Reciprocal Relations in Irreversible Processes. II.

Abstract

A general reciprocal relation, applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility. In the derivation, certain average products of fluctuations are considered. As a consequence of the general relation $S=k logW$ between entropy and probability, different (coupled) irreversible processes must be compared in terms of entropy changes. If the displacement from thermodynamic equilibrium is described by a set of variables ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$, and the relations between the rates ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ and the "forces" $\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{1}},\ensuremath{\cdots},\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{n}}$ are linear, there exists a quadratic dissipation-function, $2\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}{\ensuremath{\rho}}_{j}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{\mathrm{ij}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{i}=\frac{\mathrm{dS}}{\mathrm{dt}}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}(\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{j}}){\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{j}$ (denoting definition by $\ensuremath{\equiv}$). The symmetry conditions demanded by microscopic reversibility are equivalent to the variation-principle $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{-}\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})=\mathrm{maximum},$ which determines ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ for prescribed ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$. The dissipation-function has a statistical significance similar to that of the entropy. External magnetic fields, and also Coriolis forces, destroy the symmetry in past and future; reciprocal relations involving reversal of the field are formulated.