Statistical Dynamics of Simple Cubic Lattices. Model for the Study of Brownian Motion. II

Abstract

New results concerning the statistical dynamics of a heavy particle in an n-dimensional (nD) cubic lattice are presented. It is demonstrated that this model exhibits many properties which are familiar in the phenomenological theory of Brownian motion. In a well-defined sense, the random thermal motions of a heavy particle in a 1D lattice and a 3D lattice are accurately described by Kramers' equation for a free particle and a harmonically bound particle, respectively. A related, but not independent, result is that the velocity v(t) and position u(t) of a heavy particle in a 1D lattice and a 3D lattice constitute two-dimensional stationary Gaussian Markoff processes. It is definitely established that in the case of a 2D lattice the stationary Gaussian process {v(t),u(t)} is non-Markoffian. In the course of the analysis, several interesting connections between solutions of the discrete lattice equations of motion and solutions of the corresponding continuum equation of motion (the nD wave equation) are uncovered.