Langevin equation

In statistical physics, a Langevin equation (Paul Langevin, 1908) is a describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation.

The original Langevin equation describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,

The degree of freedom of interest here is the position ${\displaystyle \mathbf {x} }$ of the particle, ${\displaystyle m}$ denotes the particle's 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 noise term ${\displaystyle {\boldsymbol {\eta }}\left(t\right)}$ (the name given in physical contexts to terms in stochastic differential equations which are ) representing the effect of the collisions with the molecules of the fluid. The force ${\displaystyle {\boldsymbol {\eta }}\left(t\right)}$ has a Gaussian probability distribution with correlation function

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