Brownian Motion in a Classical Ideal Gas: a Microscopic Approach to Langevin's Equation

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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 (m x=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.