Int. J. of Thermodynamics ISSN 1301-9724 Vol. 8 (No. 4), pp. 159-166, December 2005

Generalized Gibbs , Irreversibility and Stationary States

A. Pérez-Madrid Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona. Diagonal 647, 08028 Barcelona. Spain E-mail: [email protected]

Abstract A generalization of the Gibbs entropy postulate is proposed, based on the BBGKY hierarchy as the non-equilibrium entropy for a system of N interacting particles. This entropy satisfies the basic principles of thermodynamics in the sense that it reaches its maximum at equilibrium and is coherent with the second law. By using this entropy and the methods of non-equilibrium thermodynamics in the phase space, a generalization of the Liouville equation describing the evolution of the distribution vector in the form of a master equation is obtained. After neglecting correlations in this master equation, the was obtained. Moreover, this entropy remains constant in non- equilibrium stationary states and leads to macroscopic hydrodynamics. Non-equilibrium Green-Kubo type relations and the probability for the non-equilibrium fluctuations are also derived Keywords: Non-equilibrium , irreversible thermodynamics, stationary states 1. Introduction In contrast to the Boltzmann entropy, the Gibbs entropy is not a function of the individual According to the mechanistic interpretation microstate but rather a function of the probability of the physical world, the basic laws of nature distribution in a statistical ensemble of systems are deterministic and time reversible. However, with both coinciding at equilibrium. As at the macroscopic level, we observe irreversible a consequence of the incompressible character of processes related to energy degradation which the flow of points representing the natural generate entropy. How do we reconcile the evolution of the statistical ensemble in phase ‘spontaneous production of entropy’ with the space, the Gibbs entropy is a constant of motion. time reversibility of the microscopic equations of Thus, it has been argued that the relevant entropy motion? At the end of the nineteenth century, for understanding thermodynamic irreversibility Boltzmann tried to answer this question from a is the Boltzmann entropy and not the Gibbs probabilistic point of view. According to him, entropy (Lebowitz, 1999a; Lebowitz, 1999b; entropy is a measure of the lack of knowledge of Goldstein, 2001). the precise state of matter and can be defined as a In addition, the problem of the diverging function of the probability of a given state of character of the Gibbs entropy related to the matter. This function associates a number negative sign of the entropy production in non- S(X)logBM(X)=Γ to each microstate X of a equilibrium stationary states apparently excludes macroscopic system, with ΓM being the the use of the Gibbs entropy in the statistical volume of the region of the phase space ΓM description of non-equilibrium systems (Andrey, corresponding to the macrostate MM(X)= . 1985; Hoover, 1992). This raises the question as The macrostate M is all of a group of states Y to how to define the non-equilibrium entropy and such that M(Y)== M(X) M . In this sense, the if possible, to give a thermodynamic description Boltzmann entropy is a function of the of non-equilibrium fluctuations. In other words: microstate which at equilibrium coincides with can thermodynamics describe systems far from the thermodynamic entropy. All systems in their equilibrium (Gallavotti, 2004; Ruelle, 1999)? irreversible evolution tend to a state of maximum Thus, from the moment when Gibbs first probability or maximum entropy -the state of postulated his entropy formula, the definition of equilibrium. the non-equilibrium entropy, and its relation to

Int. J. of Thermodynamics, Vol. 8 (No. 4) 159 irreversibility, has been an outstanding problem, equation. Nonetheless, an alternative description now compounded by the fact that the entropy of the state of the system can be given in terms production is negative in non-equilibrium of the distribution vector (Balescu, 1975) stationary states in apparent violation of the second law of thermodynamics (Evans, 1994; f ≡ {f,f(011ΓΓ ),f( 22 ), Wang, 2002). This constitutes an open problem (2) which must be solved...... , fNN (Γ )}

