Fluctuations and nonlinear irreversible processes. II

Abstract

This paper forms the second part of a study which reexamines the relationship between fluctuations and nonlinear irreversible processes. The scope of the previous paper is generalized to include macroscopic variables which transform odd under time reversal. The fluxes of some of the variables may be purely reversible so that the diffusion matrix may be singular. The deterministic equations for nonlinear irreversible processes can again be derived from a minimum principle. The fluctuations of the macroscopic variables are treated on the basis of a Fokker-Planck equation which has the form derived from statistical mechanics by one of us. The conditional probability of the fluctuations is constructed as a path integral. The connection between the deterministic and the stochastic descriptions of the macroscopic dynamics is formulated in a covariant way, independent of the frame of coordinates. For that purpose, a metric tensor in the space of state variables is introduced. The form of the metric tensor is particularly simple in frames where the macroscopic variables are sums of molecular variables.