Int. J. of Thermodynamics ISSN 1301-9724 Vol. 8 (No. 4), pp. 159-166, December 2005 Generalized Gibbs Entropy, 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 Boltzmann equation 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 statistical mechanics, 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 entropies 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.