Langevin Equation

The Langevin equation is a stochastic differential equation widely used in nonequilibrium statistical physics.

\[ m \ddot{x} = -\gamma \dot{x} - \nabla U(x) + \eta(t) \]

For overdamped motion, one often writes

(1)\[ \gamma \dot{x} = - \nabla U(x) + \eta(t) \]

Equation (1) is the starting point for many coarse-grained descriptions.

Noise correlation

A common assumption is

\[ \langle \eta(t)\eta(t') \rangle = 2 \gamma k_B T \delta(t-t'). \]