Langevin equation

Remarks

In physics notation, $\beta = 1 / (k_B T)$ for the temperature $T$ and the Boltzmann constant $k_B$. The equation is often written $m \ddot{X} = -\nabla U(X) - \gamma \dot{X} + \xi$, where $\xi$ is white noise. Example: for $U = 0$ and $d = 1$, the velocity solves $dV = -(\gamma/m) V \, dt + (\sqrt{2\gamma/\beta}/m) \, dB$. This is an Ornstein–Uhlenbeck process, and its stationary law is normal with mean $0$ and variance $1/(\beta m)$. Non-example: the equation $m \, dV = -\gamma V \, dt$ lacks the random force, so its solutions decay to $0$ and never reach thermal equilibrium. The overdamped Langevin equation $dX_t = -\nabla U(X_t) \, dt + \sqrt{2/\beta} \, dB_t$ is the limit of large friction after a change of time scale. When $e^{-\beta U}$ is integrable, the overdamped equation has a stationary law with density proportional to $e^{-\beta U(x)}$. Under the same condition, the full equation has a stationary law with density proportional to $e^{-\beta (U(x) + m \abs{v}^2 / 2)}$ in $(x, v)$.