Local variational description for a dense gas

Physica A 269 (1999) 211–221 www.elsevier.com/locate/physa Local variational description for a dense gas L. Romero-Salazar a;∗ , M. Mayorga a , R.M. Velasco b a Facultad de Ciencias de la Universidad Autonoma del Estado de Mexico, C.P. 50000 Toluca, Edo.Mex, Mexico b Departamento de Fsica, Universidad Aut onoma Metropolitana-Iztapalapa, C.P. 09340, Mexico D.F., Mexico Received 15 January 1999 Abstract We present the study of a dense gas at a hydrodynamical level of description. We start from the entropy functional obtained at a mesoscopic regime (L. Romero-Salazae et al.; Physica A 237 (1997) 150.) (MEF1) and we apply the Maximum Entropy Formalism where the restrictions are now the hydrodynamic variables and the density uctuations. An entropy functional is obtained both, in the local and the equilibrium inhomogeneous situation. c 1999 Elsevier Science B.V. All rights reserved. PACS: 05.20.-y; 47.10.+g Keywords: Maximum entropy formalism; Hydrodynamic variables; Correlation distribution function 1. Introduction The search for a scheme describing the behavior of a dense gas has known several approaches, going from the heuristic point of view such as the one given by the Enskog kinetic theory (SET) [1], up to numerical calculations for the transport coecients which start with the equations of motion for each particle in the system [2]. It is well known that all these theories give us some understanding of the properties of a dense gas, binary mixtures, etc. though the problem is far from solved. Recently [3] the kinetic variational theories (KVT) [4 –8] have been developed providing us with a consistent frame to study the behavior of dense gases. In particular, the scheme we have called as the maximum entropy formalism (MEF) [8] has been taken as a Corresponding author. E-mail address: lors@coatepec.uaemex.mx (L. Romero-Salazar) ∗ 0378-4371/99/$ - see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 0 4 - 1 212 L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 cornerstone to study di erent steps in the description of the time evolution of a dense gas to the equilibrium state. This scheme starts with the Gibbs entropy functional de ned in terms of the N -particle distribution function which satis es the Liouville equation. In a rst step the extremal principle takes this entropy functional and the restrictions imposed in the procedure correspond to the knowledge of the two rst reduced distribution functions, besides the normalization condition. This means that the description we looked for in this formalism considers the one-particle and two-particle reduced distributions functions as the main pieces to describe the behavior of the system. The new entropy functional, the kinetic equations as well as the entropy balance [9,10] were constructed in a way that in the scheme given by Lewis [3] we have a complete description. In order to have this scheme as reference we will call it as MEF-I [8]. In this work we will take the MEF-I as a starting point, since it has all the elements proper to do that, it means that we have three functions in terms of which the description can be made, an entropy functional, and the time evolution written in terms of some kinetic operators. Let us make some emphasis about the meaning of the relevant functions considered in this description, in fact there is a nonequilibrium partition function coming from the normalization condition, and the two reduced distribution functions for one and two particles [8]. The equations of motion are generalized kinetic equations which become a closed set when the maximum entropy closure is substituted in the dissipative terms [11]. Also the new entropy functional can be written in terms of the corresponding relevant functions, this functional coincides with some expressions proposed in the literature [12–14], but now it is a result of the KVT or MEF-I formalisms. Notice should be made that MEF-I as well as