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.
Brownian motion as a prototype The original Langevin equation describes Brownian motion, the apparent random movement of a particle in a fluid due to collisions with the molecules of the fluid,
m d v d t = − λ v + η ( t ) , {\displaystyle m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=-\lambda \mathbf {v} +{\boldsymbol {\eta }}(t),}
where v {\displaystyle \mathbf {v} } is the velocity of the particle, λ {\displaystyle \lambda } is its damping coefficient, and m {\displaystyle m} 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 η ( t ) {\displaystyle {\boldsymbol {\eta }}(t)} representing the effect of the collisions with the molecules of the fluid. The force η ( t ) {\displaystyle {\boldsymbol {\eta }}(t)} has a Gaussian probability distribution with correlation function
⟨ η i ( t ) η j ( t ′ ) ⟩ = 2 λ k B T δ i , j δ ( t − t ′ ) , {\displaystyle \langle \eta _{i}(t)\,\eta _{j}(t')\rangle =2\lambda k_{\text{B}}T\delta _{i,j}\delta (t-t'),}
where k B {\displaystyle k_{\text{B}}} is the Boltzmann constant, T {\displaystyle T} is the temperature, and η i ( t ) {\displaystyle \eta _{i}(t)} is the i-th component of the vector η ( t ) {\displaystyle {\boldsymbol {\eta }}(t)} . The δ {\displaystyle \delta } -function form of the time correlation means that the force at a time t {\displaystyle t} 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 δ {\displaystyle \delta } -correlation and the Langevin equation becomes virtually exact. This approximation of δ {\displaystyle \delta } -correlated (white) noise breaks down under certain conditions, especially in the presence of external driving forces that bring the system out of equilibrium, such as shear flow or for dense charged systems under AC external drive. This is also the case of the Langevin equation for relativistic systems, where the δ {\displaystyle \delta } -correlated noise assumption is in tension with the principle of locality. Another common feature of the Langevin equation is the occurrence of the damping coefficient λ {\displaystyle \lambda } in the correlation function of the random force, which in an equilibrium system is an expression of the Einstein relation.
Mathematical aspects A strictly δ {\displaystyle \delta } -correlated fluctuating force η ( t ) {\displaystyle {\boldsymbol {\eta }}(t)} is not a function in the usual mathematical sense, and even the derivative d v / d t {\displaystyle \mathrm {d} \mathbf {v} /\mathrm {d} t} is not defined in this limit. This problem disappears when the Langevin equation is written in integral form
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