Basic microscopic plasma physics from N-body mechanics

A tribute to Pierre-Simon de Laplace
  • D. F. EscandeEmail author
  • D. Bénisti
  • Y. Elskens
  • D. Zarzoso
  • F. Doveil
Review Paper


Computing is not understanding. This is exemplified by the multiple and discordant interpretations of Landau damping still present after 70 years. For long deemed impossible, the mechanical N-body description of this damping, not only enables its rigorous and simple calculation, but makes unequivocal and intuitive its interpretation as the synchronization of almost resonant passing particles. This synchronization justifies mechanically why a single formula applies to both Landau growth and damping. As to the electrostatic potential, the phase mixing of many beam modes produces Landau damping, but it is unexpectedly essential for Landau growth too. Moreover, collisions play an essential role in collisionless plasmas. In particular, Debye shielding results from a cooperative dynamical self-organization process, where “collisional” deflections due to a given electron diminish the apparent number of charges about it. The finite value of exponentiation rates due to collisions is crucial for the equivalent of the van Kampen phase mixing to occur in the N-body system. The N-body approach incorporates spontaneous emission naturally, whose compound effect with Landau damping drives a thermalization of Langmuir waves. O’Neil’s damping with trapping typical of initially large enough Langmuir waves results from a phase transition. As to Coulomb scattering, there is a smooth connection between impact parameters where the two-body Rutherford picture is correct, and those where a collective description is mandatory. The N-body approach reveals two important features of the Vlasovian limit: it is singular and it corresponds to a renormalized description of the actual N-body dynamics.


N-body dynamics Debye shielding Landau damping Wave–particle interaction Spontaneous emission Coulomb scattering 



D. F. E. is grateful to the members of Equipe Turbulence Plasma in Marseilles, since the theory reviewed in this paper is the result of three decades of collaboration with them. He thanks Professor M. Kikuchi for suggesting him to write this review. He also thanks Professor A. Sen for many useful suggestions, and Professor P. Huneman for pointing out to him the book “Reductionism, emergence and levels of reality” by Chibbaro et al. He thanks Drs F. Bonneau, M.-C. Firpo, and F. Sattin for helpful comments on the manuscript. Also D. F. G. Minenna who brought the precious views of a newcomer in the field. One of the authors (D. Z.) has been supported by the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR).

Compliance with ethical standards

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. R. Abe, Giant cluster expansion theory and its application to high temperature plasma. Prog. Theor. Phys. 22(2), 213–226 (1959)ADSMathSciNetzbMATHGoogle Scholar
  2. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables/edited by Milton Abramowitz and Irene A. Stegun (Dover, New York, 1967)zbMATHGoogle Scholar
  3. J.C. Adam, G. Laval, D. Pesme, Reconsideration of quasilinear theory. Phys. Rev. Lett. 43(22), 1671–1675 (1979)ADSGoogle Scholar
  4. M. Antoni, Y. Elskens, D.F. Escande, Explicit reduction of N-body dynamics to self-consistent particle–wave interaction. Phys. Plasmas 5(4), 841–852 (1998)ADSMathSciNetGoogle Scholar
  5. D.R. Baker, N.R. Ahern, A.Y. Wong, Ion-wave echoes. Phys. Rev. Lett. 20(7), 318–321 (1968)ADSGoogle Scholar
