Asymptotic of Grazing Collisions and Particle Approximation for the Kac Equation without Cutoff
| Autor: | Nicolas Fournier, David Godinho |
|---|---|
| Přispěvatelé: | Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), UMR8050, Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM) |
| Rok vydání: | 2012 |
| Předmět: |
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
Complex system FOS: Physical sciences Kac equation grazing collisions 01 natural sciences 010104 statistics & probability [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Convergence (routing) FOS: Mathematics Order (group theory) Cutoff Kinetic Theory 0101 mathematics Diffusion (business) Mathematical Physics Mathematics Particle systems Probability (math.PR) 010102 general mathematics Mathematical analysis MSC 82C40 60K35 Statistical and Nonlinear Physics Mathematical Physics (math-ph) Term (time) [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Rate of convergence Particle Mathematics - Probability |
| Zdroj: | Comm Math Phys Comm Math Phys, 2012, 316 (2), pp.207--344 Communications in Mathematical Physics Communications in Mathematical Physics, Springer Verlag, 2012, 316 (2), pp.207-344. ⟨10.1007/s00220-012-1578-9⟩ |
| ISSN: | 1432-0916 0010-3616 |
| DOI: | 10.1007/s00220-012-1578-9 |
| Popis: | The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence. Comment: 37 pages, 6 figures |
| Databáze: | OpenAIRE |
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