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 sh...

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Дата:	2016
Автор:	Humenyuk, Y.A.
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Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2016
Назва видання:	Condensed Matter Physics
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Цитувати:	Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state / Y.A. Humenyuk // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43002: 1–12. — Бібліогр.: 58 назв. — англ.

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spelling	irk-123456789-1565312019-06-19T01:26:47Z Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state Humenyuk, Y.A. 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. В рамках пiдходу суцiльного середовища розраховано термодинамiчнi величини системи твердих кульок у стацiонарному станi з малим тепловим потоком. Аналiтичнi вирази для тиску, внутрiшньої енергiї та ентропiї знайдено в наближеннi четвертого порядку за ґрадiєнтами температури. Показано, що ґрадiєнтнi внески до внутрiшньої енергiї залежать вiд об’єму, а ентропiя задовольняє II-е начало термодинамiки для нерiвноважних процесiв. Розрахунки проведено для вимiрностей 3D, 2D та 1D. 2016 Article Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state / Y.A. Humenyuk // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43002: 1–12. — Бібліогр.: 58 назв. — англ. 1607-324X PACS: 05.70.Ln, 51.30.+i, 44.10.+i DOI:10.5488/CMP.19.43002 arXiv:1612.07166 http://dspace.nbuv.gov.ua/handle/123456789/156531 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Humenyuk, Y.A.
spellingShingle	Humenyuk, Y.A. Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state Condensed Matter Physics
author_facet	Humenyuk, Y.A.
author_sort	Humenyuk, Y.A.
title	Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state
title_short	Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state
title_full	Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state
title_fullStr	Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state
title_full_unstemmed	Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state
title_sort	pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/156531
citation_txt	Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state / Y.A. Humenyuk // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43002: 1–12. — Бібліогр.: 58 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT humenyukya pressureandentropyofhardspheresintheweaklynonequilibriumheatconductionsteadystate
first_indexed	2025-07-14T08:52:15Z
last_indexed	2025-07-14T08:52:15Z
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fulltext	Condensed Matter Physics, 2016, Vol. 19, No 4, 43002: 1–12 DOI: 10.5488/CMP.19.43002 http://www.icmp.lviv.ua/journal Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state Y.A. Humenyuk Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St., 79011 Lviv, Ukraine Received April 15, 2016, in final form July 15, 2016 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 contri- butions to the internal energy depend on the volume, while the entropy satisfies the second law of thermody- namics for nonequilibrium processes. The calculations are performed for dimensions 3D, 2D, and 1D. Key words: heat flux, temperature gradients, equation of state, nonequilibrium entropy, steady state thermodynamics PACS: 05.70.Ln, 51.30.+i, 44.10.+i 1. Introduction It is known from statistical mechanics that the interparticle interaction manifests itself in thermody- namic 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 nonequilib- rium 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 [1–4] as well as to calculations of the linear thermal conductivity coeffi- cient [5–7]. Such nonequilibrium macroscopic quantities as pressure, internal energy, and entropy have remained less studied. Interests in the entropy weremainly associated with calculations of its production [1–7] 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 aremade to suggest some general formalism analogous to the equilibrium Gibbs relation (the basic thermodynamic equality). For the hard- sphere model as one of the simplest interparticle interactions, the Enskog