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

TL;DR

This paper derives the Langevin equation and fluctuation dissipation theorem for a heavy particle in an ideal gas, using a microscopic collision-based approach, clarifying the connection between microscopic dynamics and macroscopic stochastic behavior.

Contribution

It provides a microscopic derivation of the Langevin equation and fluctuation dissipation theorem for a particle in an ideal gas, emphasizing collision kinematics.

Findings

Average force proportional to velocity damping

Force fluctuation correlation related to damping force

Microscopic collision analysis supports Langevin dynamics

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