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.
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- 1.
Here we assume that c denotes the number of particles per unit volume (or length, in this 1D case), but it could be mass or moles, instead.
- 2.
For a spherical particle of radius a, the Stokes law states that ζ=6πηa, where η is the fluid viscosity (see Sect. F.2.2).
- 3.
Here and in the following we denote by x both the position of the Brownian particle and the local coordinate. When this is confusing, the particle trajectory will be indicated by X(t).
- 4.
Here we see that macroscopic regressions and microscopic fluctuations are governed by the same linear force, indicating that Onsager’s postulate is automatically satisfied.
- 5.
References
Einstein, A.: Brownian Movement. Dover, New York (1956), Chap. 5
Langevin, P.: C. R. Acad. Sci. (Paris) 146, 530 (1908)
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Mauri, R. (2013). Langevin Equation. In: Non-Equilibrium Thermodynamics in Multiphase Flows. Soft and Biological Matter. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5461-4_3
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DOI: https://doi.org/10.1007/978-94-007-5461-4_3
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