Uniform Propagation of Chaos for Kac’s 1D Particle System

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• 作者：Roberto Cortez
• 关键词：Kinetic theory ; Kac particle system ; Propagation of chaos
• 刊名：Journal of Statistical Physics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：165
• 期：6
• 页码：1102-1113
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics;
• 出版者：Springer US
• ISSN：1572-9613
• 卷排序：165

文摘

In this paper we study Kac’s 1D particle system, consisting of the velocities of N particles colliding at constant rate and randomly exchanging energies. We prove uniform (in time) propagation of chaos in Wasserstein distance with explicit polynomial rates in N, for both the squared (i.e., the energy) and non-squared particle system. These rates are of order \(N^{-1/3}\) (almost, in the non-squared case), assuming that the initial distribution of the limit nonlinear equation has finite moments of sufficiently high order (\(4+\epsilon \) is enough when using the 2-Wasserstein distance). The proof relies on a convenient parametrization of the collision recently introduced by Hauray, as well as on a coupling technique developed by Cortez and Fontbona.