  6. D. Baldwin, C. Watson, Magnetized plasma kinetic theory. II. Derivation of Rosenbluth potentials for a uniform magnetized plasma. Plasma Phys. 19(6), 517–528 (1977)ADSGoogle Scholar
  7. R. Balescu, Statistical Mechanics of Charged Particles (Interscience, London, 1963)zbMATHGoogle Scholar
  8. R. Balescu, Statistical Dynamics: Matter Out of Equilibrium (Imperial Coll, London, 1997)zbMATHGoogle Scholar
  9. M. Baus, J.P. Hansen, Statistical mechanics of simple Coulomb systems. Phys. Rep. 59(1), 1–94 (1980)ADSMathSciNetGoogle Scholar
  10. G. Bekefi, Collective emission processes in plasmas, in Plasma Physics, ed. by C. DeWitt, J. Peyraud (Gordon and Breach, Philadelphia, 1975), pp. 1–111Google Scholar
  11. D. Bénisti, Self-consistent theory for the linear and nonlinear propagation of a sinusoidal electron plasma wave. Application to stimulated Raman scattering in a non-uniform and non-stationary plasma. Plasma Phys. Control. Fusion 60(1), 014040 (2018)ADSGoogle Scholar
  12. D. Bénisti, D.F. Escande, Origin of diffusion in Hamiltonian dynamics. Phys. Plasmas 4(5), 1576–1581 (1997)ADSMathSciNetGoogle Scholar
  13. D. Bénisti, D.F. Escande, Finite range of large perturbations in Hamiltonian dynamics. J. Stat. Phys. 92, 909–972 (1998)ADSzbMATHGoogle Scholar
  14. D. Bénisti, L. Gremillet, Nonlinear plasma response to a slowly varying electrostatic wave, and application to stimulated Raman scattering. Phys. Plasmas 14(4), 042304 (2007)ADSGoogle Scholar
  15. D. Bénisti, D.J. Strozzi, L. Gremillet, O. Morice, Nonlinear Landau damping rate of a driven plasma wave. Phys. Rev. Lett. 103(15), 155002 (2009)ADSzbMATHGoogle Scholar
  16. D. Bénisti, O. Morice, L. Gremillet, The various manifestations of collisionless dissipation in wave propagation. Phys. Plasmas 19(6), 063110 (2012)ADSGoogle Scholar
  17. J.T. Berndtson, J.A. Heikkinen, S.J. Karttunen, T.J.H. Pättikangas, R.R.E. Salomaa, Analysis of velocity diffusion of electrons with Vlasov-Poisson simulations. Plasma Phys. Control. Fusion 36(1), 57–71 (1994)ADSGoogle Scholar
  18. N. Besse, Y. Elskens, D.F. Escande, P. Bertrand, On the validity of quasilinear theory. in Proceeding 38th EPS Conference on Controlled Fusion and Plasma Physics, Strasbourg, p P2.009, (2011a) Accessed 3 Apr 2018
  19. N. Besse, Y. Elskens, D.F. Escande, P. Bertrand, Validity of quasilinear theory: refutations and new numerical confirmation. Plasma Phys. Control. Fusion 53(2), 025012 (2011b)ADSGoogle Scholar
  20. N. Boers, P. Pickl, On mean field limits for dynamical systems. J. Stat. Phys. 164(1), 1–16 (2016)ADSMathSciNetzbMATHGoogle Scholar
  21. W.J. Bos, R. Rubinstein, L. Fang, Reduction of mean-square advection in turbulent passive scalar mixing. Phys. Fluids 24(7), 075104 (2012)ADSGoogle Scholar
  22. T.J.M. Boyd, J.J. Sanderson, The Physics of Plasmas (Cambridge University Press, Cambridge, 2003)zbMATHGoogle Scholar
  23. W. Braun, K. Hepp, The Vlasov dynamics and its fluctuations in the 1/\(N\) limit of interacting classical particles. Commun. Math. Phys. 56(2), 101–113 (1977)ADSMathSciNetzbMATHGoogle Scholar
  24. J.D. Callen, Coulomb collision effects on linear Landau damping. Phys. Plasmas 21(5), 052106 (2014)ADSGoogle Scholar
  25. N. Carlevaro, D. Fanelli, X. Garbet, P. Ghendrih, G. Montani, M. Pettini, Beam-plasma instability and fast particles: the Lynden-Bell approach. Plasma Phys. Control. Fusion 56(3), 035013 (2014)ADSGoogle Scholar
  26. J.R. Cary, D.F. Escande, A.D. Verga, Non quasilinear diffusion far from the chaotic threshold. Phys. Rev. Lett. 65, 3132–3135 (1990)ADSGoogle Scholar
