Brownian motion in a classical ideal gas: A microscopic approach to Langevin's equation

摘要

We present an insightful 'derivation' of the Langevin equation and the fluctuation dissipation theorem in the specific context of a heavier particle moving through an ideal gas of much lighter particles. The Newton's law of motion (mxe = F) for the heavy particle reduces to a Langevin equation (valid on a coarser time-scale) with the assumption that the lighter gas particles follow a Boltzmann velocity distribution. Starting from the kinematics of the random collisions we show that (1) the average force < F > ∝ - x and (2) the correlation function of the fluctuating force η = F — < F > is related to the strength of the average force.