Langevin Equation

  • Roberto Mauri
Part of the Soft and Biological Matter book series (SOBIMA)


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.


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

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Roberto Mauri
    • 1
  1. 1.Department of Chemical Engineering, Industrial Chemistry and Material ScienceUniversity of PisaPisaItaly

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