A huge amount of work has been done on with Γs12,s= (x , x ...... , x ) and this subject, trying to address the problem. On x(,)jjj≡ qp. Additionally, the set of the one hand, there have been attempts to extend quantities Γs can be grouped as the vector the equilibrium entropy (Rondoni, 2000; Ruelle, ΓΓΓ≡ { 12, ,...... , Γ N} , and 2003; Tuckerman, 1997; Gaspard, 1997) to non- correspondingly Η ≡ {H12 , H ,...... , H N} equilibrium situations in order to avoid the can be defined with Hs being the s-particle divergence of the Gibbs entropy. On the other Hamiltonian. The distribution vector represents hand, work has been done to establish fluctuation the set of all the s-particle reduced distribution theorems for the probability of the entropy functions f(Γ ) (s= 0,...., N) , defined production fluctuations (Evans, 2002). In a ss through previous work (Pérez-Madrid, 2004), we showed a way to circumvent the difficulty of reconciling N! fF(x,..,x,s1s= the second law of thermodynamics with the (N− s)!∫ (3) reversible microscopic equations of motion in the x ,.., x ) dx ...dx framework of the BBGKY hierarchy. We s1++ N s1 N proposed a functional of the set of s-particle The evolution equations of these functions reduced distribution functions as the entropy for can be obtained by integrating the Liouville a system of N interacting particles. This entropy equation, thereby constituting a set of coupled does not enter into contradiction with equations: the BBGKY hierarchy which can be thermodynamics, and as shown here, in addition written in a compact way as a generalized to being time-dependent, it enables the Liouville equation (Pérez-Madrid, 2004) performance of a thermodynamic analysis of the stationary non-equilibrium states. In this sense, ∂ −=PL f(t) QLf (t) (4) our theory constitutes an extension of the scope ∂t of thermodynamics to systems away from with equilibrium.  ss We begin this contribution introducing in  o ′  Section 2 the representation of the state of the 〈〉=δ+ssPL ′ s, s′ ∑∑ L j L j, n  (5) isolated system in terms of the hierarchy of  j1=<= jn1  reduced distribution functions. Afterwards in =δ H ,... s, s′ [] s P Section 3 we develop the thermodynamic analysis and derive the entropy production which and enables us to draw kinetic equations, in particular s the Boltzmann equation. Section 4 is devoted to ′ the analysis of the non-equilibrium stationary 〈〉=δssQL ′ s,s1′ + ∫ ∑ L j,s1++ dx s1 (6) states of the system. Finally in Section 5 we j1= stress the main conclusions. where |s〉 represents the s-particle state defined through ΓΓsf= ( ) and where the 2. Hamiltonian Dynamics sss projection operators P and Q , its complement Let's consider a dynamical system of N with respect to the identity, give the diagonal and identical particles whose Hamiltonian nondiagonal part of the generalized Liouvillian H(qpNN , ) is given by L , respectively. Here, LH,...oo=  where N {} jj ...,... is the Poisson bracket, H2mo2=Pp , [ ]P jj NN2 ''  ' p j 1 and LH,...j, n= j, n , with H/2j, n=φ j, n . In H=+φ−qq (1)  P Njk∑∑2m 2 () the next section we will show that irreversibility j1=≠= jk1 is manifested in the dynamics of the system when the adequate description, i.e. in terms of where m is the mass and φ()qqjk−≡φ jk is the interaction potential. The state of the system the distribution vector, is used. is completely specified at a given time by the N- particle distribution function F,({}qpNN ;t) 3. Non-Equilibrium Thermodynamics which evolves in time according to the Liouville As the expression for the non-equilibrium 160 Int. J. of Thermodynamics, Vol. 8 (No. 4) entropy, we propose 11Nn σ=−1nj∑∑∫f p ⋅ −1 Tn!n1== j1 Sktrln=−Beqeq{}fff() + S ∇−jB()k T ln f eq,n1 dx ....dx n (9) N 1 =−kf (7) Bn∑ ∫ 11Nn n1= n! −⋅fpF dx .....dx Tn!∑∑∫ ni ij1 n f n1=≠ ji lnn dx .....dx + S f 1neq eq, n where Fij= −∇ i φ ij is the force on particle i due to particle j and a functional of f , analogous with the Gibbs Nn entropy postulate (de Groot, 1984; van Kampen, 11 σ2nj=−∑∑∫f F ⋅ 1990), is based on the fact that the distribution Tn!n1== j1 (10) vector determines the state of the system. Here, p j1dx .....dx n kB is the Boltzmann constant, feq= {}f eq, 0 , f eq, 1 , f eq, 2 ,.... is assumed to be where Fj is the force on particle j pertaining to the equilibrium distribution vector which the nth− cluster from outside this cluster corresponds with the equilibrium entropy f(q)nFF j= ∫ j,n1n1++ fdx n1 + (11) SSeq= B , satisfying Lfeq = 0 (i.e. the Ybon- j −1 Born-Green hierarchy) whereas feq denotes the Both contributions, σ1 and σ2 , vanish vector whose components are the inverse of the when the distribution function f coincides with reduced equilibrium distribution functions, n −1 its equilibrium value feq, n . In any other case feq, n . Moreover, S is maximum at equilibrium these should not necessarily be zero. One has, when ff= eq giving SS= eq , which can be then proven by taking the first and second variation of σ =σ +σ (12) S with respect to f while Seq and feq remain 12 fixed. This entropy is also coherent with the For small deviations from equilibrium, equation second law according to which S increases in (8) becomes irreversible processes such as the relaxation to ∂f equilibrium from an initially non-equilibrium σ=−ktrB X (13) ∂t state. This expression based on the BBGKY −1 hierarchy is different from other developments in where Xf=−eq( ff eq ) is the thermodynamic the literature, such as the paper by Green and force conjugated with the current ∂f ∂t . Nettleton (Nettleton, 1958). Therefore, the rate Following the methods of non-equilibrium of change of S which can be obtained by thermodynamics (de Groot, 1984; Bedeaux and differentiating equation (7) with the help of Mazur, 2001) in the phase space, from equation equation (4) is (13) the linear relation ∂f dS ∂f −1 =−MX (14) ≡σ=−ktrBeq ln()ff ∂t dt∂ t (8) −1 is inferred, where the master or mobility matrix =−ktrBeq{}Lf ln() f f ≥ 0 M acts on an arbitrary vector Y according to 〈〉〈sss|M(s|s)MY′ ′′= ∫ where use has been made of the fact that (15) PL+= QL L . Thus, equation (8) constitutes the ′ ′′ ′ Ys1′ (x ,...., x s′′ )dx 1 .....dx s entropy production corresponding to the relaxation passing from a non-equilibrium state Note that in view of the orthogonal to equilibrium. character of the operators P and Q , equation (14) leads to QMX= QLf and PMX= PLf . The explicit expression of σ can be Substituting the linear relation equation obtained by using equations (5), (6) and (8). We (14) into equation (13) and using the cyclic get two contributions invariance of the trace, the entropy production is

