Entropy and macroscopic disequilibrium. II. The information theoretic characterization of Markovian relaxation processes

Abstract

This paper considers the characterization of disequilibrium in uniform systems in terms of the time evolution of the macroscopic observables. Using these as the independent variables the procedure of maximal entropy is employed to generate a new, complementary set of independent variables, the thermodynamic forces. An equation for the time rate of change of the macroscopic observables (i.e., for the thermodynamic fluxes) which is valid also for a finite displacement from equilibrium is derived. The kinetic coefficients satisfy reciprocity relations and reduce to the Onsager coefficients in the linear regime. An equation of motion which is linear in the thermodynamic forces throughout the relaxation process is then obtained using the Legendre transform technique. The two possible equations of motion are strictly equivalent and provide (complementary) generating functions for the thermodynamic forces and fluxes. Among all possible population distributions consistent with the magnitude of the macroscopic observables, the ’’most probable’’ distribution is shown to be characterized by having the lowest rate of entropy production throughout the relaxtion process. The Rayleigh–Onsager principle of least dissipation of energy is extended to the nonlinear regime. It is then employed as a variational principle to derive the equations of motion, which inherently satisfy the reciprocity relations and which are valid throughout the relaxation process. An evolution criterion for the system is thereby provided.