Langevin Equation - Brownian Motion As A Prototype

Brownian Motion As A Prototype

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 of the particle, 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 η(t) (the name given in physical contexts to terms in stochastic differential equations which are stochastic processes) representing the effect of the collisions with the molecules of the fluid. The force η(t) has a Gaussian probability distribution with correlation function

where k_B is Boltzmann's constant and T is the temperature. The δ-function of the time difference is an approximation, the actual random force has a finite 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 δ-correlation and the Langevin equation become exact. Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient λ in the correlation function of the random force.