kinetic equation [5, 7, 8] is often used in the both cases. Applications of kinetic theory. In order to determine the effect of heat transfer on the weakly nonequi- librium pressure or entropy, it is necessary to take into account the terms of higher orders in temperature gradient than the linear ones. Marques and Kremer [9] solved the Enskog kinetic equation in the higher approximations using both Grad’s and Chapman-Enskog’s methods. The calculated pressure tensor con- tains contributions from the temperature gradients of the second order, which is referred to the Burnett level [10]. The Grad method was also used in solving the Enskog equation for two-dimensional hard disks [11]. The set of transport equations of the derived extended hydrodynamics was applied to two problems: © Y.A. Humenyuk, 2016 43002-1 http://dx.doi.org/10.5488/CMP.19.43002 http://www.icmp.lviv.ua/journal Y.A. Humenyuk 1) the first one [12] was on the pressure difference between the equilibrium and nonequilibrium sta- tionary heat-conduction hard-disk gases separated by a porous wall; the phenomenological conclusion on the pressure difference claimed in [13] had not been confirmed; 2) the second problem concerned the description of hard disks between two parallel walls with different temperatures [11]; for the weakly nonequilibrium case, the pressure correction was estimated to be quadratic in the heat flux. These results follow from the nonstationary Enskog equation. In application to the steady case, the pressure calculations were determined by both stationary and time-dependent parts of the nonequilib- rium distribution function. Besides, this function was sought as a normal solution, while the appropriate boundary conditions were not taken into consideration explicitly. As a consequence, one can deal with local quantities, e.g., the energy and entropy densities and the intensity of entropy production. The total energy, entropy, and its production cannot be found, unless explicit solutions for the hydrodynamic fields of the local number density and temperature are obtained. Extensions of the thermodynamic formalism. The idea of extension, introduced by Grad [14] for the hydrodynamic level, is known to be applied to the construction of the local thermodynamic description of nonequilibrium states, which is expected to go beyond the domain of the assumption of local equilibrium. As a starting equation, the generalized Gibbs-like relation is chosen in the form written for the local entropy density which depends on the extended set of hydrodynamic variables (e.g., including the stress tensor and the heat flux.) An approach referred to as Extended Thermodynamics (developed by Liu, Müller, and Ruggeri with co- workers in works [15–20]), is aimed at analysing the transport equations for an extended set of variables, to search for the ways of closure procedures for them, and to produce the criteria of choices of closure relations. The analysis of the constitutive closure relations is realized on the grounds of phenomeno- logical principles of invariance of the local hydrodynamic description, increasing the local entropy, and concavity of the entropy density functional [15, 16, 18–20]. However, regardless of the use of entropic (that is thermodynamic) criteria, the notion “extended hydrodynamics” seems to be more suitable for this approach. Banach proposed [21] microscopic justification of this scheme for the hard sphere system based on the Enskog kinetic equation of the RET variant [8]. A recent study [22] is concerned with the phenomeno- logical analysis and closure procedure for the extended set of hydrodynamic equations for dense gases, from which the hard sphere result is derived as a particular case. Extended Irreversible Thermodynamics [23–25] develops a formalism of local thermodynamics for nonequilibrium processes, which does not exploit the local equilibrium assumption and interprets the heat flux as an additional thermodynamic degree of freedom. One expects that it is capable of catching the effects that are unattainable in the linear irreversible thermodynamics [2, 3]. This approach has been applied to hard spheres [26]. Using the Enskog kinetic equation and the method of molecular