the KVT theories give a description in kinetic terms, from which we need to calculate the macroscopic properties through averages on the corresponding phase space. A characteristic that should be remarked about these schemes, is their consistency with the macroscopic balance equations [4,5,8]. Here we will take the MEF-I scheme and apply the maximization procedure once more, with the restrictions imposed by the knowledge of the macroscopic variables [15]. In Section 2 we will give the details of the maximization procedure, Section 3 is devoted to the study of the Lagrange multipliers needed in the scheme, and their corresponding limiting value in the equilibrium state is given in Section 4. Lastly in Section 5 we give some concluding remarks. 2. Maximum entropy formalism: hydrodynamic regime The system we want to describe is a dense monatomic gas which is modeled as a set of N particles interacting via an additive pairwise short range potential. The kinetic description of such a system has been done in terms of the MEF-I formalism [8], where we have used the maximum entropy formalism to obtain the entropy functional, the distribution functions (d. .) and the evolution equations for the d. . for a kinetic L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 213 level of description. Such a description had as restrictions: the normalization of the N particle d.f., the one f(1) = f(1) (x1 ; t) and two particle f(2) = f(2) (x1 ; x2 ; t) d. ., here the vector xi = (ri ; pi) denotes the position and momentum dependence of the d. . On the same basis as the one used before [8], we can obtain a complete description as de ned by Lewis [3], at the hydrodynamic level. The relevant functions that describe the system will be the conserved variables: mass density, momentum density, internal energy density and additionally the normalization condition for the two-particle correlation function g(2) which is represented by the density uctuations as we will show later on. Now we wish to follow the maximum entropy formalism with the hydrodynamic conserved variables and the density uctuations as restrictions. In order to proceed with the variational principle we need the entropy functional that corresponds to the MEF-I, namely, Z S[f(1) ; f(2) ] − Sexc (t) = −kB f(1) (x1 ; t)ln (h3 f(1) (x1 ; t)) dx1 Z kB f(2) (x1 ; x2 ; t)ln (g(2) (x1 ; x2 ; t)) dx1 dx2 ; (1) − 2 where the rst term is a Boltzmann-like one for an ideal gas and the second term takes into account the correlations between particles through the conditional probability g(2) (x1 ; x2 ; t) = f(2) (x1 ; x2 ; t)=f(1) (x1 ; t)f(1) (x2 ; t). The second addend on the left-hand side (l.h.s.) of Eq. (1) is an excess contribution [8] which does not contribute to the next level of description as we will show later on. The restrictions at this level are the following: • mass density written as Z (2) (r1 ; t) = mf(1) (x1 ; t) dp1 ; • momentum density de ned as Z (r1 ; t)u(r1 ; t) = p1 f(1) (x1 ; t) dp1 ; • the internal energy density stated as Z [p1 − mu(r1 ; t)]2 (1) f (x1 ; t) dp1 (r1 ; t) = 2m Z 1 (r1 ; r2 )f(2) (x1 ; x2 ; t) dx2 dp1 ; + 2 (3) (4) where the total internal energy has been chosen consistently with the model in MEF-I as a pairwise additive function of the pair intermolecular potential (r1 ; r2 ), • and nally, the normalization of the correlation function g(2) , de ned as follows: Z (r1 ; t) (5)
(r1 ; t) = f(1) (x1 ; t)f(1) (x2 ; t)[g(2) (x1 ; x2 ; t) − 1] dx2 dp1 ; m which physically concerns density uctuations [16]. 214 L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 In order to continue with the maximum entropy formalism we need to construct a functional I, with the previous information, i.e.    Z Z (1) (r ; t) (r ; t) − mf (x ; t) dp I = S[f(1) ; f(2) ] + 1 1 1 1 1 dr1 + Z    Z (1) (r ; t) · (r ; t)u(r ; t) − p f (x ; t) dp 2 1 1 1 1 1 1 dr1  Z [p1 − mu(r1 ; t)]2 (1) f (x1 ; t) dp1 (r1 ; t) − 2m   Z 1 (r1 ; r2 )f(2) (x1 ; x2 ; t) dx2 dp1 dr1 − 2  Z (r1 ; t)
