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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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.

Article information

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

Dates
First available in Project Euclid: 19 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1287494553

Digital Object Identifier
doi:10.1214/09-AAP671

Mathematical Reviews number (MathSciNet)
MR2759727

Zentralblatt MATH identifier
1215.82048

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1287494553


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