  27. J.R. Cary, I. Doxas, D.F. Escande, A. Verga, Enhancement of the velocity diffusion in longitudinal plasma turbulence. Phys. Fluids B Plasma Phys. 4(7), 2062–2069 (1992)Google Scholar
  28. K.M. Case, Plasma oscillations. Ann. Phys. 7(3), 349–364 (1959)ADSMathSciNetzbMATHGoogle Scholar
  29. C. Chandre, G. Ciraolo, F. Doveil, R. Lima, A. Macor, M. Vittot, Channeling chaos by building barriers. Phys. Rev. Lett. 94, 074101 (2005). (4 pp)ADSGoogle Scholar
  30. S. Chibbaro, L. Rondoni, A. Vulpiani, Reductionism, Emergence and Levels of Reality (Springer, Heidelberg, 2014)zbMATHGoogle Scholar
  31. J.M. Dawson, Plasma oscillations of a large number of electron beams. Phys. Rev. 118(2), 381–389 (1960)ADSMathSciNetzbMATHGoogle Scholar
  32. R.L. Dobrushin, Vlasov equations. Funct. Anal. Appl. 13(2), 115–123 (1979)zbMATHGoogle Scholar
  33. F. Doveil, A. Macor, Two regimes of self-consistent heating of charged particles. Phys. Rev. E 84(4), 045401 (2011)ADSGoogle Scholar
  34. F. Doveil, M.C. Firpo, Y. Elskens, D. Guyomarc’h, M. Poleni, P. Bertrand, Trapping oscillations, discrete particle effects and kinetic theory of collisionless plasma. Phys. Lett. A 284(6), 279–285 (2001)ADSGoogle Scholar
  35. F. Doveil, K. Auhmani, A. Macor, D. Guyomarc’h, Experimental observation of resonance overlap responsible for Hamiltonian chaos. Phys. Plasmas 12, 010702 (2005a). (4 pp)ADSGoogle Scholar
  36. F. Doveil, D.F. Escande, A. Macor, Experimental observation of nonlinear synchronization due to a single wave. Phys. Rev. Lett. 94(8), 085003 (2005b)ADSGoogle Scholar
  37. I. Doxas, J.R. Cary, Numerical observation of turbulence enhanced growth rates. Phys. Plasmas 4(7), 2508–2518 (1997)ADSGoogle Scholar
  38. W.E. Drummond, D. Pines, Nonlinear stability of plasma oscillations. Nucl. Fusion Suppl. 3, 1049–1057 (1962)Google Scholar
  39. Y. Elskens, Finite-\(N\) dynamics admit no travelling-waves solutions for the Hamiltonian XY model and single-wave collisionless plasma model. ESAIM: Proceedings, EDP Sciences, vol 10, pp 221–215 (2001)zbMATHGoogle Scholar
  40. Y. Elskens, Irreversible behaviours in Vlasov equations and many-body Hamiltonian dynamics: Landau damping, chaos and granularity. in Topics in Kinetic Theory, American Mathematical Soc., Fields Institute Communications Series, ed. by T. Passot, C. Sulem, P.L. Sulem, vol 46, pp 89–108 (2005)Google Scholar
  41. Y. Elskens, Quasilinear limit for particle motion in a prescribed spectrum of random waves. Phys. AUC 17, 109–121 (2007)Google Scholar
  42. Y. Elskens, Nonquasilinear evolution of particle velocity in incoherent waves with random amplitudes. Commun. Nonlinear Sci. Numer. Simul. 15(1), 10–15 (2010)ADSGoogle Scholar
  43. Y. Elskens, Gaussian convergence for stochastic acceleration of \(N\) particles in the dense spectrum limit. J. Stat.Phys. 148, 591–605 (2012)ADSMathSciNetzbMATHGoogle Scholar
  44. Y. Elskens, D.F. Escande, Microscopic Dynamics of Plasmas and Chaos (Institute of Physics, Bristol, 2002)Google Scholar
  45. Y. Elskens, E. Pardoux, Diffusion limit for many particles in a periodic stochastic acceleration field. Ann. Appl. Probab. 20, 2022–2039 (2010)MathSciNetzbMATHGoogle Scholar
  46. Y. Elskens, D.F. Escande, F. Doveil, Vlasov equation and \(N\)-body dynamics : how central is particle dynamics to our understanding of plasmas ? Eur. Phys. J. D 68(8), 218 (2014). (7 pages)ADSGoogle Scholar
  47. D.F. Escande, Description of Landau damping and weak Langmuir turbulence through microscopic dynamics, in Nonlinear World, vol. 2, ed. by V.G. Bar’yakhtar, et al. (World Scientific, Singapore, 1989), pp. 817–836Google Scholar
