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Journal of Statistical Physics

, Volume 165, Issue 6, pp 1102–1113 | Cite as

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

  • Roberto Cortez
Article
  • 140 Downloads

Abstract

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.

Keywords

Kinetic theory Kac particle system Propagation of chaos 

Mathematics Subject Classification

82C40 60K35 

Notes

Acknowledgements

The author thanks Joaquin Fontbona and Jean-François Jabir for very useful suggestions and corrections of earlier versions of this manuscript. This work was supported by Fondecyt Postdoctoral Project 3160250.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.CIMFAV, Facultad de IngenieríaUniversidad de ValparaísoValparaísoChile

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