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

Rangan Lahiri; Arvind; Anirban Sain

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

Rangan Lahiri·Arvind·Anirban Sain

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Abstract

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 ($m{\ddot x}=F$) 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 $ \propto -{\dot x}$ and (2) the correlation function of the fluctuating force $η= F-< F>$ is related to the strength of the average force.

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Rangan Lahiri, Arvind, Anirban Sain. 2002-07-10. Brownian Motion in a Classical Ideal Gas: a Microscopic Approach to Langevin's Equation. https://arxiv.org/abs/cond-mat/0207246

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