  48. D.F. Escande, Large scale structures in kinetic plasma turbulence, in Large Scale Structures in Nonlinear Physics, ed. by J.D. Fournier, P.L. Sulem (Springer, Berlin, 1991), pp. 73–104Google Scholar
  49. D.F. Escande, How to face the complexity of plasmas?, in From Hamiltonian Chaos to Complex Systems, ed. by X. Leoncini, M. Leonetti (Springer, Berlin, 2013), pp. 109–157Google Scholar
  50. D.F. Escande, Complexity and simplicity of plasmas, in AIP Conference Proceedings, AIP, vol. 1582, ed. by A. Das, A. Sh Sharma (Melville, 2014), pp. 22–34Google Scholar
  51. D.F. Escande, Contributions of plasma physics to chaos and nonlinear dynamics. Plasma Phys. Control. Fusion 58(11), 113001 (2016)ADSGoogle Scholar
  52. D.F. Escande, From thermonuclear fusion to Hamiltonian chaos. Eur. Phys. J. H (2017). CrossRefGoogle Scholar
  53. D.F. Escande, Y. Elskens, Proof of quasilinear equations in the chaotic regime of the weak warm beam instability. Phys. Lett. A 302, 110–118 (2002a)ADSGoogle Scholar
  54. D.F. Escande, Y. Elskens, Proof of quasilinear equations in the chaotic regime of the weak warm beam instability. Phys. Lett. A 302(2–3), 110–118 (2002b)ADSGoogle Scholar
  55. D.F. Escande, Y. Elskens, Quasilinear diffusion for the chaotic motion of a particle in a set of longitudinal waves. Acta Phys. Pol. B 33, 1073–1084 (2002c)ADSGoogle Scholar
  56. D.F. Escande, Y. Elskens, Proof of quasilinear equations in the strongly nonlinear regime of the weak warm beam instability. Phys. Plasmas 10, 1588–1594 (2003)ADSMathSciNetGoogle Scholar
  57. D.F. Escande, F. Sattin, When can the Fokker-Planck equation describe anomalous or chaotic transport? Phys. Rev. Lett. 99, 185005 (2007). (4 pages)ADSGoogle Scholar
  58. D.F. Escande, F. Sattin, When can the Fokker-Planck equation describe anomalous or chaotic transport? Intuitive aspects. Plasma Phys. Control. Fusion 50, 124023 (2008). (8 pp)ADSGoogle Scholar
  59. D.F. Escande, S. Zekri, Y. Elskens, Intuitive and rigorous derivation of spontaneous emission and Landau damping of Langmuir waves through classical mechanics. Phys. Plasmas 3(10), 3534–3539 (1996)ADSGoogle Scholar
  60. D.F. Escande, N. Besse, F. Doveil, Y. Elskens, Application of Picard iteration technique to self-consistent wave-particle interaction in plasmas. In 39th EPS Conference and 16th International Congress on Plasma Physics, Stockholm, P4.161 (2012) Accessed 3 Apr 2018
  61. D.F. Escande, Y. Elskens, F. Doveil, Corrigendum: direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction (2015 plasma phys. control. fusion 57 025017). Plasma Phys. Control. Fusion 57(6), 069501 (2015a)ADSGoogle Scholar
  62. D.F. Escande, Y. Elskens, F. Doveil, Direct path from microscopic mechanics to Debye shielding, Landau damping and wave-particle interaction. Plasma Phys. Control. Fusion 57(2), 025017 (2015b)ADSGoogle Scholar
  63. D.F. Escande, Y. Elskens, F. Doveil, Uniform derivation of Coulomb collisional transport thanks to Debye shielding. J. Plasma Phys. 81(1), 305810101 (2015c)Google Scholar
  64. D.F. Escande, F. Doveil, Y. Elskens, \(N\)-body description of Debye shielding and Landau damping. Plasma Phys. Control. Fusion 58(1), 014040 (2016)ADSGoogle Scholar
  65. M.C. Firpo, Y. Elskens, Kinetic limit of \(N\)-body description of wave-particle self-consistent interaction. J. Stat. Phys. 93(1–2), 193–209 (1998)ADSMathSciNetzbMATHGoogle Scholar
