Time Evolution in Macroscopic Systems. II. The Entropy

Abstract

Abstract. The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined. IntroductionThe empirical statement of the Second Law of thermodynamics by Clausius (1865) iswhere S is the total entropy of everything taking part in the process under consideration, and the entropy for a single closed system is defined to within an additive constant bywhere C(T ) is a heat capacity. The integral in (2) is to be taken over a reversible path, a locus of thermal equilibrium states connecting the macroscopic states 1 and 2, which is necessary because the absolute temperature T is not defined for other than equilibrium states; dQ represents the net thermal energy added to or taken from the system in the process. As a consequence, entropy is defined in classical thermodynamics only for states of thermal equilibrium. Equation (1) states that in the change from one state of thermal equilibrium to another along a reversible path the total entropy of all bodies involved cannot decrease; if it increases, the process is irreversible. That is, the integral provides a lower bound on the change in entropy. This phenomenological entropy is to be found from experimental measurements with calorimeters and thermometers, so that by construction it is a function only of the macroscopic parameters defining the macroscopic state of a system, S(V, T, N ), say, where V and N are the system volume and particle number, respectively. It makes no reference to microscopic variables or probabilities, nor can any explicit time dependence be justified in the context of classical thermodynamics. Equation (1) is a statement of macroscopic phenomenology that cannot be proved true solely as a consequence of the microscopic dynamical laws of physics, as appreciated already by 1 Boltzmann (1895): "The Second Law can never be proved mathematically by means of the equations of dynamics alone." (Nor, for that matter, can the First Law!) Theoretical definitions of entropy were first given in the context of statistical mechanics by Boltzmann and Gibbs, and these efforts culminated in the formal definition of equilibrium ultimately given by Gibbs (1902) in terms of his variational principle. In Part I (Grandy, 2003, preceding paper † ) we observed that the latter is a special case of a more general principle of maximum information entropy (PME), and in equilibrium it is that maximum subject to macroscopic constraints that is identified with the experimental entropy of (2). One of the dominant concerns in statistical mechanic...

Predictive Statistical Mechanics and Macroscopic Time Evolution: Hydrodynamics and Entropy Production

“…Sharp, definite predictions of macroscopic behavior are possible only because certain behavior is characteristic for each of the overwhelming majority of microstates compatible with data, and therefore, this is just the behavior that is reproduced experimentally under those constraints. That was confirmed also by the conclusions reached in the framework of MaxEnt formalism by Grandy [12][13][14][15][16].…”

“…The dynamical variables J(r), J Pα (r) and J h (r) are the flux densities of conserved quantities whose densities are n(r), P α (r) i h(r). Equations (14) are standard expressions; explicit derivations of the flux densities are found in the literature [16,17,23,25].…”

“…are equivalent to the local macroscopic conservation laws (18). By using the divergence theorem in expressions (19) along with the vanishing of the contribution of the boundary of the set M , in the way described in the derivation of (15), and then using the right hand side of (14), we obtain the equivalent expressions…”

“…are satisfied then the local microscopic conservation laws are valid in the form which is identical to (14). The condition (24) is more restrictive for H ni (x, p, t) than (23); if the condition (24) is satisfied then also the condition (23) is satisfied.…”

Isotropic submanifolds generated by the Maximum Entropy Principle and Onsager reciprocity relations

“…In this section, we introduce briefly a model for a statistical description of a classical continuum system which is derived from the one that was introduced by Jaynes ( [12,13], see also [6][7][8][9][10]) to give a complete formulation (based on the MEP) of non-equilibrium statistical mechanics for a quantum system. The aim of this paper, however, is to investigate some geometric features of this probabilistic model in its general form, without reference to specific applications.…”