Diffusion limit for many particles in a periodic stochastic acceleration field

Details Diffusion limit for many particles in a periodic stochastic acceleration field Diffusion limit for many particles in a periodic stochastic acceleration field

2008-11-05

arxiv.org/abs/0811.0801v1

Mathematics

Probability

20 pages

Scientific paper

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.

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