(r1 ; t) + 4 (r1 ; t) m   Z (2) (1) (1) − [f (x1 ; x2 ; t) − f (x1 ; t)f (x2 ; t)] dx2 dp1 dr1 : + Z 3 (r1 ; t) (6) The functional I is written as a function of the one and two particle d. . and it contains 1 (r1 ; t); 2 (r1 ; t); 3 (r1 ; t); and 4 (r1 ; t) which are the Lagrange functions acquainted to their corresponding restrictions: mass density, momentum density, internal energy density and density uctuations, respectively. Once we have constructed the functional I we need to make two variations over the functional, one from f(1) to the state f(1) + f(1) and a second variation from the conditional probability g(2) to the state g(2) +g(2) . These variations are denoted by (1) I and (2) I, respectively, and must be equal to zero, this gives us a coupled pair of equations and their solution will give us the corresponding distribution functions, f(1) and g(2) that maximize the functional I. The variation respect to g(2) then leads us to the following equation:  Z 1 f(1) (r1 ; p1 ; t)f(1) (r2 ; p2 ; t) ln g(2) (r1 ; p1 ; r2 ; p2 ; t) + 1 (2) I = −kB 2  2 4 (r1 ; t) 3 (r1 ; t)(r1 ; r2 ) g(2) dp1 dp2 dr2 + + kB kB =0: (7) Which allows us to obtain an expression for the conditional probability g(2) , namely, g(2) (x1 ; x2 ; t) → g(2) (r1 ; r2 ; t)  = exp −1 − 2  4 (r1 ; t) = Y (r1 ; t)exp − kB   exp − 3 (r1 ; t)(r1 ; r2 ) kB 3 (r1 ; t)(r1 ; r2 ) kB  :  (8) L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 215 The result shown in Eq. (8) deserves some attention, rst of all the conditional probability becomes independent of the momentum coordinates of both particles, giving a dynamic correlation function g(2) which is related to the interaction potential and the cavity function Y (r1 ; t) = exp(−1 − 2( 4 (r1 ; t)=kB )): The cavity function as de ned in Eq. (8) is given in terms of the Lagrange multiplier associated to the normalization of the two-particle correlation function. The complete expression 1 + 2( 4 (r1 ; t)=kB ) + ( 3 (r1 ; t)=kB )(r1 ; r2 ) corresponds to the potential of the mean force in kB T units, where 1 + 2( 4 (r1 ; t)=kB ) is the contribution due to the surrounding particles in the system, and (r1 ; r2 ) stands for the direct interaction between particles. This result has given us a qualitative physical insight on the Lagrange multipliers 3 (r1 ; t) and 4 (r1 ; t); which appear as energy and density uctuations restrictions, respectively. On the other hand, since all restrictions are averages in momentum space, hence neither the multipliers nor the two particle correlation d.f. have momentum dependence. In order to continue with the maximum entropy formalism, we proceed with the variation of I from f(1) to f(1) , as follows:  Z 1 (r1 ; t)m 2 (r1 ; t) · p1 + (1) I = −kB (xk − x1 ) ln h3 f(1) (x1 ; t) + 1 + kB 2kB  [p − mu(r1 ; t)]2 fk(1) dx1 + 3 (r1 ; t) 1 2m  Z kB 3 (r1 ; t) (1) f (x2 ; t)(xk − x1 ) ln g(2) (r1 ; r2 ; t) + − (r1 ; r2 ) 2 2kB  2 4 (r1 ; t) g(2) (r1 ; r2 ; t) fk(1) dx1 dx2 + kB  Z kB 3 (r1 ; t) f(1) (x1 ; t)(xk − x2 ) ln g(2) (r1 ; r2 ; t) + − (r1 ; r2 ) 2 2kB  2 4 (r1 ; t) g(2) (r1 ; r2 ; t) fk(1) dx1 dx2 + kB Z 2 4 (r1 ; t) kB {(xk − x1 )f(1) (x2 ; t) + f(1) (x1 ; t) + 2 kB ×(xk − x2 )} fk(1) dx2 dx1 : (9) In order to nd the d.f. f(1) we substitute Eq. (9) in Eq. (12), afterwards the variation (1) I is equal to zero and the ensuing distribution function is the following:   1 (r1 ; t)m (1) f (x1 ; t) = exp − 1 + kB   [p − mu(r1 ; t)]2 2 (r1 ; t) · p1 + 3 (r1 ; t) 1 × exp − 2kB 2m 216 L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 ×h −3 exp Z 0 Rc  (r2 ; t) (2) (2) [g (r1 ; r2 ; t) −(ln Y (r1 ; t)+ 1)] dr2 m (10) The one-particle distribution function thus obtained looks like a Maxwellian distribution function in the momentum coordinates, however the coordinates dependence of Lagrange multipliers and the factor containing the probability prevents us from interpreting it in a simple way. The latter integration in Eq. (10) can be limited to the space of an “action sphere” of radius Rc , that represents the radius of the“action sphere” where the dynamic correlations take place. 