  66. M.C. Firpo, Y. Elskens, Phase transition in the collisionless damping regime for wave-particle interaction. Phys. Rev. Lett. 84(15), 3318 (2000)ADSGoogle Scholar
  67. M.C. Firpo, F. Doveil, Y. Elskens, P. Bertrand, M. Poleni, D. Guyomarc’h, Long-time discrete particle effects versus kinetic theory in the self-consistent single-wave model. Phys. Rev. E 64(2), 026407 (2001)ADSGoogle Scholar
  68. M.C. Firpo, F. Leyvraz, G. Attuel, Equilibrium statistical mechanics for single waves and wave spectra in Langmuir wave-particle interaction. Phys. Plasmas 13(12), 122302 (2006)ADSGoogle Scholar
  69. A.A. Galeev, R.Z. Sagdeev, V.D. Shapiro, V.I. Shevchenko, Is renormalization necessary in the quasi-linear theory of Langmuir oscillations? Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 79, 2167–2174 (1980). [English trans. Sov. Phys.-JETP 52: 1095–1099 (1980)]ADSGoogle Scholar
  70. S. Gasiorowicz, M. Neuman, R.J. Riddell, Dynamics of ionized media. Phys. Rev. 101, 922–934 (1956)ADSzbMATHGoogle Scholar
  71. R. Godement, Mathematical Analysis III, vol. 3 (Springer, Berlin, 2015)zbMATHGoogle Scholar
  72. D.A. Hartmann, C.F. Driscoll, T.M. O’Neil, V.D. Shapiro, Measurements of the weak warm beam instability. Phys. Plasmas 2(3), 654–677 (1995)ADSGoogle Scholar
  73. M. Hassan, C. Watson, Magnetized plasma kinetic theory. I. Derivation of the kinetic equation for a uniform magnetized plasma. Plasma Phys. 19(3), 237–247 (1977a)ADSGoogle Scholar
  74. M. Hassan, C. Watson, Magnetized plasma kinetic theory. III. Fokker-Planck coefficients for a uniform magnetized plasma. Plasma Phys. 19(7), 627–649 (1977b)ADSGoogle Scholar
  75. M. Hauray, P.E. Jabin, Particle approximation of Vlasov equations with singular forces: propagation of chaos. Ann. Sci. Ec Norm. Sup. 48, 891–940 (2015)MathSciNetzbMATHGoogle Scholar
  76. R.D. Hazeltine, The Framework of Plasma Physics (CRC Press, Boca Raton, 2018)Google Scholar
  77. R.D. Hazeltine, F.L. Waelbroeck, The Framework of Plasma Physics (Westview, London, 2004)Google Scholar
  78. J. Hubbard, The friction and diffusion coefficients of the Fokker-Planck equation in a plasma. II. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 261(1306), 371–387 (1961)ADSMathSciNetzbMATHGoogle Scholar
  79. S. Ichimaru, M.N. Rosenbluth, Relaxation processes in plasmas with magnetic field. Temperature relaxations. Phys. Fluids 13(11), 2778–2789 (1970)ADSzbMATHGoogle Scholar
  80. P.E. Jabin, Z. Wang, Mean field limit and propagation of chaos for Vlasov systems with bounded forces. J. Funct. Anal. 271(12), 3588–3627 (2016)MathSciNetzbMATHGoogle Scholar
  81. N.G. van Kampen, On the theory of stationary waves in plasmas. Physica 21(6–10), 949–963 (1955)ADSMathSciNetGoogle Scholar
  82. N.G. van Kampen, The dispersion equation for plasma waves. Physica 23(6–10), 641–650 (1957)ADSMathSciNetzbMATHGoogle Scholar
  83. A.N. Kaufman, Reformulation of quasi-linear theory. J. Plasma Phys. 8(1), 1–5 (1972)ADSGoogle Scholar
  84. M.K.H. Kiessling, The microscopic foundations of Vlasov theory for jellium-like Newtonian \(N\)-body systems. J. Stat. Phys. 155(6), 1299–1328 (2014)ADSMathSciNetzbMATHGoogle Scholar
  85. R.H. Kraichnan, R. Panda, Depression of nonlinearity in decaying isotropic turbulence. Phys. Fluids 31(9), 2395–2397 (1988)ADSzbMATHGoogle Scholar
  86. S.M. Krivoruchko, V.A. Bashko, A.S. Bakai, Experimental investigations of correlation phenomena in the relaxation of velocity-spread beam in a plasma. Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 80, 579–581 (1981). [English trans. Sov. Phys.-JETP 53: 292–298 (1981)]ADSGoogle Scholar
