆S (2) < 0, while the sign of ∆S (4) is undetermined and depends on the values of the fourth-order gradients. However, the restrictions (2.2) imposed on the weak nonequilibrium ensure that |∆S (4) | ≪ |∆S (2) |. For this reason, we conclude that the nonequilibrium entropy found is less than the entropy of the corresponding equilibrium state and as a consequence it satisfies the second law of thermodynamics for nonequilibrium processes [2,3,57].

Conclusions
We have considered the pressure, internal energy, and entropy of the hard-sphere system in the weakly nonequilibrium heat-conduction steady state. They are calculated in the continuous media approach using integrations of the proper local densities.
The results are obtained in the form of expansions in the temperature gradients evaluated in the geometrical middle of the system up to the fourth order. They describe the effect of the particle size on thermodynamic quantities at intermediate densities. The coefficients of the expansions depend on the packing parameter (referred to the uniform equilibrium), revealing dependence of the nonequilibrium corrections on the volume of the system. The entropy calculated is shown to obey the second law of thermodynamics for nonequilibrium processes.
The results are applicable for dimensions D = 1; 2; 3. The van der Waals approximation for hard spheres used restricts the applicability to the domain of not high densities for three-and two-dimensional systems where this approximation is valid for the equilibrium. In the one-dimensional case, the equilibrium van der Waals equation of state is exact [58]. For this reason, we expect that our results can be used at high densities while the probable inaccuracy may be caused only by the method used rather than by the local equation of state.

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Our calculations do not go beyond the scope of thermodynamic ideas, since no external results coming from other nonequilibrium theories (e.g., kinetic theory, informational theory, or the approach of fluctuation theorems) have been used. The simplicity and explicit analytical description can be also regarded as positive features.