Langevin equation

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):

$$ \tag{a1 } \frac{dv ( t) }{dt} = - \gamma v ( t) + L ( t). $$

Here $ v ( t) $ denotes the velocity at time $ t $ along one of the coordinate axes of the Brownian particle, $ \gamma > 0 $ is a friction coefficient due to the viscosity of the liquid, and $ L ( t) $ 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

$$ \mathbf E ( L ( t)) = 0 \ \ \textrm{ and } \ \ \mathbf E ( L ( t) L ( s)) = D \cdot \delta ( t - s). $$

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

$$ \tag{a2 } { \frac \partial {\partial t } } \rho _ {t} ( v) = \ \gamma \frac \partial {\partial v } ( v \rho _ {t} ( v)) + { \frac{1}{2} } D ^ {2} \frac{\partial ^ {2} }{\partial v ^ {2} } \rho _ {t} ( v). $$

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 $ \gamma $ and $ D $, namely $ D = 2 \gamma kT/M $( where $ T $ is the temperature and $ M $ the particle's mass), gave the first estimate of Boltzmann's constant $ k $, and thereby of Avogadro's number.

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

$$ dv ( t) = - \gamma u ( t) dt + D dw ( t), $$

where $ w ( t) $ 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