Diffusion limit for many particles in a periodic stochastic acceleration field

TL;DR

This paper demonstrates that in a periodic stochastic acceleration field, the momenta of many particles behave like independent Brownian motions in the limit of vanishing particle mass, justifying the diffusion approximation.

Contribution

It provides a rigorous proof that particle momenta converge to independent Brownian motions under strong stochastic forcing, even with periodic noise.

Findings

Particle momenta converge to independent Brownian motions

Convergence holds for periodic stochastic fields

Supports the diffusion approximation in stochastic acceleration models

Abstract

The one-dimensional motion of any number $\cN$ 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 ${\mathfrak{m}} \to 0$, or equivalently of large noise intensity, we show that the momenta of all $N$ particles converge weakly to $N$ 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.