Elementary derivation of nonlinear transport equations from statistical mechanics

Abstract

Exact closed nonlinear transport equations for a set of macroscopic variablesa are derived from classical statistical mechanics. The derivation involves only simple manipulations of the Liouville equation, and makes no use of projection operators or graphical expansions. It is based on the Chapman-Enskog idea of separating the distribution function into a constrained equilibrium part, obtained from information theory, and a small remainder. The resulting exact transport equations involve time convolutions over the past history of botha(t) anda(t). However, if the variablesa provide a complete macroscopic description, the equations may be simplified. This is accomplished by a systematic expansion procedure of Chapman-Enskog type, in which the small parameter is the natural parameter of slowness relevant to the problem. When carried out to second order, this expansion leads to approximate nonlinear transport equations that are local in time. These equations are valid far from equilibrium. They contain nonlinear (i.e., state-dependent) transport coefficients given by integrals of time correlation functions in the constrained equilibrium ensemble. Earlier results are recovered when the equations are linearized about equilibrium. As an illustrative application of the formalism, an expression is derived for the nonlinear (i.e., velocity-dependent) friction coefficient for a heavy particle in a bath of light particles.