Path large deviations for kinetic theories: beyond the Boltzmann, the Landau, and the Balescu—Lenard—Guernsey kinetic equations

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• Freddy Bouchet (ENS - Lyon, CNRS (Centre national de la recherche scientifique))
• Monday 25 April 2022, 14:30-15:30
• Seminar Room 1, Newton Institute.

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FKTW03 - Frontiers in kinetic equations for plasmas and collective behaviour

In many physical systems one seeks to describe effectively mesoscopic or macroscopic variables. Kinetic theories and kinetic equations are examples where the average mesoscopic dynamics is obtained through very clear theoretical procedures and can possibly lead to mathematical proofs, for instance the Landau or the Balescu—Guernsey—Lenard equations in plasma physics. A few works go beyond the average evolution and describe, for instance, Gaussian fluctuations. However, for many physical systems, rare events can be of importance, and Gaussian fluctuations are not relevant. This is the case for instance if one wants to understand the irreversibility paradox associated to the kinetic equations, or to understand the dynamics that leads to rare events with big impact.            The aim of this presentation is to describe recent results where we derived explicitly the functional that describes the path large deviations for the empirical measure of dilute gases, plasma and systems of particles with long range interactions. The associated kinetic equations (the average evolution) are then either the Boltzmann, the Landau or the Balescu—Lenard—Guernsey equations.  After making the classic assumptions in theoretical physics textbooks for deriving the kinetic equation, our derivation of the large deviation functional is exact. These path large deviation principles give a very nice and transparent new interpretation of the classical irreversibility paradox. This new explanation is fully compatible with the classical one, but it gives a deeper insight. References: For the large deviations associated to the Boltzmann equation (dilute gazes), and a general introduction (published in J. Stat. Phys. in 2020): F. Bouchet, 2020, Is the Boltzmann equation reversible? A large deviation perspective on the irreversibility paradox and the Boltzmann equation, Journal of Statistical Physics, 181, 515–550, https://link.springer.com/article/10.1007/s10955-020-02588-y, https://arxiv.org/abs/2002.10398 For the large deviations associated to the Landau equation (plasma below the Debye length, accepted for publication in J. Stat. Phys. in March 2021): O. Feliachi and F. Bouchet, 2021, Dynamical large deviations for plasma below the Debye length and the Landau equation, Journal of Statistical Physics, 183, 42, https://link.springer.com/article/10.1007/s10955-021-02771-9, https://arxiv.org/abs/2101.04455. For the large deviations associated with the Balescu—Guernsey—Lenard equation (plasma and systems with long range interactions): O. Feliachi and F. Bouchet, 2022, Dynamical Large Deviations for Homogeneous Systems with Long Range Interactions and the Balescu–Guernsey–Lenard Equation, Journal of Statistical Physics 186, 22, https://link.springer.com/article/10.1007/s10955-021-02854-7 ;and https://arxiv.org/abs/2105.05644 Joint works with Ouassim Feliachi

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