## Abstract

The *Langevin equation* was proposed in 1908 by Paul Langevin (C. R. Acad. Sci. (Paris) 146, 530, 1908) 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 (Sect. 3.1), first we review the original Langevin approach in 1D (Sect. 3.2), then we generalize it to study the evolution of a set of random variables with linear phenomenological forces (Sect. 3.3). The most general case, with non-linear phenomenological forces, represents a non-trivial generalization of the Langevin equation and is studied in Chap. 5 within the framework of the theory of stochastic differential equations.

## References

- 1.Einstein, A.: Brownian Movement. Dover, New York (1956), Chap. 5 zbMATHGoogle Scholar
- 2.Langevin, P.: C. R. Acad. Sci. (Paris)
**146**, 530 (1908) zbMATHGoogle Scholar