Fluctuation Theory in the Boltzmann–Grad Limit

Abstract

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics.

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Notes

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    For previous quantitative investigations of the correlation error, we refer to [14, 26].

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Acknowledgements

We are very grateful to H. Spohn, F. Bouchet, F. Rezakhanlou, G. Basile, D. Benedetto, and L. Bertini for many enlightening discussions on the subjects treated in this text. T.B. acknowledges the support of ANR-15-CE40-0020-01 Grant LSD.

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Correspondence to Laure Saint-Raymond.

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This paper is dedicated to Joel Lebowitz for his 90th birthday.

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Communicated by Eric A. Carlen.

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Bodineau, T., Gallagher, I., Saint-Raymond, L. et al. Fluctuation Theory in the Boltzmann–Grad Limit. J Stat Phys 180, 873–895 (2020). https://doi.org/10.1007/s10955-020-02549-5

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Keywords

  • Kinetic theory
  • Hard sphere dynamics
  • Low density limit
  • Boltzmann equation
  • Fluctuations
  • Large deviations
  • cumulants