3. The Lagrange multipliers analysis In this section we will search for a better understanding of the corresponding Lagrange multipliers for the system, and we will try to identify them. This means that we must insert the d. . Eqs. (8) and (10) in the established restrictions (Eqs. (2) – (5)). Starting with Eq. (10), once we substitute f(1) in the momentum density de nition we can realize that 2 (r1 ; t) is zero. For the local mass density we have  3=2   2mkB (r1 ; t) 1 (r1 ; t)m = exp −1 − m kB 3 (r1 ; t)   Z Rc 1 (2) (r2 ; t)[g (r1 ; r2 ; t) − (ln Y (r1 ; t) + 1)] dr2 ; (11) × exp m 0 which represents the density pro le of a dense gas around a test particle. This equation shows how the density pro le is a local function which contains the spatial correlations between particles, measured through the correlation function g(2) . When the correlations become unimportant and the e ective potential does not have e ect we can de ne the ideal gas limit density,    3=2 2mkB id (r1 ; t) 1 (r1 ; t)m = exp −1 − : (12) 3 m kB 3 (r1 ; t)h These last equations can be used to express the density pro le in a di erent manner, namely,  Z Rc  1 (2) (r2 ; t)[g (r1 ; r2 ; t) − 1 − ln Y (r1 ; t)] dr2 : (r1 ; t) = id (r1 ; t)exp m 0 (13) The remaining product in the density pro le, besides the ideal one, takes into account the potential interactions. As the addition of two main contributions, the rst one accounts for the pair interactions in the “action sphere” of radius Rc ; and it is represented by the following: Z kB T kB T Rc (14) (r2 ; t)[g(2) (r1 ; r2 ; t) − 1] dr2 ≈ no ; 2 m 0 m L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 217 where no = NVo =V is the number density in the nonequilibrium action sphere and Vo is the volume of the nonequilibrium action sphere. The second one Z N kB T Rc (15) (r2 ; t)ln Y (r1 ; t) dr2 ≈ exc − 2 m 0 m accounts the remaining potential contribution due to the interactions within the nonequilibrium action sphere and is a mean eld contribution. On the other hand, substituting Eqs. (8) and (10) in the internal energy density de nition Eq. (4), we obtain Z (r1 ; t) 1 kB (r2 ; t) (2) 3 (r1 ; t) (r1 ; r2 ) + g (r1 ; r2 ; t) dr2 ; (r1 ; t) = 2 m m 2 m 3 (r1 ; t) (16) this reminds us the standard expression for the internal energy [15], the rst addend represents the kinetic energy and the second takes into account the potential interaction between the particles that compose the system. This allows us to identify the Lagrange multiplier 3 (r1 ; t) as follows: 3 (r1 ; t) = 1 ; T (r1 ; t) (17) this means that it is the inverse of the local temperature de ned through the total internal energy density. A very important remark must be done at this point, the g(2) (r1 ; r2 ; t) is a function of the local temperature T (r1 ; t) as a result of the substitution of the Lagrange multiplier 3 (r1 ; t) in Eq. (8) and of course it has a density dependence as can be seen from expression (13). This means that the local temperature is in fact de ned through the total internal energy density, taking into account the contribution of the potential energy and not only the kinetic energy, as sometimes used in dense gas theory [6,7]. On the other hand, with the identi cation of the Lagrange multiplier 3 (r1 ; t) as the inverse of the temperature we can go back to the ideal density expression, Eq. (12), and we can rewrite it as   2mkB T (r1 ; t) m 1 (r1 ; t) id (r1 ; t) 3 − ln = −1 − : (18) ln m 2 mh2 kB This equation leads to the identi cation of the Lagrange multiplier 1 in terms of thermodynamic parameters and on the other hand, it reminds us to the chemical potential for a non uniform uid. Having this in mind we de ne the ideal part of the chemical potential k   m 1 (r1 ; t) kB k T (r1 ; t) (19)  (r1 ; t) = −1 − kB m such that the Lagrange multiplier 1 concerns with the intensity of kinetic chemical potential in terms of kB T . On