Langevin Methods

The lecture outlines the most important mathematical facts about stochastic processes which are described by a Langevin equation (stochastic differential equation, SDE) or (equivalently) a Fokker-Planck equation, (FPE) comprising both drift and diffusion terms. The importance of the short-time behavior of the moments (mean displacement, mean square displacement) is stressed, and the problem of interpretation of SDEs (Ito vs. Stratonovich) is explained. The simplest integration scheme (Euler) is a straightforward consequence of this theory. For the simulation of thermal systems, drift and diffusion must balance each other in a well-defined way which fixes the temperature (fluctuation-dissipation theorem). The application of the general framework is then discussed for various methods commonly used in classical statistical physics (Brownian dynamics, stochastic dynamics, dissipative particle dynamics, force-based Monte Carlo)