tr(XMX) ≥ 0 (16) according to the second law of thermodynamics. Therefore, M is a positive semidefinite matrix. In addition, the master matrix should be

Int. J. of Thermodynamics, Vol. 8 (No. 4) 161 Hermitian ∂ −=PL f(t) QWf (t) (23) M(s|s)′′′== M† (s|s) M∗ (s|s) (17) ∂t where as predicted by the Onsager symmetry relations. Here † refers to the Hermitian conjugate and ∗ 〈〉〈sss|QW′′ f stands for the complex conjugated. Furthermore, ′′ = δ ′′{f (x ,..., x ′ )W(s′ | s+ 1) due to the fact that f should be normalized, s,s+ 1∫ s 1 s (24) tr(f ) is a constant quantity which is a function of −fs1++ (x 1 ,..., x s1 ) N . It can be inferred from equation (14) that W(s+ 1 | s′ )}dx′′ ...dx dx tr(MX )= 0 which, when the Hermitian 1ss1′ + character of the master matrix is taken into The introduction of the transition matrix account, leads to the following constraints enables us to write equation (19) as a master equation. ∫M(s | s′ )dx1s .....dx = (18) A particularly interesting case corresponds † ′′ ∫M (s | s′ )dx1s .....dx′ = 0 to s1= where equations (23) and (24) reduce to Hence, by using equation (14), the generalized Liouville equation. equation (4), can ∂ o be written as − Lf(x)111 ∂t ′′ ′′ ∂ = {f (x , x )W(x , x | x , x ) (25) −=PL f QMX (19) ∫ 21 2 1 2 1 2 ∂t −f(x,x)21 2 with ′′ ′′ W(x1212 , x | x , x )}dx 122 dx dx 〈〉〈sss|QM′′ X This is not yet a kinetic equation; however, its =δs,s′ + 1∫M(s + 1| s′ ) (20) uncorrelated part can be written as ′′ ∂ Xs1′′ dx .....dx ss1 dx+ o ′′ −=L111 f(x) {f(x)f(x) 1112 ∂t ∫ written in analogy with equation (6). It should be  ′′ mentioned that the presence of the master matrix W(x,x12121112 |x, x )− f(x)f(x) (26) in equation (19) notably simplifies the BBGKY ′′ ′′ hierarchy. In fact, the master matrix introduces a W(x,x1212 |x, x )}dxdxdx 122 relaxation time scale. Hence, our theory is which is the famous Boltzmann equation. equivalent to a relaxation time approach to the Analogous procedure can be followed to obtain study of the BBGKY hierarchy valid when there the kinetic equations for the correlations coming is a broad separation between the hydrodynamic from the components of f of an order higher and microscopic scales. than one. Here, equation (19) can be put into a more Although the generalized Liouville common form by introducing a new equation.(4) is reversible, the generalized Gibbs function W defined through (Meixner, 1957) entropy (7) we propose is a monotonous functional of the distribution vector. We have fWs|sMs|s()′′=− () o, s′ shown that unlike the conventional belief an ′′ external manipulation of the system like coarse +δ()()x11 − x ..... δ x s′ − x s (21) graining procedures is not necessary in order to ψs1()x ,...., x s obtain irreversibility. This is based on the fact that the generalized Gibbs entropy given by where the auxiliary function ψ is not arbitrarily s equation (7), which is formally analogous to the selectable because of the constraints given by relative or Kullback entropy, is a concave equation (18). Instead functional of the distribution vector. In addition ψ=s1()x ,..., x s f o,s1 (x ,..., x s ) the generalized Liouville dynamics introduces a (22) trace preserving linear transformation in the set ′ ′′ ∫W(s | s )dx1s ...dx′ of distribution vectors related to a compressible can be applied. Thus, equation (19) may be flow in Γ -space. Thus, in general, the written in terms of the transition matrix as compressible character of the flow associated to follows the generalized Liouville equation causes a non- vanishing rate of change of the generalized entropy that will never decrease under a trace

