Transport, Collective Motion, and Brownian Motion

Abstract

A theory of many-particle systems is developed to formulate transport, collective motion, and Brownian motion from a unified, statistical-mechanical point of view. This is done by, first, rewriting the equation of motion in a generalized form of the Langevin equation in the stochastic theory of Brownian motion and then, either studying the average evolution of a non-equilibrium system or calculating the linear response function to a mechanical perturbation. (1) An expression is obtained for the damping function φ(t), the real part of whose Laplace transform gives the damping constnat of collective motion. (2) A general equation of motion for a set of dynamical variables At) is derived, which takes the form where is a frequency matrix determining the collective oscillation of A(t). The quantity f(t) consists of those terms which are either non-linear in A(s), t ≧s ≧0, or dependent on the other degrees-of-freedom explicitly, and its time-correlation function is connected with the damping function φ(t) by (f(t1), f(t2)*) = φ(t1 − t2)·(A, A*). (3) An expression is obtained for the linear after-effect function to thermal disturbances such as temperature gradient and strain tensor. Both the conjugate fluxes and the time dependence differ from those of the mechanical response function. The conjugate fluxes are random parts of the fluxes of the state variables, thus depending on temperature. (4) The difference in the time dependence arises from a special property of the time evolution of f(t) and ensures that the damping function and the thermal after-effect function are determined by the microscopic processes in strong contrast to the mechanical response function. The difficulty of the plateau value problem in the previous theories of Brownian motion and transport coefficients is thus removed. (5) The theory is illustrated by dealing with the motion of inhomogeneous magnetization in ferromagnets and the Brownian motion of the collective coordinates of fluids. (6) Explicit expressions are derived for the thermal after-effect functions and the transport coefficients of multi-component systems.