Gibbs Entropy and Irreversibility

This contribution is dedicated to dilucidating the role of the Gibbs entropy in the discussion of the emergence of irreversibility in the macroscopic world from the microscopic level. By using an extension of the Onsager theory to the phase space we obtain a generalization of the Liouville equation describing the evolution of the distribution vector in the form of a master equation. This formalism leads in a natural way to the breaking of the BBGKY hierarchy. As a particular case we derive the Boltzmann equation. Typeset using REVTEX 1 PACS numbers: 05.20.-y, 05.20.Dd, 05.70.Ln Introduction.—According to the mechanicistic interpretation of the physical world, the basic laws of nature are deterministic and time reversible. However, at the macroscopic level we observe irreversible processes related to energy degradation, which generate entropy. How do we reconcile the ‘spontaneous production of entropy’ with the time reversibility of the microscopic equations of motion?. At the end of nineteenth century Boltzmann tried to answer this question from a probabilistic point of view. According to him, entropy is a measure of the lack of knowledge of the precise state of matter, and can be defined as a function of the probability of a given state of matter, i.e. it is a function of the microstate. All systems in their irreversible evolution tend to a state of maximum probability or maximum entropy -the state of equilibrium-. In contrast to the Boltzmann entropy, the Gibbs entropy is not a function of the individual microstate but a function of the probability distribution in a statistical ensemble of systems, both coinciding at equilibrium. As a consequence of the incompressible character of the flow of points representing the natural evolution of the statistical ensemble in phase space , the Gibbs entropy is a constant of motion. Thus, it has been argued that the relevant entropy for understanding thermodynamic irreversibility is the Boltzmann entropy and not the Gibbs entropy [1], [2], [3]. But in our opinion, this is not the end of the story. It is our contention here to show that depending on the representation we use to describe the state of the system, the Gibbs entropy is a good definition of the nonequilibrium entropy which increases in the approach to equilibrium and is compatible with the Boltzmann’s account for irreversibility . To accomplish that goal, we will use the methods of mesoscopic nonequilibrium thermodynamics MNET [4], [5] in addition to the generalized Liouville description of dynamical systems in terms of the distribution vector. The paper is organized as follows. In Section II we introduce the representation of the state of the isolated system in terms of the hierarchy of reduced distribution functions. Sec2 tion III, is devoted to developing the thermodynamic analysis, and to deriving the entropy production. Here, we draw kinetic equations as one of the consequences, in particular the Boltzmann equation. In section IV, we stress our main conclusions. Distribution vector dynamics.—Let us think of a dynamical system having N interacting degrees of freedom (q1, ......, q2, p1, ......, pN), where qi and pi are the generalized coordinate and conjugated momentum corresponding to the i-th degree of freedom. Let H( { q , p } ) be the Hamiltonian of the system, given by H = N ∑ j=1 { p2j 2m + φ(qj) }