Abstract
A system of N particles \(\xi ^N = x_1 ,\upsilon_1,...,x_N ,\upsilon _N )\) interacting self-consistently with one wave Z = A exp(iφ) is considered. Given initial data (Z (N)(0), ξN(0)), it evolves according to Hamiltonian dynamics to (Z (N)(t), ξN(t)). In the limit N → ∞, this generates a Vlasov-like kinetic equation for the distribution function f(x, v, t), abbreviated as f(t), coupled to the envelope equation for Z: initial data (Z (∞)(0), f(0)) evolve to (Z (∞)(t), f(t)). The solution (Z, f) exists and is unique for any initial data with finite energy. Moreover, for any time T>0, given a sequence of initial data with N particles distributed so that the particle distribution f N(0) → f(0) weakly and with Z (N)(0) → Z(0) as N → ∞, the states generated by the Hamiltonian dynamics at all times 0 ≤ t ≤ T are such that (Z (N)(t), f N(t)) converges weakly to (Z (∞)(t), f(t)).
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REFERENCES
F. King, BBGKY hierarchy for positive potentials, Ph.D. thesis (University of California, Berkeley, 1975).
O. E. Lanford III, Time evolution of large classical systems, in Dynamical Systems, Theory and Applications, J. Moser, ed., Lect. Notes Phys., Vol. 38 (Springer, Berlin, 1975), pp. 1–111.
C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases (Springer, New York, 1994).
J. Piasecki, Echelles de temps en théorie cinétique (Presses universitaires romandes, Lausanne, 1997).
H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation, C. Cercignani, ed., Lect. Notes Math., Vol. 1048 (Springer, Berlin, 1984), pp. 60–110.
H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991).
M. Antoni, Dynamique microscopique des plasmas: de “N corps” à “M modes et N q particules,” Thèse de doctorat de l'université de Provence (Marseille, 1993).
M. Antoni, Y. Elskens, and D. F. Escande, Reduction of N-body dynamics to particle-wave interaction in plasmas, in Dynamics of Transport in Plasmas and for Charged Beams, G. Maino and M. Ottaviani, eds. (World Scientific, Singapore, 1996), pp. 1–17.
M. Antoni, Y. Elskens, and D. F. Escande, Explicit reduction of N-body dynamics to self-consistent particle-wave interaction, Phys. Plasmas 5:841–852 (1998).
J. R. Cary, I. Doxas, D. F. Escande, and A. D. Verga, Enhancement of the velocity diffusion in longitudinal plasma turbulence, Phys. Fluids B 4:2062–2069 (1992).
I. Doxas and J. R. Cary, Numerical observation of turbulence enhanced growth rates, Phys. Plasmas 4:2508–2518 (1997).
D. F. Escande, Description of Landau damping and weak Langmuir turbulence through microscopic dynamics, in Nonlinear World, V. G. Bar'yakhtar, V. M. Chernousenko, N. S. Erokhin, A. G. Sitenko, and V. E. Zakharov, eds. (World Scientific, Singapore, 1990), pp. 817–836.
D. F. Escande, Large scale structures in kinetic plasma turbulence, in Large Scale Structures in Nonlinear Physics, J. D. Fournier and P. L. Sulem, eds., Lect. Notes Phys., Vol. 392 (Springer, Berlin, 1991), pp. 73–104.
D. Guyomarc'h, F. Doveil, Y. Elskens, and D. Fanelli, Warm beam-plasma instability beyond saturation, in Transport, Chaos and Plasma Physics 2, S. Benkadda, F. Doveil, and Y. Elskens, eds. (World Scientific, Singapore, 1996), pp. 406–410.
H. E. Mynick and A. N. Kaufman, Soluble theory of nonlinear beam-plasma interaction, Phys. Fluids 21:653–663 (1978).
T. M. O'Neil, J. H. Winfrey, and J. H. Malmberg, Nonlinear interaction of a small cold beam and a plasma, Phys. Fluids 14:1204–1212 (1971).
T. M. O'Neil and J. H. Winfrey, Nonlinear interaction of a small cold beam and a plasma, II, Phys. Fluids 15:1514–1522 (1972).
J. L. Tennyson, J. D. Meiss, and P. J. Morrison, Self-consistent chaos in the beam-plasma instability, Physica D 71:1–17 (1994).
S. Zekri, Approche hamiltonienne de la turbulence faible de Langmuir, Thèse de doctorat de l'université de Provence (Marseille, 1993).
D. F. Escande, S. Zekri, and Y. Elskens, Intuitive and rigorous derivation of spontaneous emission and Landau damping of Langmuir waves through classical mechanics, Phys. Plasmas 3:3534–3539 (1996).
P. Bertrand, A. Ghizzo, S. J. Karttunen, T. J. H. Pättikangas, R. R. E. Salomaa, and M. Shoucri, Two-stage acceleration by simultaneous stimulated Raman backward and forward scattering, Phys. Plasmas 2:3115–3129 (1995).
A. Ghizzo, M. Shoucri, P. Bertrand, T. Johnston, and J. Lebas, Trajectories of trapped particles in the field of a plasma wave excited by a stimulated Raman scattering, J. Comput. Phys. 108:373–376 (1993).
P. Chaix and D. Iracane, Stochastic electronic motion and high efficiency free electron lasers, in Transport, Chaos and Plasma Physics, S. Benkadda, F. Doveil, and Y. Elskens, eds. (World Scientific, Singapore, 1994), pp. 370–373.
D. A. Hartmann, C. F. Driscoll, T. M. O'Neill, and V. D. Shapiro, Measurements of the weak warm beam instability, Phys. Plasmas 2:654–677 (1995).
G. Brodin, Nonlinear Landau damping, Phys. Rev. Lett. 78:1263–1266 (1997).
M. B. Isichenko, Nonlinear Landau damping in collisionless plasma and inviscid fluid, Phys. Rev. Lett. 78:2369–2372 (1997).
G. Manfredi, Long-time behavior of nonlinear Landau damping, Phys. Rev. Lett. 79:2815–2818 (1997).
G. J. Morales, Interaction of a low density runaway beam with cavity modes, Phys. Fluids 22:1359–1371 (1979).
G. J. Morales, Effect of a dc field on the trapping dynamics of a cold electron beam, Phys. Fluids 23:2472–2484 (1980).
H. L. Berk, B. N. Breizman, and M. Pekker, Numerical simulation of bump-on-tail instability with source and sink, Phys. Plasmas 2:3007–3016 (1995).
D. Fanelli, Regime non lineare dell'instabilità fascio di elettroni/plasma (interazione onde/particelle), tesi di laurea, università studi di Fírenze (1996).
B. D. Fried, C. S. Liu, R. W. Means, and R. Z. Sagdeev, Nonlinear evolution of an unstable electrostatic wave, Ccalifornia, Los Angeles, plasma physics group, report PPG-93 (1971).
Y. Elskens and M. C. Firpo, Kinetic theory and large-N limit for wave-particle self-consistent interaction, Physica Scripta, in press.
J. R. Cary and I. Doxas, An explicit symplectic integration scheme for plasma simulation, J. Comput. Phys. 107:98–104 (1993).
D. Guyomarc'h, Un tube à onde progressive pour l'étude de la turbulence plasma, Thèse de doctorat de l'université de Provence (Marseille, 1996).
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Firpo, MC., Elskens, Y. Kinetic Limit of N-Body Description of Wave–Particle Self-Consistent Interaction. Journal of Statistical Physics 93, 193–209 (1998). https://doi.org/10.1023/B:JOSS.0000026732.51044.87
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DOI: https://doi.org/10.1023/B:JOSS.0000026732.51044.87