Derivation of the Langevin Equation

Abstract

In this chapter, we derive the Langevin equation from a simple mechanical model for a small system (which we will refer to as a Brownian particle) that is in contact with a thermal reservoir that is at thermodynamic equilibrium at time t = 0. The full dynamics, Brownian particle plus thermal reservoir, are assumed to be Hamiltonian. The derivation proceeds in three steps. First, we derive a closed stochastic integrodifferential equation for the dynamics of the Brownian particle, the generalized Langevin equation (GLE). In the second step, we approximate the GLE by a finite-dimensional Markovian equation in an extended phase space. Finally, we use singular perturbation theory for Markov processes to derive the Langevin equation, under the assumption of rapidly decorrelating noise. This derivation provides a partial justification for the use of stochastic differential equations, in particular the Langevin equation, in the modeling of physical systems.