Diffusion limit for many particles in a periodic stochastic acceleration field

Abstract

The one-dimensional motion of any number ${\mathcal{N}}$ of particles in the field of many independent waves (with strong spatial correlation) is formulated as a second-order system of stochastic differential equations, driven by two Wiener processes. In the limit of vanishing particle mass ${\mathfrak{m}}\to0$, or, equivalently, of large noise intensity, we show that the momenta of all ${\mathcal{N}}$ particles converge weakly to ${\mathcal{N}}$ independent Brownian motions, and this convergence holds even if the noise is periodic. This justifies the usual application of the diffusion equation to a family of particles in a unique stochastic force field. The proof rests on the ergodic properties of the relative velocity of two particles in the scaling limit.