Editing Langevin equation (section)

== Brownian motion as a prototype ==
The original Langevin equation<ref>{{cite journal | title = Sur la théorie du mouvement brownien [On the Theory of Brownian Motion] | journal = C. R. Acad. Sci. Paris | year = 1908 | first = P. | last = Langevin | volume = 146 | pages = 530–533}}</ref><ref>{{cite journal | last1=Lemons | first1=Don S. | last2=Gythiel | first2=Anthony | title=Paul Langevin's 1908 paper "On the Theory of Brownian Motion" &#91;"Sur la théorie du mouvement brownien," C. R. Acad. Sci. (Paris) 146, 530–533 (1908)&#93; | journal=American Journal of Physics | publisher=American Association of Physics Teachers (AAPT) | volume=65 | issue=11 | year=1997 | issn=0002-9505 | doi=10.1119/1.18725 | pages=1079–1081| bibcode=1997AmJPh..65.1079L }}</ref> describes [[Brownian motion]], the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,
<math display="block">m\frac{\mathrm d\mathbf{v}}{\mathrm d t}=-\lambda \mathbf{v}+\boldsymbol{\eta}\left( t\right).</math>

Here, <math>\mathbf{v}</math> is the velocity of the particle, <math>\lambda</math> is its damping coefficient, and <math>m</math> is its mass. The force acting on the particle is written as a sum of a viscous force proportional to the particle's velocity ([[Stokes' law]]), and a [[Wiener process|''noise term'']] <math>\boldsymbol{\eta}\left( t\right)</math> representing the effect of the collisions with the molecules of the fluid. The force <math>\boldsymbol{\eta}\left( t\right)</math> has a [[Gaussian distribution|Gaussian probability distribution]] with correlation function
<math display="block">\left\langle \eta_{i}\left( t\right)\eta_{j}\left( t'\right) \right\rangle =2\lambda k_\text{B}T\delta _{i,j} \delta \left(t-t'\right) ,</math>
where <math>k_\text{B}</math> is the [[Boltzmann constant]], <math>T</math> is the temperature and <math>\eta_i\left( t\right)</math> is the i-th component of the vector <math>\boldsymbol{\eta}\left( t\right)</math>. The [[Dirac delta function|<math>\delta</math>-function]] form of the time correlation means that the force at a time <math>t</math> is uncorrelated with the force at any other time.  This is an approximation: the actual random force has a nonzero correlation time corresponding to the collision time of the molecules. However, the Langevin equation is used to describe the motion of a "macroscopic" particle at a much longer time scale, and in this limit the <math>\delta</math>-correlation and the Langevin equation becomes virtually exact.

Another common feature of the Langevin equation is the occurrence of the damping coefficient <math>\lambda</math> in the correlation function of the random force, which in an equilibrium system is an expression of the [[Einstein relation (kinetic theory)|Einstein relation]].