the other hand, we observe that expression (11) for the density pro le, contains an excess contribution due to the density uctuations hence it can be identi ed with 218 L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 the excess part of the chemical potential, namely Z Rc kB (r2 ; t)[g(2) (r1 ; r2 ; t) − 1 − ln Y (2) (r1 ; t)] dr2 : ex (r1 ; t) = 2 T m 0 (20) After substituting Eqs. (15) and (18) in Eq. (13), we obtain the following density pro le:   ex m (r1 ; t) : (21) (r1 ; t) = id exp kB T (r1 ; t) We still have to substitute our d. . in the last restriction, namely the density uctuations, (=m)  = (=m)
+ 1, hence from Eq. (5), we obtain     Z Rc (r1 ; r2 ) (r2 ; t) (r1 ; t) Y (r1 ; t)exp − − 1 dr2 (22)  (r1 ; t) − 1 = m m kB T (r1 ; t) 0 the compressibility equation in a local form. Using the density expression (11) in Eq. (10), we can rewrite the local momentum distribution function of particles:
2 ! p1 − mu(r1 ; t) (r1 ; t) −3=2 (1) ; (23) (2mkB T (r1 ; t)) exp − f (p1 ; r1 ; t) = m 2mkB T (r1 ; t) which resembles the usual local Maxwellian d.f. In fact, it should be mentioned that it contains the local temperature de ned through the total energy density and the local density carries with it the nonideal contributions coming from correlations between particles. In the same manner, with identi cation (15), we have an expression for the two-particle dynamic correlation d.f.   (r1 ; r2 ) : (24) g(2) (r1 ; r2 ; t) = Y (2) (r1 ; t)exp − kB T (r1 ; t) To continue with the formal description, we need to obtain the maximized entropy functional in terms of the above imposed restrictions. Hence we substitute d. . (22) and (23) that maximize the entropy equation (1), and we obtain the next expression (r1 ; t) h id 3 i (r1 ; t) − kB ln  S[; u; ] = T (r1 ; t) m m Z kB (r1 ; t) Rc (r2 ; t) (25) ln [g(r1 ; t) − 1] dr2 ; − 2 m m 0 where we have used the de nition of the internal energy density, Eq. (16). The entropy expression (24) is the last part of the complete description at the hydrodynamic level, in terms of the conserved variables and the density uctuations. In addition to the ideal-like contribution, the last term involves the density uctuations  (r1 ; t) (see Eq. (21)), that guarantees the additive nature of the entropy, because it weights the term proportional to the number of particles of the system to the square. The previous assertion will become clearer in the equilibrium limit. L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 219 4. Equilibrium limits In order to complete the hydrodynamic description we will discuss the equilibrium values that must be reached by all quantities introduced in the maximization procedure. This is an important step when we try to solve the proper evolution equations to this scheme (i.e. the mass, momentum and energy density balance equations), besides it allows us to show the internal consistency of the theory. In this limit the one-particle distribution function (22) is a Maxwellian, i.e.,   eq (r1 ) |p1 |2 (1) (2mkB T eq )−3=2 exp − = ; (26) feq m 2mkB T eq it is important to notice that the density pro le is the equilibrium limit of the pro le showed in Eq. (13). Also, the Lagrange function associated to the internal energy is the inverse of the temperature of the system at equilibrium, 3eq = (T eq )−1 . As we have stated earlier 2eq = 0. In the next step we analyze the information given by the pair correlation (23) at equilibrium, which is written as follows:   (r1; r2 ) (2) ; (27) (r1; r2 ) = Yeq (r1 )exp − geq kB Teq (2) (r1; r2 ) is the usual radial distribution function of dense uids and Yeq (r1 ) aci.e., geq counts for the deviation from the low-density behavior. It must be noticed that using a cluster expansion it can be written as Yeq (r1 ) = 1 + g1 + O(2 ); where g1 is expressed in terms of Mayer functions [17]. On the other hand, this expression leads us to the normalization of the pair correlation (2) (r1 ; r2 ): function geq     Z rc eq (r1 ; r2 )  (r2 ) eq eq Y (r1 )exp − − 1 dr2 ; (28)  −1= m kB T eq 0 i.e., the compressibility equation [16], and rc is the radii of the equilibrium “action sphere”. The radii Rc in Eq. (11) and this last radii rc may be di erent due to nonequilibrium e ects. Now we will continue with the mass density, substituting the known Lagrange multiplier we obtain    3=2 eq 2mkB T eq eq (r1 ) 1 (r1 )m = exp −1 − m h3 kB  Z rc  eq  (r2 ) (2) [g (r1; r2 ) − 1] dr2 ×exp m 0  Z rc eq  (r2 ) (29) ln Y eq (r1 ) dr2 ; ×exp m 0 the density equilibrium pro le for an inhomogeneous system. 