Langevin Equation

Abstract

According to Langevin, a particle performing a random movement, which we call Brownian motion, is subject to two forces. One dissipative, which we assume to be proportional to its velocity, and another of random character due to the impact of the particle with the molecules of the medium. Considering the simple case of a one-dimensional motion along a straight line, the equation of motion for a particle of mass m is given by

$$\displaystyle{ m\frac{\mathit{dv}} {\mathit{dt}} = -\alpha v + F_{\mathrm{a}}(t), }$$

(3.1)

where

$$\displaystyle{ v = \frac{\mathit{dx}} {\mathit{dt}} }$$

(3.2)

is the velocity and x the position of the particle.