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

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 (mxx = 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) oc -x x and (2) the correlation function of the fluctuating force rl = F - (F) is related to the strength of the average force