The Annals of Applied Probability

Diffusion limit for many particles in a periodic stochastic acceleration field

Yves Elskens and Etienne Pardoux

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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.

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Ann. Appl. Probab., Volume 20, Number 6 (2010), 2022-2039.

First available in Project Euclid: 19 October 2010

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Primary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03] 60H10: Stochastic ordinary differential equations [See also 34F05] 82C05: Classical dynamic and nonequilibrium statistical mechanics (general) 82D10: Plasmas
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60K40: Other physical applications of random processes

Quasilinear diffusion weak plasma turbulence propagation of chaos wave-particle interaction stochastic acceleration Fokker–Planck equation Hamiltonian chaos


Elskens, Yves; Pardoux, Etienne. Diffusion limit for many particles in a periodic stochastic acceleration field. Ann. Appl. Probab. 20 (2010), no. 6, 2022--2039. doi:10.1214/09-AAP671.

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