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)


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)).



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.


  1. 1.
    Neunzert H (1984) An introduction to the nonlinear Boltzmann-Vlasov equation. In: Cercignani C (ed) Kinetic theories and the Boltzmann equation. Lect notes math, vol 1048. Springer, Berlin, pp 60–110 CrossRefGoogle Scholar
  2. 2.
    Spohn H (1991) Large scale dynamics of interacting particles. Springer, Berlin zbMATHCrossRefGoogle Scholar
  3. 3.
    Firpo M-C, Elskens Y (1998) Kinetic limit of N-body description of wave-particle self-consistent interaction. J Stat Phys 93:193–209 MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Elskens Y, Kiessling MK-H, Ricci V (2008) The Vlasov limit for a system of particles which interact with a wave field. Commun Math Phys 285:673–712 MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Elskens Y, Escande D (2003) Microscopic dynamics of plasmas and chaos. IOP Publishing, Bristol zbMATHCrossRefGoogle Scholar
  6. 6.
    Kiessling MK-H (2008) Microscopic derivations of Vlasov equations. Commun Nonlinear Sci Numer Simul 13:106–113 MathSciNetADSzbMATHCrossRefGoogle Scholar

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

Personalised recommendations