Derivation of the Langevin Equation from the Microcanonical Ensemble

Abstract

When writing down a Langevin equation for the time evolution of a “system” in contact with a thermal bath, one typically makes the implicit (and often tacit) assumption that the thermal environment is in equilibrium at all times. Here, we take this assumption as a starting point to formulate the problem of a system evolving in contact with a thermal bath from the perspective of the bath, which, since it is in equilibrium, can be described by the microcanonical ensemble. We show that the microcanonical ensemble of the bath, together with the Hamiltonian equations of motion for all the constituents of the bath and system together, give rise to a Langevin equation for the system evolution alone. The friction coefficient turns out to be given in terms of auto-correlation functions of the interaction forces between the bath particles and the system, and the Einstein relation is recovered. Moreover, the connection to the Fokker–Planck equation is established.

Keywords

• Langevin equation
• Brownian motion
• driven diffusion
• microcanonical ensemble
• fluctuation–dissipation relation