Onsager Relations in Statistical Dynamics

Abstract

We study two models of Brownian motion in a thermal fluid, one of them new, undergoing diffusion coupled with heat transport. Each model is described by a coupled system of two parabolic equations for the density f(x, t) and the temperature field Θ(x, t). These equations are obtained by taking the continuum limit of discrete micoscopic models, with the help of MAPLE. We show that the generalised forces {Xα} in the sense of Onsager can be uniquely defined, and that the fluxes {jα} are ambiguous up to curls, but can be chosen so that the (nonlinear) dynamics possesses Onsager symmetry. This answers one of Truesdell's criticisms of Onsager's theory. We show that Onsager symmetry is violated if some other choices of the fluxes are made. In these cases the microscopic dynamics, when written in Kossakowski-Lindblad form, has a Hamiltonian term. Thus Onsager symmetry follows rather from the absence of the Hamiltonian term and does not follow from time- reversal invariance per se.