# Langevin equation

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In 1908 P. Langevin [a1] proposed the following equation to describe the natural phenomenon of Brownian motion (the irregular vibrations of small dust particles suspended in a liquid):

 (a1)

Here denotes the velocity at time along one of the coordinate axes of the Brownian particle, is a friction coefficient due to the viscosity of the liquid, and is a postulated "Langevin forceLangevin force" , standing for the pressure fluctuations due to thermal motion of the molecules comprising the liquid. This Langevin force was supposed to have the properties

The Langevin equation (a1) leads to the following diffusion (or "Fokker–Planck" ) equation (cf. Diffusion equation) for the probability density on the velocity axis:

 (a2)

The equations (a1) and (a2) provided a conceptual and quantitative improvement on the description of the phenomenon of Brownian motion given by A. Einstein in 1905. The quantitative understanding of Brownian motion played a large role in the acceptance of the theory of molecules by the scientific community. The numerical relation between the two observable constants and , namely (where is the temperature and the particle's mass), gave the first estimate of Boltzmann's constant , and thereby of Avogadro's number.

The Langevin equation may be considered as the first stochastic differential equation. Today it would be written as

where is the Wiener process (confusingly called "Brownian motion" as well). The solution of the Langevin equation is a Markov process, first described by G.E. Uhlenbeck and L.S. Ornstein in 1930 [a2] (cf. also Ornstein–Uhlenbeck process).

The Langevin equation is a heuristic equation. The program to give it a solid foundation in Hamiltonian mechanics has not yet fully been carried through. Considerable progress was made by G.W. Ford, M. Kac and P. Mazur [a3], who showed that the process of Uhlenbeck and Ornstein can be realized by coupling the Brownian particle in a specific way to an infinite number of harmonic oscillators put in a state of thermal equilibrium.

In more recent years, quantum mechanical versions of the Langevin equation have been considered. They can be subdivided into two classes: those which yield Markov processes and those which satisfy a condition of thermal equilibrium. The former are known as "quantum stochastic differential equations" [a4], the latter are named "quantum Langevin equations" [a5].

#### References

 [a1] P. Langevin, "Sur la théorie de mouvement Brownien" C.R. Acad. Sci. Paris , 146 (1908) pp. 530–533 [a2] G.E. Uhlenbeck, L.S. Ornstein, "On the theory of Brownian motion" Phys. Rev. , 36 (1930) pp. 823–841 [a3] G.W. Ford, M. Kac, P. Mazur, "Statistical mechanics of assemblies of coupled oscillators" J. Math. Phys. , 6 (1965) pp. 504–515 [a4] C. Barnett, R.F. Streater, I.F. Wilde, "Quasi-free quantum stochastic integrals for the CAR and CCR" J. Funct. Anal. , 52 (1983) pp. 19–47 [a5] R.L. Hudson, K.R. Parthasarathy, "Quantum Itô's formula and stochastic evolutions" Commun. Math. Phys. , 93 (1984) pp. 301–323
How to Cite This Entry:
Langevin equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Langevin_equation&oldid=16399
This article was adapted from an original article by H. Maassen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article