Langevin original approach and Ornstein–Uhlenbeck-type processes

Abstract

This paper applies Langevin idea to describe the Brownian motion of a particle characterized by an Ornstein–Uhlenbeck-type process. The original and clever method proposed by Langevin is based on Newton’s second law plus a fluctuating force whose solution for the mean square displacement consists in separating the fluctuating force from his equation to obtain a deterministic equation for the relevant physical variable. In this work the Langevin original idea is applied to calculate the mean square velocity for a field free particle case; then it is extended for a charged particle in a constant magnetic field. In a similar way, the strategy is also applied to calculate the mean square displacement for a Brownian harmonic oscillator in the overdamped regime, and also when a magnetic field is present. In particular, it is shown in the field free case that Langevin’s original strategy leads to the same results as those obtained using the statistical properties of a Gaussian white noise. All the theoretical results are compared with both the numerical simulation of the Langevin equation, and numerical solution of the corresponding deterministic differential equations.