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Lectures Year : 2022

Mean field kinetic theory

Abstract

This note presents the theoretical framework to predict the evolution of a mean distribution function in an Hamiltonian dynamical system. Assumptions are an integrable unperturbed Hamiltonian and a scale separation between mean and perturbed distribution functions. The first result shows that that the evolution of the mean distribution function, which depends on action only, can be written in a conservative form. In other words the advection contribution to the kinetic equation can be written as the divergence of a flux in the action space. Moreover fluxes can be computed via a principle of minimum entropy production rate. The special case of a single perturbation is treated first. It appears that the set of constant energy surfaces bear the shape of an island near the resonant surface in the phase space. The distribution function gets flat within the island, while strong gradients develop near its separatrix. Not surprisingly the entropy production rate is maximum in the separatrix region. The case of multiple perturbations is then addressed. A Chirikov overlap parameter is defined that measures the degree of island overlap, i.e. compares the island width with the distance between adjacent resonant surfaces. Whenever the Chirikov overlap parameter exceeds 1, trajectories become chaotic. Under some conditions a quasilinear transport theory can then be used. Finally the case where two perturbations are involved, but do not produce chaos is also investigated. Sections labelled with a star "*" can be skipped in a first reading.
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hal-03974995 , version 1 (06-02-2023)

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Attribution - NonCommercial - NoDerivatives

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  • HAL Id : hal-03974995 , version 1

Cite

Xavier Garbet. Mean field kinetic theory. Doctoral. France. 2022. ⟨hal-03974995⟩
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