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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 285–317 | Cite as

On the Size of Chaos in the Mean Field Dynamics

  • Thierry Paul
  • Mario PulvirentiEmail author
  • Sergio Simonella
Article

Abstract

We consider the error arising from the approximation of an N-particle dynamics with its description in terms of a one-particle kinetic equation. We estimate the distance between the j-marginal of the system and the factorized state, obtained in a mean field limit as \({N \to \infty}\) . Our analysis relies on the evolution equation for the “correlation error” rather than on the usual BBGKY hierarchy. The rate of convergence is shown to be \({O(j^2/ N)}\) in any bounded interval of time (size of chaos), as expected from heuristic arguments. Our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data, covering (a) stochastic jump processes converging to the homogeneous Boltzmann and the Povzner equation and (b) quantum systems giving rise to the Hartree equation.

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Notes

Acknowledgements

The authors are grateful to Joaquin Fontbona and Stéphane Mischler for helpful discussions. This work has been partially carried out thanks to the support of the LIA AMU-CNRS-ECM-INdAM Laboratoire Yapatia des Sciences Mathématiques (LYSM) and the A*MIDEX Project (no. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR). T.P. also thanks the Dipartimento di Matematica, Sapienza Università di Roma, for its kind hospitality during the elaboration of this work. S. S. acknowledges the support of the German Research Foundation (DFG no. 269134396).

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Authors and Affiliations

  1. 1.CMLS, Ecole polytechnique, CNRS, Université Paris-SaclayPalaiseau CedexFrance
  2. 2.International Research Center on the Mathematics and Mechanics of Complex Systems, MeMoCSUniversity of L’AquilaL’AquilaItaly
  3. 3.Zentrum Mathematik, TU MünchenGarchingGermany

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