162 Int. J. of Thermodynamics, Vol. 8 (No. 4) preserving linear transformation (Schlögl, 1980; i.e. the energy is constant. Moreover, the total Wehrl, 1978). rate of change of the entropy is given by

dSst∂f st −1  4. Non-Equilibrium Stationary States =−ktrBeqst ln()ff  dt∂ t Let us assume that a nonconservative field = ktrBst{Gf (33) ho acts on the system, modifying the Hamilton equations of motion −1 ln()ffeq st} +σ st = 0 ∂HN which is also constant, showing that entropy does qC& ii1No=+(x ,...x )h ∂pi not diverge. Hence, the stationary state coincides (27) with the state of constant and ∂HN pD& ii1No=− + (x ,...x )h entropy. In fact, in general the stationary average ∂q i of any phase function remains constant. where Cij(x ) and Dij(x ) are coupling functions. According to equation (33), the entropy This drive introduces a compressible production (8) reduces to contribution to the flow in the phase space which σ=−ktrGf ln f−1 f is reflected in the Liouville equation st B{ st() eq st } characterizing the phase space flow (34)  nc∂ − 1  = ktrBstf Γ& ln()ff eqst ∂∂ ∂Γ FH,F−=−⋅Γnc F (28) []NNP & ∂∂t ΓN Here in analogy with Brownian motion nc nc nc (Meixner, 1957) , the entropy production given where Γ& N1= (x&& ,...., x N ) , nc by equation (34) describes a diffusion process in x& iijijo= CD (x), (x) h and F is assumed nc () Γ -space for which fstΓ& can be interpreted as to be an smooth function. To equation (28) the diffusion current J and corresponds −1 ln(ffost) =µ k Bst T , µ being the chemical ∂ potential and Tst the stationary temperature in −−PL() G f = QLf (29) such a way that ∂µ∂Γ plays the role of the ∂t  corresponding thermodynamic force for the distribution vector, where XJ conjugated to the current J . Thus, the Gf = ∂∂ΓΓ& ncf and entropy production can be rewritten as nc 〈〉=δ∂∂s|′ PGf |ss, s′ ΓΓ& s f s s . Equation (28) admits a stationary solution satisfying 1 σ=sttr{}JX J (35) Tst ∂ nc []H,FNst=⋅Γ& Nst F (30) >From this equation, the following pheno- P ∂Γ N menological law can be inferred or equivalently 1 JLX= JX J (36) ∂∂nc Tst ΓΓ&&NstN⋅−⋅=ln F 0 (31) ∂∂ΓΓNN where LJX is a phenomenological coefficient. Up to order one with respect to the external The phenomenological law (36) enables us to force, equation (31) becomes express the generalized velocity Γ& nc as []H,lnF =∂ΓΓ& nc ∂ . To this stationary NstNNP nc∂ − 1 distribution function F , there is a corresponding Γ& = kTBff ln (37) st Bst∂Γ () eqst stationary distribution vector f which can be st −1 obtained as a solution of the stationary version of where Bf= ()stTL st JX is a mobility matrix. equation. (29) or computed directly from Fst by Moreover, kTBstB is a diffusion coefficient D . using equation (3), ffst≠ eq . Thus, In this stationary state the rate of change of ∂ −1-1nc ln (ffeq st ) = DΓ& (38) the internal energy UH(x,...x)= ∫ N1 N ∂Γ F(x ,...x )dx ....dx , is given by 1N1 N and equation (34) becomes dUst nc -1 nc 2 = ∫HN1 (x ,.., x N ) σ=stktr B{f stΓ&&D Γ } h o (39) dt (32) ∂ Thus, we can assume that for small deviations ×=F (x ,.., x )dx ...dx 0 ∂t st 1 N 1 N from the stationary state