hydrodynamics [27], explicit expressions for the phenomenological coefficients ap- pearing in the extended Gibbs relation are calculated. However, the extended irreversible thermodynamics and specifically its notion of the nonequilib- rium temperature [28, 29] were criticized from the viewpoint of computer simulations [30, 31], using phenomenological ideas [32, 33], and even on the grounds of its own internal methodology [34]. To our knowledge, the discussion has not been resolved as well as no crucial experiment which would approve or refute the main assumptions of this approach has been proposed. Computer simulations. The hard spheres or disks are investigated in computer simulations with re- gard to their properties in the heat-conduction state and behaviour of local quantities. The measure of deviation of the results for the heat flux from the linear Fourier law is also studied. Using the nonequilibrium molecular dynamics, the temperature and number density profiles for the hard-disk system are obtained [35], which are shown to be in agreement with the results of the continuum media approach. However, the thermal conductivity coefficient is found [35] to differ appreciably from that given by the Enskog theory. The steady states of hard spheres with a heat flux are explored by numerical methods of solving the Enskog equation. One of them is the generalization [36] of the direct simulation Monte Carlo, proposed 43002-2 Hard spheres in the heat-conduction steady state and developed by Bird [37–39]. 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 configuration— the quasi-one-dimensional sys- tem 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 [41–43]. The temperature profile, the local entropy den- sity, 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. How- ever, they do not solve the problem of establishing theoretical interrelations between different macro- scopic quantities. In recent works by del Pozo et al. [47, 48] computer simulation data for the two- dimensional 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 pro- files 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 contin- uous 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 mak- ing 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. 2. Heat-conduction state of the hard spheres Our aim is to study the effect of the size ofmolecules 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 andΩ (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 T0 and values {G1, . . . ,Gr } 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 T0 ≡ T (z) ∣ ∣ ∣ z=0 , Gk ≡ ∂k ∂zk T (z) ∣ ∣ ∣ z=0 can approximately determine the local temperature: T (z) = T0 +G1z + 1 2! G2z2 + . . .+ 1 r ! Gr zr . (2.1) The approximation is defined by the number of the gradients in equation (2.1). 43002-3 Y.A. Humenyuk L Ω N q HEAT FLUX X Y Z O Figure 1. The hard-sphere system in the heat- conduction steady state. O Z T(z) T0 N,Ω,L −L/2 L/2 qHEAT FLUX z − −dz1 z + −dz1 22 Figure 2. The local temperature profile along the hard-sphere system. The weak nonequilibriummeans that any two neighbouring terms in equation (2.1) differ by an order 1 k \|Gk+1z\|≪ \|Gk \| for all z. For boundary values z =±L/2, these inequalities read: 1 2(k +1) \|Gk+1\|L ≪\|Gk \|, (2.2) in particular, 1 2 \|G1\|L≪T0, 1 4 \|G2\|L≪\|G1\|, etc. 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): T (z)= T0 [ 1+δg1z + . . .+δr gr zr ] , (2.3) where reduced gradients {g } ≡ {g1, . . . , gr } defined by gk ≡ 1 T0 1 k ! Gk are used instead of {G} ≡ {G1, . . . ,Gr }. According to this δ-expansion, any macroscopic quantity A will be represented below as a series in pow- ers of δ: A = A0 + r ∑ i=1 δi Ai , (2.4) where Ai 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 nonequi- librium 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]: PV dW −HS = N kBT V −N b , (2.5) 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: Eeq ≡ 1 2 DN kBT, (2.6) Seq ≡ N kB [ − ln N + ln(V −N b)+ 1 2 D ln T +ξ(D) S ] , (2.7) with D being the dimensionality and ξ(D) S ≡ 1 2 D ln(2πkBm/h2)+1+ 1 2 D , where m is themass 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 equa- tion (2.5) to get the local pressure assumption for the weakly