  87. C. Lancellotti, From Vlasov fluctuations to the BGL kinetic equation. Il Nuovo cimento della Società italiana di fisica C 33(1), 111–119 (2010)Google Scholar
  88. L.D. Landau, On the vibrations of the electronic plasma. J. Phys. (USSR) 10(1), 25–34 (1946). Russian original: Zh.Eksp.Teor.Fiz. 16 (1946) 574-586MathSciNetzbMATHGoogle Scholar
  89. P.S. Laplace, Essai philosophique sur les probabilités. Bachelier, Paris (1840)Google Scholar
  90. G. Laval, D. Pesme, Breakdown of quasilinear theory for incoherent 1-D Langmuir waves. Phys. Fluids 26(1), 52–65 (1983a)ADSzbMATHGoogle Scholar
  91. G. Laval, D. Pesme, Inconsistency of quasilinear theory. Phys. Fluids 26(1), 66–68 (1983b)ADSzbMATHGoogle Scholar
  92. G. Laval, D. Pesme, Self-consistency effects in quasilinear theory: a model for turbulent trapping. Phys. Rev. Lett. 53(3), 270–273 (1984)ADSGoogle Scholar
  93. G. Laval, D. Pesme, Controversies about quasi-linear theory. Plasma Phys. Control. Fusion 41(3A), A239 (1999)ADSGoogle Scholar
  94. Y.M. Liang, P. Diamond, Revisiting the validity of quasilinear theory. Phys. Fluids B Plasma Phys. 5(12), 4333–4340 (1993a)Google Scholar
  95. Y.M. Liang, P.H. Diamond, Weak turbulence theory of Langmuir waves: a reconsideration of the validity of quasilinear theory. Comments Plasma Phys. Control. Fusion 15, 139–149 (1993b)Google Scholar
  96. E.M. Lifshitz, L.P. Pitaevskii, Fizicheskaya kinetika, (Moscow, Nauka, 1979) [English trans. Landau and Lifshitz Course of Theoretical Physics 10: Physical Kinetics, Transl. Sykes JB and Franklin RN (Pergamon, Oxford, 1981)]Google Scholar
  97. A. Macor, F. Doveil, Y. Elskens, Electrons climbing a “devils staircase” in wave-particle interaction. Phys. Rev. Lett. 95, 264102 (2005). (4pp)ADSGoogle Scholar
  98. W.D. McComb, Renormalization Methods: A Guide for Beginners (Oxford University Press, Oxford, 2004)zbMATHGoogle Scholar
  99. N. Meyer-Vernet, Aspects of Debye shielding. Am. J. Phys. 61(3), 249–257 (1993)ADSGoogle Scholar
  100. H.E. Mynick, A.N. Kaufman, Soluble theory of nonlinear beam-plasma interaction. Phys. Fluids 21(4), 653–663 (1978)ADSzbMATHGoogle Scholar
  101. H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation. in Kinetic Theories and the Boltzmann Equation, ed. by C. Cercignani, Springer, Berlin, no. 1048 in Lect. Notes Math., pp 60–110 (1984)Google Scholar
  102. H. Neunzert, J. Wick, Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen. in Numerische Behandlung nichtlinearer Integrodifferential- und Differentialgleichungen, ed. by R. Ansorge, W. Törnig, Springer, Berlin, no. 395 in Lect. Notes Math., pp 275–290 (1974)Google Scholar
  103. C.S. Ng, A. Bhattacharjee, F. Skiff, Kinetic eigenmodes and discrete spectrum of plasma oscillations in a weakly collisional plasma. Phys. Rev. Lett. 83(10), 1974–1977 (1999)ADSGoogle Scholar
  104. C.S. Ng, A. Bhattacharjee, F. Skiff, Weakly collisional Landau damping and three-dimensional Bernstein-Greene-Kruskal modes: new results on old problems. Phys. Plasmas 13(5), 055903 (2006)ADSGoogle Scholar
  105. D.R. Nicholson, Introduction to Plasma Theory (Wiley, New York, 1983)Google Scholar
  106. T.M. O’Neil, Collisionless damping of nonlinear plasma oscillations. Phys. Fluids 8(12), 2255–2262 (1965)ADSMathSciNetGoogle Scholar
  107. T.M. O’Neil, J.H. Malmberg, Transition of the dispersion roots from beam-type to Landau-type solutions. Phys. Fluids 11(8), 1754–1760 (1968)ADSGoogle Scholar
  108. T.M. O’Neil, J.H. Winfrey, J.H. Malmberg, Nonlinear interaction of a small cold beam and a plasma. Phys. Fluids 14(6), 1204–1212 (1971)ADSGoogle Scholar