Int. J. of Thermodynamics, Vol. 8 (No. 4) 163 dS particular cases (Pérez-Madrid, 1994; Mayorga, σ≡) −σ=−ktrf Γ&&ncD -1Γ nc (40) dt B { } 2002), we obtain a linear relation between the irreversible rate of change of the distribution Therefore, following Boltzmann, the probability vector and the conjugated thermodynamic force of the non-equilibrium fluctuations can be which introduces a master matrix which should defined as satisfy the properties required by the Onsager nc ) theory. This result simplifies the BBGKY P(τ Γ& , t)∼στ= exp() (41) hierarchy and enables us to break the hierarchy exp(− tr f Γ&&ncD -1Γ nc by neglecting higher order correlations, thus {}) obtaining the Boltzmann equation for the one- where τ is a macroscopic time. This probability particle distribution function. This last result enables us to compute the average value of any constitutes a test of our theory. Hence, based on phase function ψ(Γ& nc ) , as for example the the MNET, we have founded a way for deriving compressibility factor G . kinetic equations. Finally, by approximating D by its average Our approach enables us to describe macroscopic stationary states. In this sense we value Dtr= fD and using equation (8), we {}st have also shown that the entropy defined in find terms of the distribution vector remains constant nc nc in a non-equilibrium stationary state; hence, tr{}fstΓΓ&& divergences reported in the recent literature are D = (42) avoided and the second law is satisfied. Also a −trLf ln f−1 f {}st() eq st kind of Green-Kubo relation was derived for a transport coefficient that can be defined in the which constitutes a kind of Green-Kubo relation. system relating the power supplied to the system 5. Conclusions with the entropy production with the aid of a stationary temperature. Finally, following the Here, we have proposed a generalization of Boltzmann principle, we derive the expression of the Gibbs entropy postulate based on the the probability for the non-equilibrium BBGKY-hierarchy as the non-equilibrium fluctuations. Therefore, the answer to the entropy for an N-body system. This entropy, question raised at the beginning: can which at equilibrium coincides with the thermodynamics describe systems far from thermodynamic entropy, is a concave functional equilibrium? is obviously yes if one works at the of the distribution vector. In addition, the adequate level of description, which corresponds distribution vector representing the set of all the to the distribution vector. s-particle reduced distribution functions evolves according to the generalized Liouville equation, Nomenclature a compact way of writing the BBGKY hierarchy position vector of the j-th point particle of equations which introduces a trace-preserving q j linear transformation in the set of distribution p j momentum of the j-th point particle vectors. x(,)jjj≡ qp φjk interaction potential between the j-th Our generalized Gibbs entropy is formally and k-th particles analogous to the relative or Kullback entropy. F full space distribution function Thus, due to the compressible character of the f distribution vector flow in Γ -space induced by the generalized Γ 6s-dimensional phase space Liouville equation and to the fact that this s H s-particle Hamiltonian equation is a trace-preserving linear s transformation, the generalized Gibbs entropy fs s-particle reduced distribution function will never decrease. This is crucial to show that L generalized Liouvillian our approach constitutes the most appropriate P projection operator onto the diagonal description leading to irreversibility at the part of the generalized Liouvillian macroscopic level. No coarse graining Q projection operator onto the non- procedures are necessary in order to obtain diagonal part of the generalized irreversibility. Liouvillian S entropy On the other hand, by applying the methods of non-equilibrium thermodynamics in phase Seq equilibrium entropy space, i.e. in the framework of the mesoscopic SB Boltzmann entropy non-equilibrium thermodynamics MNET, a feq equilibrium distribution vector theory which has been proved to be successful in feq, n equilibrium n-particle reduced deriving kinetic transport equations in some distribution function σ entropy production 164 Int. J. of Thermodynamics, Vol. 8 (No. 4) T temperature Bricmont et al., Lectures Notes in Physics, Vol. M mobility matrix 574, p. 39. X thermodynamic force Hoover, W.G., Posch, H.A., Hoover, C.G., 1992, M(s | s′ ) s,s′ component of the mobility Fractal dimension of steady non-equilibrium matrix flows, Chaos, Vol. 2, p. 245. 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