# Kinetic Theory for Plasmas and Wave-Particle Hamiltonian Dynamics

• Yves Elskens
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 602)

## Abstract

The plasma limit is characterized by the fact that a particle interacts with many partners. The dynamics leads to the Vlasov equation in the mean-field limit. Collective behaviour in the N-body system is naturally described by M $$M{\mathbf{ }} \ll {\mathbf{ }}\mathcal{N}$$ N wavelike degrees of freedom, which behave as harmonic oscillator interacting with a population of N $$N{\mathbf{ }} \ll {\mathbf{ }}\mathcal{N}$$ N particles. The self-consistent hamiltonian dynamics with M waves is also mean-field for the N particles, and a Vlasov equation applies for N → ∞. The careful understanding of the limit N → ∞ requires taking into account a continuous spectrum of singular excitations of van Kampen-Case type. For M → ∞, the motion of a single particle obeys a Fokker-Planck equation in the strong resonance overlap limit, and the full particle-wave system obeys the quasilinear evolution equations in the weak turbulence regime.

## Keywords

Kinetic Theory Vlasov Equation Langmuir Wave Hamiltonian Dynamic Chaotic Regime
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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