Diffusion limit for many particles in a periodic stochastic acceleration field

Elskens, Yves ; Pardoux, Etienne

Ann. Appl. Probab., Tome 20 (2010) no. 1, p. 2022-2039 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2010-12-15
Classification:  Quasilinear diffusion,  weak plasma turbulence,  propagation of chaos,  wave-particle interaction,  stochastic acceleration,  Fokker–Planck equation,  Hamiltonian chaos,  34F05,  60H10,  82C05,  82D10,  60J70,  60K40