Langevin Equation

The Langevin equation was proposed in 1908 by Paul Langevin, to describe Brownian motion, that is the apparently random movement of a particle immersed in a fluid, due to its collisions with the much smaller fluid molecules. As the Reynolds number of this movement is very low, the drag force is proportional to the particle velocity; this, so called, Stokes law represents a particular case of the linear phenomenological relations that are assumed to hold in irreversible thermodynamics. In this Chapter, after a brief description of Brownian motion, first we review the original Langevin approach in 1D (Section, then we generalize it to study the evolution of a set of random variables with linear phenomenological forces. The most general case, with non-linear phenomenological forces, represents a non-trivial generalization of the Langevin equation and must be studied within the framework of the theory of stochastic differential equations.