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Kinetic Limit of Dynamical Description of Wave-Particle Self-consistent Interaction in an Open Domain

  • Bruno Vieira RibeiroEmail author
  • Yves Elskens
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

In a closed domain Ω of space, we consider a system of N particles σ N =(x 1,v 1,…,x N ,v N ) interacting via a pair potential U. In this region, particles also interact self-consistently with a wave Z=Aexp(iϕ). We consider injection of particles in Ω, so N varies in time.

Given initial data (Z N (0),σ N (0)) and a boundary source/sink, the system evolves according to a Hamiltonian dynamics to (Z N (t),σ N (t)). In the limit of infinitely many particles (kinetic limit), this generates a Vlasov-like kinetic equation for the distribution function f(x,v,t) coupled to an envelope equation for Z(t)=Z (t). The solution (Z ,f) exists and is unique for any initial data with finite energy, provided that Ω has smooth enough boundaries.

Further, for any finite time t, given a sequence of initial data such that σ N (0)→f(0) weakly and Z N (0)→Z(0) as N→∞, the states generated by the Hamiltonian dynamics (Z N (t),σ N (t)) are such that lim N→∞(Z N (t),σ N (t))=(Z (t),f(x,v,t)).

Notes

Acknowledgements

The authors thank N. Dubuit for fruitful discussion and Marco A. Amato for initiating and collaborating in the ongoing researches.

B. Vieira Ribeiro is supported by a grant from CAPES Foundation through the PDSE program, process number: 8510/11-3.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Instituto de FísicaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Equipe Turbulence Plasma, case 321, PIIMUMR 7345 CNRS, Aix-Marseille UniversitéMarseille Cedex 13France
  3. 3.CAPES FoundationMinistry of Education of BrazilBrasíliaBrazil

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