Antireciprocity and Memory in the Statistical Approach to Irreversible Thermodynamics

Abstract

The macroscopic state of a large, closed system is specified by the ensemble averages αj of a set of even dynamical variables Aj, together with the averages vj of the quantities iLAj, obtained by operating on the Aj with the self-adjoint Liouville operator L. Phenomenological equations for the time rates of change, α̇j and v̇j, are derived by a technique due to Zwanzig, in which one operates on Liouville's equation with a projection operator which projects out the part which is ``relevant'' to the phenomenological description employed. These phenomenological equations are shown to exhibit the Onsager—Casimir reciprocity relations, including antisymmetry relations whose derivation is found to require a slight modification of Zwanzig's mathematical assumptions. Since vj=α̇j, these equations also show that irreversible thermodynamics can be extended to the case where second-order time derivatives appear representing memory effects, as well as nonlinear terms in the αj, provided the equations are still required to be linear in the vj. Furthermore, Onsager's equations are obtained while allowing the phenomenological matrices to be functions of the variables αj, and not merely of constants of the motion. This serves to generalize and extend Zwanzig's earlier treatment.