nonequilibrium heat-conduction steady state: P (z) = n(z)kBT (z) 1−bn(z) . (2.8) 43002-4 Hard spheres in the heat-conduction steady state The local densities of the internal energy and entropy can be obtained from the equilibrium counterparts Eeq/V and Seq/V in the same way: ε(z) ≡ 1 2 D n(z)kBT (z), (2.9) s(z) ≡ kBn(z) ( ln { [n(z)]−1 −b } + 1 2 D ln T (z)+ξ(D) S ) . (2.10) 3. Baric and caloric equations of state Next, we turn to calculation of the pressure. The fact that the hard spheres are maintained in a me- chanical equilibriummeans that the pressure has the same value all over the vessel: P (z) = const. (3.1) This statement is a natural condition for the heat-conduction steady state. It is involved in the statistical- mechanical description of light scattering [51–53] 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: Ω L/2 ∫ −L/2 dz n(z)= N , (3.2) where integrations with respect to transverse coordinates x and y have been performed1 in the integral over the volumeΩ×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 (2.8): n(z) =C/[T (z)+bC ], where C ≡ P/kB is a constant. Substitution of the expansion for the local temperature, equation (2.3), yields: n(z) = n0 1 1+δγ1z + . . .+δrγr zr , (3.3) with n0 ≡ C T +bC , γk ≡ 1 k! 1 T +bC Gk ; (3.4) here and below, the middle-point temperature value is denoted by T (in place of T0). It follows from the weak nonequilibrium conditions (2.2) that ∑ δkγk zk ≪ 1 and the fraction in equa- tion (3.3) can be expanded (up to the r -th order): n(z) = n0 [ ν0 +δν1z + . . .+δrνr zr + . . . ] , (3.5) with coefficients dependent on C through the parameters {γ}, equation (3.4): ν0 = 1, ν3(C ) =−γ3 +2γ2γ1 −γ3 1, ν1(C ) =−γ1, ν4(C ) =−γ4 +2γ3γ1 +γ2 2 −3γ2γ 2 1 +γ4 1, ν2(C ) =−γ2 +γ2 1, . . . In what follows, we restrict ourselves to the fourth order though it is not so hard to derive higher-order contributions. 1 Ω= 1 in the 1D case, while for the 2D caseΩ means the linear size in the perpendicular direction. 43002-5 Y.A. Humenyuk Perturbations for the pressure. Equation (3.5) inserted into the normalization condition (3.2) can be integrated explicitly n(T +bC )=C [ 1+δ2 κ2(C )+δ4 κ4(C )+ . . . ] , (3.6) where equation (3.4) has been used for n0; here, n ≡ N /ΩL is the number density in the state of thermal equilibrium, while the coefficients introduced read: κ2(C )≡ 1 12 ν2(C )L2, κ4(C ) ≡ 1 80 ν4(C )L4. (3.7) Expression (3.6) is an equation to determine the constant C . Since ν2, ν4, . . . depend on C too, equa- tion (3.6) is highly nonlinear. Its solution for the weak nonequilibrium can be sought by perturbations: C =C (0) +δ2C (2) +δ4C (4) + . . . . (3.8) Even orders in δ are absent here because they are absent in equation (3.6). We note that the coefficients κ2, κ4, . . . are also to be expanded in δ: κi (C )= κ(0) i +δ2 κ (2) i +δ4 κ (4) i + . . . , i = 2; 4; . . . , (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: 1+δ2 κ2 +δ4 κ4 + . . . = 1+δ2 κ(2) +δ4 κ(4) + . . . , (3.10) where κ(2) ≡ κ (0) 2 and κ(4) ≡ κ (0) 4 +κ (2) 2 contain terms of their own orders in the gradients. In accordance with equations (3.7), they are related to ν(2) ≡ ν(0) 2 , ν(4) ≡ ν(0) 4 + 20 3 L−2ν(2) 2 , . . . (3.11) Explicit expressions for these coefficients are given in the Appendix. Insertion of equations (3.8) and (3.10) into expression (3.6) gives equations for C (k): 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: C (0) = n T η̃−1, C (2) =C (0)η̃−1[−κ(2)], C (4) =C (0)η̃−1 [ −κ(4) + η̃−1 κ 2 (2) ] . Here, η̃≡ 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 equa- tion of the weakly nonequilibrium heat-conduction steady state for hard spheres in the van der Waals approximation [56]: P (N ,Ω,L;T, g1 , . . . , g4) = N kBT ΩL−N b (p0 +p2 +p4 + . . .), (3.12) where powers of δ are omitted, and p0 ≡ 1, p2 ≡ 1 12 ( g2 − g 2 1 η̃ ) L2, and p4 ≡ 1 80 [ g4 −2g3g1η̃− 4 9 g 2 2 η̃+ 17 9 g2g 2 1 η̃ ( 1− 12 17 η ) − 4 9 g 4 1 η̃ 2 ( 1+ 1 4 η )] L4. The quantities p2 and p4 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 43002-6 Hard spheres in the heat-conduction steady state combination η̃≡ 1−η. Tending b → 0 causes η→ 0, η̃→ 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 n0 = 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 ( g2η̃− g 2 1 η̃ 2 ) L2, and n(4) 0 = n 1 80 [ g4η̃−2g3g1η̃ 2 − 4 9 g 2 2 η̃ ( 1+ 1 4 η ) + 17 9 g2g 2 1 η̃ 2 ( 1− 2 17 η ) − 4 9 g 4 1 η̃ 3 ( 1+ 3 2 η )] L4. The internal energy is calculated by integration of its density ε(z), equation (2.9): E ≡Ω L/2 ∫ −L/2 dz ε(z). (3.14) It follows from the local equation of state (2.8) that ε(z) = 1 2 DP [1−bn(z)] is not constant along the heat flux, contrary to the low-density gas case [49]. Normalization condition (3.2) simplifies the integration in equation (3.14) resulting in E = 1 2 DPΩLη̃. Using expression for P yields: E (N ,Ω,L;T, g1, . . . , g4) = 1 2 DN kBT (1+e2 +e4 + . . .) (3.15) with coefficients ek = pk 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 low- density result [49], while the equilibrium energies of these systems are known to be identical and inde- pendent on volume. 4. Entropy Expression (2.10) for s(z) is related to a number of states in the phase space owing to the starting equilibrium entropy (2.7). For this reason, we accept an integral quantity S ≡Ω L/2 ∫ −L/2 dz s(z) (4.1) 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]. Contributions to S. Using the local equation of state (2.8) for the expression { [n(z)]−1 − b } in equa- tion (2.10), we obtain: s(z)= kBn(z) [ d1 ln T (z)− ln(P/kB)+ξ(D) S ] , (4.2) with d1 ≡ 1 2 D+1. This expression inserted into equation (4.1) produces three contributions corresponding to the terms in the square brackets: S = ST +SP +Sξ . SP and Sξ can be found by virtue of the normalization condition, equation (3.2): SP =−N kB ln(P/kB), Sξ = N kBξ (D) S . (4.3) Expanding ln(P/kB) into a series yields SP = N kB ( sP,0 + sP,2 + sP,4 + . . . ) , (4.4) where the coefficients read: sP,0 ≡− ln(n T η̃−1), sP,2 ≡ 1 12 ( − g2 + g 2 1 η̃ ) L2, and sP,4≡ 1 80 [ −g4 + g3g12η̃+ g 2 2 ( 13 18 − 4 9 η ) + g2g 2 1 η̃ ( − 22 9 + 4 3 η ) + g 4 1 η̃ 2 ( 13 18 + 1 9 η )] L4. 43002-7 Y.A. Humenyuk Calculation of ST . We insert equation (4.2) into definition (4.1): ST ≡Ωd1kB L/2 ∫ −L/2 dz w(z), (4.5) where w(z) ≡ n(z) lnT (z). After the expansion of this function is used, w(z) = n0 ( w0 +w1z +w2z2 + . . . ) , we can integrate in equation (4.5) in explicit way: ST = Ωd1kBn0L ( w0 + 1 12 w2L2 + 1 80 w4L4 + . . . ) . (4.6) Now, we use the expansion (3.5) for n(z) and the series lnT (z) = τ0 +τ1z +τ2z2 + . . . with the coeffi- cients calculated from equation (2.1), which read: τ0 = lnT, τ3 = g3 − g2g1 + 1 3 g 3 1 , τ1 = g1, τ4 = g4 − g3g1 − 1 2 g 2 2 + g2g 2 1 − 1 4 g 4 1 , τ2 = g2 − 1 2 g 2 1 , . . . The quantities wi can be identified as discrete convolutions of {ν} and {τ}: wi ≡ νiτ0 + . . .+νi−kτk + . . .+ν0τi . (4.7) The series in equation (4.6) is to be rearranged in a similar way to the pressure case given above, as wi are linear combinations of coefficients {ν}: ST =Ωd1kBn0L [ w(0) + 1 12 w(2)L 2 + 1 80 w(4)L 4 + . . . ] , (4.8) where the new coefficients read: w(0) ≡ w0, w(2) ≡ w (0) 2 , w(4) ≡ w (0) 4 + 20 3 L−2w (2) 2 , . . . ; here, w (0) i ≡ ∑i k=0 ν(0) i−k τk and w (2) 2 ≡ ν(2) 2 τ0 +ν(2) 1 τ1. After substitution of expansion (3.13) for n0 into equation (4.8), we obtain the following result: ST = N kB ( sT,0 + sT,2 + sT,4 + . . . ) , (4.9) where sT,0 ≡ d1 ln T , sT,2 ≡ d1 1 12 [ g2 − g 2 1 ( 3 2 −η )] L2, and sT,4 ≡ d1 1 26·32 ·5 [ 36g4 + (−108+72η)g3 g1 + (−34+16η)g 2 2 + + ( 148−160η+48η2 ) g2g 2 1 + ( −45+56η−16η2 −4η3 ) g 4 1 ] L4. Compatibility with the second law of thermodynamics. We collect the contributions found in equa- tions (4.3), (4.4), and (4.9) to obtain the final expression [56]: S = N kB ( s0 + s2 + s4 + . . . ) (4.10) with the coefficients defined by si ≡ sT,i + sP,i + sξ,i : s0 ≡ ln(ΩL/N −b)+ D 2 ln T +ξ(D) S , s2 ≡ 1 23 ·3 { Dg2 + [( − 3 2 +η ) D −1 ] g 2 1 } L2, s4 ≡ 1 27 ·32·5 ( σ4g4 +σ31g3g1 +σ2g 2 2 +σ21g2g 2 1 +σ1g 4 1 ) L4; 43002-8 Hard spheres in the heat-conduction steady state the σα’s multiplying the gradients in s4 are written as follows: σ4 ≡ 36D, σ21 ≡ (148−160η+48η2 )D+120−48η, σ31 ≡ (−108+72η)D −72, σ1 ≡ (−45+56η−16η2 −4η3)D −38+16η+4η2 . σ2 ≡ (−34+16η)D −16, We notice that the coefficients si for the entropy (an additive quantity) depend on the dimensionality D , contrary to the pressure ones, pi . 