220 L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 In the same manner as was made in the preceding section, we identify the total chemical potential, in this case at nonuniform equilibrium Z rc eq mk  (r2 ) (2) eq (r1 ) = + [geq (r1 ; r2 ) − 1 − ln Y (r1 ; t)] dr2 : (30) kB T eq (r1 ) kB T eq (r1 ) m 0 This completes the searched values for the Lagrange functions at equilibrium state for this thermodynamic maximum entropy formalism. An additional information to be stated is the entropy functional at equilibrium, namely   Z eq  (r2 ) kB (r1 ) (r1 ) (r1 )  ln 3 (ln Y (r1 ; t)) dr2 : − (31) Seq = − kB T m m 2m m From thermodynamics we know that the entropy functional S, the total internal energy  and the free energy A must obey the following expression: T eq Seq =  − A (32) and from the inspection of the previous two relations we can identify to the free energy   Z eq  (r2 ) kB T eq  (r1 ) (r1 )  (r1 ) + ln 3 (ln Y (r1 ; t)) dr2 ; (33) A = kB T eq m m 2m m where the rst and the second addend are the ideal and excess part, respectively. 5. Discussion The maximum entropy formalism (MEF) applied to the entropy functional obtained in MEF-I with a set of restrictions proper to a hydrodynamical level of description has allowed us to study some properties of the formalism itself. Such properties concern the successive application of the procedure to obtain a description which has less information than the preceding one. Here the calculation we have made gives us local hydrodynamical variables and the information in the momentum variables in the phase space has been lost. In spite of this particularity, we have the distributions functions consistent with this knowledge, the new entropy functional and the equations of motion corresponding to this level. Hence the new description is also a complete one. In fact, we have not discussed the equation of motion in detail, because they immediately arise from the kinetic equations in MEF-I and are the usual balance equations, as it was shown in Ref. [8]. On the other hand, the procedure used in this work has driven us to an identi cation of the Lagrange multipliers, one of them being related with the temperature, and another with the density uctuations. This restriction arose from the normalization condition in the two particle distribution function, which is not explicitly taken into account in other treatments. We have shown internal consistency in this description and it satis es all the requirements to be a complete description [3] for a dense gas in a local variational approximation. L. Romero-Salazar et al. / Physica A 269 (1999) 211–221 221 Also the density pro le plays an important role, since it corresponds to the pro le that maximizes the entropy functional in analogy as the corresponding one that would minimize the free energy in the density functional theory [18]. Acknowledgements L. Romero-Salazar and M. Mayorga acknowledge nancial support from CONACyTMexico, under projects: I25363-E and I25364-E, respectively. References [1] S. Chapmann, T.S. Cowling, The Mathematical Theory of Non-Uniform Gases, third ed., Cambridge University Press, Cambridge, 1970. [2] D.M. Heyes, The Liquid State, Applications of Molecular Simulations, Wiley, New York, 1998. [3] R.M. Lewis, J. Math. Phys. 8 (1967) 1448. [4] J. Karkheck, G. Stell, J. Chem. Phys. 75 (1981) 1475. [5] J. Karkheck, G. 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