1963•Journal of Mathematical PhysicsRequires access

Modified Langevin Equation for the Description of Brownian Motion

Hans Christian Andersen, Irwin Oppenheim

Abstract

A modified Langevin equation for the description of Brownian motion is shown to give results equivalent to those of the Langevin equation in some physical situations. The Smoluchowski equation for the probability density of the displacement of a Brownian particle is derived from the modified Langevin equation with the aid of assumptions weaker than those needed to derive the Smoluchowski equation from the ordinary Langevin equation. Sufficient conditions for the applicability of the modified Langevin equation to the calculation of configuration-space averages are obtained. Then the modified Langevin equation is applied to three simple systems, and the results are compared with those of the Langevin equation.

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Key concepts: Langevin equation, Brownian dynamics, Brownian motion, Smoluchowski coagulation equation, Stochastic differential equation, Langevin dynamics, Fokker–Planck equation, Statistical physics