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In 1908 P. Langevin [[#References|[a1]]] proposed the following equation to describe the natural phenomenon of [[Brownian motion|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|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|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|Wiener process]] (confusingly called  "Brownian motion"  as well). The solution of the Langevin equation is a [[Markov process|Markov process]], first described by G.E. Uhlenbeck and L.S. Ornstein in 1930 [[#References|[a2]]] (cf. also [[Ornstein–Uhlenbeck process|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 [[#References|[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"  [[#References|[a4]]], the latter are named  "quantum Langevin equations"  [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Langevin,   "Sur la théorie de mouvement Brownien"  ''C.R. Acad. Sci. Paris'' , '''146'''  (1908)  pp. 530–533</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.E. Uhlenbeck,   L.S. Ornstein,   "On the theory of Brownian motion"  ''Phys. Rev.'' , '''36'''  (1930)  pp. 823–841</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.W. Ford,   M. Kac,   P. Mazur,   "Statistical mechanics of assemblies of coupled oscillators"  ''J. Math. Phys.'' , '''6'''  (1965)  pp. 504–515</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.L. Hudson,   K.R. Parthasarathy,   "Quantum Itô's formula and stochastic evolutions"  ''Commun. Math. Phys.'' , '''93'''  (1984)  pp. 301–323</TD></TR></table>