  109. I.N. Onishchenko, A.R. Linetskii, N.G. Matsiborko, V.D. Shapiro, V.I. Shevchenko, Contribution to the nonlinear theory of excitation of a monochromatic plasma wave by an electron beam. ZhETF Pis Red 12, 281–285 (1970). [Engl. Transl. 1960, JETP Lett.12 281–285 12: 281–285]Google Scholar
  110. A. Piel, Plasma Physics: An Introduction to Laboratory, Space, and Fusion Plasmas (Springer, Heidelberg, 2017)zbMATHGoogle Scholar
  111. C. Roberson, K.W. Gentle, Experimental test of the quasilinear theory of the gentle bump instability. Phys. Fluids 14(11), 2462–2469 (1971)ADSGoogle Scholar
  112. Y.A. Romanov, G.F. Filippov, Interaction of fast electron beams with longitudinal plasma waves. Zh Eksp Theor. Phys. 40, 123–132 (1961). [English trans. Sov. Phys.-JETP 13: 87–92 (1961)]Google Scholar
  113. M.N. Rosenbluth, W.M. MacDonald, D.L. Judd, Fokker-Planck equation for an inverse-square force. Phys. Rev. 107, 1–6 (1957)ADSMathSciNetzbMATHGoogle Scholar
  114. N. Rostoker, Superposition of dressed test particles. Phys. Fluids 7(4), 479–490 (1964)ADSMathSciNetGoogle Scholar
  115. D.D. Ryutov, Landau damping: half a century with the great discovery. Plasma Phys. Control. Fusion 41(3A), A1–A12 (1999)ADSGoogle Scholar
  116. E.E. Salpeter, On Mayer’s theory of cluster expansions. Ann. Phys. 5(3), 183–223 (1958)ADSMathSciNetzbMATHGoogle Scholar
  117. V. Silin, On relaxation of electron and ion temperatures of fully ionized plasma in a strong magnetic field. SOVIET PHYSICS JETP 16(4), 1281–1285 (1963)ADSMathSciNetGoogle Scholar
  118. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer Science & Business Media, Heidelberg, 2012)zbMATHGoogle Scholar
  119. J.L. Tennyson, J.D. Meiss, P.J. Morrison, Self-consistent chaos in the beam-plasma instability. Phys. D 71, 1–17 (1994)MathSciNetzbMATHGoogle Scholar
  120. K. Theilhaber, G. Laval, D. Pesme, Numerical simulations of turbulent trapping in the weak beam-plasma instability. Phys. Fluids 30(10), 3129–3149 (1987)ADSGoogle Scholar
  121. S.F. Tigik, L.F. Ziebell, P.H. Yoon, Collisional damping rates for plasma waves. Phys. Plasmas 23(6), 064504 (2016)ADSGoogle Scholar
  122. S.I. Tsunoda, F. Doveil, J.H. Malmberg, Experimental test of the quasilinear theory of the interaction between a weak warm electron beam and a spectrum of waves. Phys. Rev. Lett. 58, 1112–1115 (1987a)ADSGoogle Scholar
  123. S.I. Tsunoda, F. Doveil, J.H. Malmberg, An experimental test of quasilinear theory. Phys. Scr. 40, 204–205 (1987b)ADSGoogle Scholar
  124. S.I. Tsunoda, F. Doveil, J.H. Malmberg, Experimental test of quasilinear theory. Phys. Fluids B 3, 2747–2757 (1991)ADSGoogle Scholar
  125. A.A. Vedenov, E.P. Velikhov, R.Z. Sagdeev, Quasilinear theory of plasma oscillations. Nuclear Fusion Suppl. 2, 465–475 (1962)Google Scholar
  126. A. Volokitin, C. Krafft, Velocity diffusion in plasma waves excited by electron beams. Plasma Phys. Control. Fusion 54(8), 085002 (2012)ADSGoogle Scholar
  127. P.H. Yoon, L.F. Ziebell, E.P. Kontar, R. Schlickeiser, Weak turbulence theory for collisional plasmas. Phys. Rev. E 93(3), 033203 (2016)ADSMathSciNetGoogle Scholar

Copyright information

© Division of Plasma Physics, Association of Asia Pacific Physical Societies 2018

Authors and Affiliations

  • D. F. Escande
    • 1
    Email author
  • D. Bénisti
    • 2
  • Y. Elskens
    • 1
  • D. Zarzoso
    • 1
  • F. Doveil
    • 1
  1. 1.Aix-Marseille Université, CNRS, PIIMMarseilleFrance
  2. 2.CEA, DAM, DIFArpajonFrance

Personalised recommendations