Langevin-Vladimirsky approach to Brownian motion with memory

Abstract

A number of random processes in various fields of science is described by phenomenological equations of motion containing stochastic forces, the best known example being the Langevin equation (LE) for the Brownian motion (BM) of particles. Long ago Vladimirsky (1942) in a little known paper proposed a simple method for solving such equations. The method, based on the classical Gibbs statistics, consists in converting the stochastic LE into a deterministic equation for the mean square displacement of the particle, and is applicable to linear equations with any kind of memory in the dynamics of the system. This approach can be effectively used in solving many of the problems currently considered in the literature. We apply it to the description of the BM when the noise driving the particle is exponentially correlated in time. The problem of the hydrodynamic BM of a charged particle in an external magnetic field is also solved.