Molecular theory of Brownian motion

A molecular theory of Brownian motion is presented which starts from the Hamiltonian equations of motion for a system consisting of one heavy particle of mass M and N light particles of mass m (bath). A projection operator which averages over the bath variables is utilized. Expansions in powers of λ=(m/M)12 are obtained for the equation of motion for the heavy particle. It is rigorously demonstrated that the Langevin equation is valid up to O(λ2) for all times. The magnitude of the momentum of the heavy particle is restricted to be of order λ-1. It is also shown using linear response theory that the friction constant appearing in the Langevin equation is identical to the friction coefficient which characterizes the drag on a macroscopic body moving with a prescribed velocity.