Advertisement

Kinetic Theory for Plasmas and Wave-Particle Hamiltonian Dynamics

  • Yves Elskens
Chapter
  • 780 Downloads
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Antoni, Y. Elskens and C. Sandoz: Weak turbulence and structure evolution in N-body hamiltonian systems with long range force, Phys. Rev. E 57 (1998) 5347–5357CrossRefADSGoogle Scholar
  2. 2.
    W. Appel and M.K-H. Kiessling: Mass and spin renormalization in Lorentz electrodynamics, Ann. Phys. (N.Y.) 289 (2001) 24–83zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    R. Balescu: Statistical dynamics — Matter out of equilibrium (Imperial college press, London, 1997)zbMATHGoogle Scholar
  4. 4.
    D. Bénisti and D.F. Escande: Origin of diffusion in hamiltonian dynamics Phys. Plasmas 4 (1997) 1576–1581CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    D. Bénisti and D.F. Escande: Finite range of large perturbations in hamiltonian dynamics J. Stat. Phys. 92 (1998) 909–972zbMATHCrossRefGoogle Scholar
  6. 6.
    W. Braun and K. Hepp: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Commun. Math. Phys. 56 (1977) 101–113CrossRefADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    D. del-Castillo-Negrete and M-C. Firpo: Coherent structures and self-consistent transport in a mean field hamiltonian model Chaos 12 (2002) 496–507zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    T. Dauxois, V. Latora, A. Rapisarda, S. Ruffo and A. Torcini: The hamiltonian. mean field model: from dynamics to statistical mechanics and back in “Dynamics and Thermodynamics of Systems with Long Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)Google Scholar
  9. 9.
    Y. Elskens: Dynamical and kinetic aspects of collisions Singularities in gravitational systems — Applications to chaotic transport in the solar system D. Benest and C. Froeschlé eds Lect. Notes Phys. (Springer, Berlin, 2002)Google Scholar
  10. 10.
    Y. Elskens and D.F. Escande: Microscopic dynamics of plasmas and chaos (IoP Publishing, Bristol, 2002)Google Scholar
  11. 11.
    D.F. Escande and Y. Elskens: Quasilinear diffusion for the chaotic motion of a particle in a set of longitudinal waves Acta Phys. Pol. B 33 (2002) 1073–1084ADSGoogle Scholar
  12. 12.
    D.F. Escande and Y. Elskens: Proof of quasilinear equations in the chaotic regime of the weak warm beam instability (preprint)Google Scholar
  13. 13.
    M-C. Firpo, F. Doveil, Y. Elskens, P. Bertrand, M. Poleni and D. Guyomarc’h: Long-time discrete particle effects versus kinetic theory in the self-consistent singlewave model Phys. Rev. E 64 (2001) 026407CrossRefADSGoogle Scholar
  14. 14.
    M-C. Firpo and F. Doveil: Velocity width of the resonant domain in wave-particle interaction Phys. Rev. E 65 (2002) 016411CrossRefADSGoogle Scholar
  15. 15.
    M-C. Firpo and Y. Elskens: Kinetic limit of N-body description of wave-particle self-consistent interaction J. Stat. Phys. 93 (1998) 193–209zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    M-C. Firpo and Y. Elskens: Phase transition in the collisionless damping regime for wave-particle interaction Phys. Rev. Lett. 84 (2000) 3318–3321CrossRefADSGoogle Scholar
  17. 17.
    N.G. van Kampen and B.U. Felderhof: Theoretical methods in plasma physics (North-Holland, Amsterdam, 1967)zbMATHGoogle Scholar
  18. 18.
    M. K-H. Kiessling: Mathematical vindication of the “Jeans swindle” arXiv:astro-ph/9910247Google Scholar
  19. 19.
    H. Neunzert: An introduction to the nonlinear Boltzmann-Vlasov equation Kinetic. theories and the Boltzmann equation C. Cercignani ed. Lect. Notes Math. 1048 (Springer, Berlin, 1984) 60–110CrossRefGoogle Scholar
  20. 20.
    J.L. Rouet and M.R. Feix: Relaxation for a one-dimensional plasma: test particles versus global distribution behaviour Phys. Fluids B 3 (1991) 1830–1834CrossRefADSGoogle Scholar
  21. 21.
    J.L. Rouet and M.R. Feix: Computer experiments on dynamical cloud and space time fluctuations in one-dimensional meta-equilibrium plasmas Phys. Plasmas 3 (1996) 2538–2545CrossRefADSGoogle Scholar
  22. 22.
    S. Ruffo: Hamiltonian dynamics and phase transition Transport, chaos and plasma. physics S. Benkadda, F. Doveil and Y. Elskens eds (World Scientific, Singapore, 1994) 114–119Google Scholar
  23. 23.
    H. Spohn: Large scale dynamics of interacting particles (Springer, New York, 1991)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Yves Elskens
    • 1
  1. 1.Equipe turbulence plasmaUMR 6633 CNRS-Université de ProvenceMarseille cedex 20

Personalised recommendations