Langevin equation

In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation 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. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid. 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, Here, is the velocity of the particle, is its damping coefficient, and 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 noise term representing the effect of the collisions with the molecules of the fluid. The force has a Gaussian probability distribution with correlation function where is Boltzmann's constant, is the temperature and is the i-th component of the vector . The -function form of the time correlation means that the force at a time 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 -correlation and the Langevin equation becomes virtually exact. Another common feature of the Langevin equation is the occurrence of the damping coefficient in the correlation function of the random force, which in an equilibrium system is an expression of the Einstein relation. A strictly -correlated fluctuating force is not a function in the usual mathematical sense and even the derivative is not defined in this limit.