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 Sfin 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 kBTfin, where Tfin is the temperature ascribed to the final state. We derive from equation (3.15) for the internal energy: Tfin = T ( 1+e2 +e4 + . . . ) . Substitution of this result into equation (2.6) leads to the entropy of the final equilibrium: Sfin = N kB { ln ( ΩL N −b ) + D 2 ln T + D 2 [ p2 + ( p4 − 1 2 p2 2 ) + . . . ] +ξ(D) S } , where we have used that ek = pk , while the third term in the curly brackets is an expansion of the logarithm. The entropy difference ∆S ≡ S −Sfin takes the form ∆S =∆S(2) +∆S(4) + . . . with ∆S(2) ≡ N kB D +2 48 ( − g 2 1 L2 ) , ∆S(4) ≡ N kB D +2 27 ·32 ·5 [ −36g3g1 −8g 2 2 + (60−24η)g2 g 2 1 + ( −19+8η+2η2 ) g 4 1 ] L4. 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 ther- modynamics for nonequilibrium processes [2, 3, 57]. 5. 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 ap- proach 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 equilib- rium 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. 43002-9 Y.A. Humenyuk 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 fluctu- ation theorems) have been used. The simplicity and explicit analytical description can be also regarded as positive features. Acknowledgement The author is grateful to Dr. T.M. Verkholyak for useful comments and for reporting the work on the exact solution for the equilibrium one-dimensional hard disks. This research has made use of NASA’s Astrophysics Data System. A. Expressions for coefficients {ν} and {κ}. First of all, we expand quantities {γ}, which define coefficients {ν} by the expressions given after equation (3.5). Note that γ j is of the order ∼ δ j , equation (3.4): γ j = 1 j ! G j T +bC (0) 1 1+δ2ζ(2) +δ4ζ(4) + . . . , where ζ(k) ≡ bC (k)/[T +bC (0)] ∣ ∣ k=2;4; .... For the fourth order it is sufficient to expand only γ1 and γ2, while γ3 and γ4 are to be taken in the lowest approximation: γ j ∣ ∣ j=1;2 = g j η̃ { 1+δ2[−ζ(2)]+ . . . } , γ j ∣ ∣ j=3;4 = g j η̃ ( 1+ . . . ) , where g j ≡ 1 j ! G j /T . A contribution from ζ(2) is also needed in γ2 1 = g 2 1 η̃ 2 { 1+δ22[−ζ(2)]+ . . . } . From the definitions given after equation (3.5), we find the lowest-order contributions to {ν} in the form: ν(0) 1 =−g1η̃, ν(0) 3 =−g3η̃+2g2g1η̃ 2 − g 3 1 η̃ 3, ν(0) 2 =−g2η̃+ g 2 1 η̃ 2, ν(0) 4 =−g4η̃+2g3g1η̃ 2 + g 2 2 η̃ 2 −3g2g 2 1 η̃ 3 + g 4 1 η̃ 4. The result ζ(2) =− L2 12 η̃−1ην(0) 2 gives the corrections as follows: ν(2) 1 = L2 12 η ( g2g1η̃− g 3 1 η̃ 2 ) , ν(2) 2 = L2 12 η ( g 2 2 η̃−3g2g 2 1 η̃ 2 +2g 4 1 η̃ 3 ) which are of the third and fourth orders. Due to the factor η ∼ n, ν(2) i are small in comparison with ν(0) i at low densities. Expressions for κ(0) 2 , κ(2) 2 , and κ(0) 4 can be obtained using the formulae for ν(k) i and equations (3.7). Then, the rearranged coefficients read: κ(2) = L2 12 ( − g2η̃+ g 2 1 η̃ 2 ) , κ(4) = L4 80 [ −g4η̃+2g3g1η̃ 2 + g 2 2 η̃ ( 1− 4 9 η ) −3g2g 2 1 η̃ 2 ( 1− 4 9 η ) + g 4 1 η̃ 3 ( 1+ 1 9 η )] . References 1. Balescu R., Equilibrium and non-equilibrium statistical mechanics, Wiley-Interscience, New York, 1975. 2. 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Humenyuk Тиск i ентропiя твердих кульок у слабонерiвноважному теплопровiдному стацiонарному станi Й.А. Гуменюк Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна В рамках пiдходу суцiльного середовища розраховано термодинамiчнi величини системи твердих кульок у стацiонарному станi з малим тепловим потоком. Аналiтичнi вирази для тиску, внутрiшньої енергiї та ентропiї знайдено в наближеннi четвертого порядку за ґрадiєнтами температури. Показано, що ґрадiєнтнi внески до внутрiшньої енергiї залежать вiд об’єму, а ентропiя задовольняє II-е начало термодинамiки для нерiвноважних процесiв. Розрахунки проведено для вимiрностей 3D, 2D та 1D. Ключовi слова: тепловий потiк, температурнi ґрадiєнти, рiвняння стану, нерiвноважна ентропiя, термодинамiка стацiонарного стану 43002-12 Introduction Heat-conduction state of the hard spheres Baric and caloric equations of state Entropy Conclusions Expressions for coefficients { } and { }.

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

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