Antireciprocity and Memory in the Statistical Approach to Irreversible Thermodynamics

The macroscopic state of a large, closed system is specified by the ensemble averages α_j of a set of even dynamical variables A_j, together with the averages v_j of the quantities iLA_j, obtained by operating on the A_j with the self‐adjoint Liouville operator L. Phenomenological equations for the time rates of change, ${\mathrm{\alpha ̇}}_j$ and v̇_j, 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 $v_j{\mathrm{=\alpha ̇}}_j$ , 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 α_j, provided the equations are still required to be linear in the v_j. Furthermore, Onsager's equations are obtained while allowing the phenomenological matrices to be functions of the variables α_j, and not merely of constants of the motion. This serves to generalize and extend Zwanzig's earlier treatment.