Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena

Abstract

Two principles of a statistical mechanics of time-dependent phenomena are proposed and argued for. The first states that the proper mathematical object to describe the physical situation is the stationary random process specified by the ensemble of time series ai(Xt)i=1···s and the distribution ρ(X). The set phase functions ai(X)i=1···s represent the set of grossly observable features of the system. Xt is the image of the phase X after time t. ρ(X) is a stationary distribution. The second principle is concerned with the very common case in which the phenomenological equations are of the first order in time and states that in this case the random process in question is a Markoff process. A Fokker-Planck equation is derived for the process, and an entropy is defined and is shown always to increase. Phenomenological equations are derived as a first approximation to the Markoff process. These involve a certain matrix ξij which is shown to satisfy symmetry relations which are a generalization of Onsager's. The theory is applied to the uniformization of the direction of momentum of a particle on a two-dimensional torus subject to a small perturbing potential.