Theory of many-body diffusion by the path-probability method: Conversion from ensemble averaging to time averaging

Abstract

The formalism of the path-probability method (PPM) of irreversible statistical mechanics as applied to transport processes is examined in connection with the calculation of the "correlation factor" in many-body diffusion problems. Tracer diffusion in disordered binary alloys is taken as an example for this treatment. The essential characteristic of the PPM is to evaluate the evolution of state with time under nonequilibrium conditions with the use of ensemble averaging at an instant in time. It is pointed out that time averaging rather than ensemble averaging is to be taken in order to evaluate the time correlation of the motion of a small number of particles necessary for the calculation of properties such as the correlation factor in tracer diffusion and the flow of particles in general. The conversion from ensemble averaging to time averaging is made in the "linear range," in which the Onsager equations are valid, without changing the nature of approximation of the treatment. Comparisons of results in these two different averaging methods are thus given. In particular, the percolation sensitivity of tracer diffusion in the time average is discussed.