Fluctuation and dissipation in Brownian motion

Abstract

An analysis of Brownian motion based upon a ‘‘Langevin equation’’ form of Newton’s second law provides a physically motivated introduction to the theory of continuous Markov processes, which in turn illuminates the subtle mathematical underpinnings of the Langevin equation. But the Langevin approach to Brownian motion requires one to assume that the collisional forces of the bath molecules on the Brownian particle artfully resolve themselves into a ‘‘dissipative drag’’ component and a ‘‘zero-mean fluctuating’’ component. A physically more plausible approach is provided by a simple discrete-state jump Markov process that models in a highly idealized way the immediate effects of individual molecular collisions on the velocity of the Brownian particle. The predictions of this jump Markov process model in the continuum limit are found to precisely duplicate the predictions of the Langevin equation, thereby validating the critical two-force assumption of the Langevin approach.