Bernstein–Greene–Kruskal modes

Bernstein–Greene–Kruskal modes (a.k.a. BGK modes) are nonlinear electrostatic waves that propagate in an unmagnetized, collisionless plasma. They are nonlinear solutions to the Vlasov–Poisson equation in plasma physics,[1] and are named after physicists Ira B. Bernstein, John M. Greene, and Martin D. Kruskal, who solved and published the exact solution for the one-dimensional case in 1957.[2]

Not to be confused with Bhatnagar–Gross–Krook in statistical physics, also abbreviated as BGK.

BGK modes have been studied extensively in numerical simulations for two- and three-dimensional cases,[1][3][4][5] and are believed to be produced by the two-stream instability.[6][7] They have been observed as electron phase space holes (electrostatic solitary structures)[8][9][10][11] and double layers[12] in space plasmas, as well as in scattering experiments in the laboratory.[13]

Small-amplitude limit: Van Kampen modes ?

It is generally claimed that in the linear limit BGK modes (e.g. in the small amplitude approximation) reduce to what is known as Van Kampen modes,[14] named after Nico van Kampen who derived the solutions in 1955.[15]

This is wrong, however, since such a transition from a nonlinear to a linear mode does not take place even in the infinitesimal amplitude limit. A harmonic hole equilibrium of the Vlasov-Poisson system, which is correctly described as a complete solution, i.e. inclusively its phase velocity, by the Schamel method,[16] shows that nonlinearity persists even in the small amplitude limit. The area of trapped particles in the phase space never vanishes in this limit and there is no moment in which the distribution of trapped particles is transformed (or collapses) into a delta-function.[17][18][19][20] Another indication that this claim is unfounded is that such nonlinear modes prove to be unconditionally marginal stable in current-carrying plasmas regardless of the drift velocity between electrons and ions. Landau's theory, as a linear wave theory, is obviously not applicable in case of coherent waves such as BGK modes.[21][22]

Quantum BGK (QBGK) modes

BGK modes have been generalized to quantum mechanics, in which the solutions (called quantum BGK modes) solve the quantum equivalent of the Vlasov–Poisson system known as the Wigner–Poisson system, with periodic boundary conditions.[23] The solutions for the QBGK modes were put forth by Lange et al. in 1996,[24] with potential applications to quantum plasmas.[25]

References
  1. Ng, C. S.; Bhattacharjee, A. (2005). "Bernstein-Greene-Kruskal Modes in a Three-Dimensional Plasma". Physical Review Letters. 95 (24): 245004. Bibcode:2005PhRvL..95x5004N. doi:10.1103/physrevlett.95.245004. ISSN 0031-9007. PMID 16384391.
  2. Bernstein, Ira B.; Greene, John M.; Kruskal, Martin D. (1957). "Exact Nonlinear Plasma Oscillations". Physical Review. 108 (3): 546–550. Bibcode:1957PhRv..108..546B. doi:10.1103/PhysRev.108.546. hdl:2027/mdp.39015095115203.
  3. Demeio, Lucio; Holloway, James Paul (1991). "Numerical simulations of BGK modes". Journal of Plasma Physics. 46 (1): 63–84. Bibcode:1991JPlPh..46...63D. doi:10.1017/S0022377800015956. ISSN 1469-7807. S2CID 123050224.
  4. Manfredi, Giovanni; Bertrand, Pierre (2000). "Stability of Bernstein–Greene–Kruskal modes". Physics of Plasmas. 7 (6): 2425–2431. Bibcode:2000PhPl....7.2425M. doi:10.1063/1.874081. ISSN 1070-664X.
  5. Berk, H. L.; Breizman, B. N.; Candy, J.; Pekker, M.; Petviashvili, N. V. (1999). "Spontaneous hole–clump pair creation". Physics of Plasmas. 6 (8): 3102–3113. Bibcode:1999PhPl....6.3102B. doi:10.1063/1.873550. ISSN 1070-664X.
  6. Omura, Y.; Matsumoto, H.; Miyake, T.; Kojima, H. (1996). "Electron beam instabilities as generation mechanism of electrostatic solitary waves in the magnetotail". Journal of Geophysical Research: Space Physics. 101 (A2): 2685–2697. Bibcode:1996JGR...101.2685O. doi:10.1029/95ja03145. ISSN 0148-0227.
  7. Dieckmann, M. E.; Eliasson, B.; Shukla, P. K. (2004). "Streaming instabilities driven by mildly relativistic proton beams in plasmas". Physics of Plasmas. 11 (4): 1394–1401. Bibcode:2004PhPl...11.1394D. doi:10.1063/1.1649996. ISSN 1070-664X.
  8. H. Schamel, Theory of Electron Holes,Phys. Scripta 20, 336 (1979)
  9. Turikov, V. A. (1984). "Electron Phase Space Holes as Localized BGK Solutions". Physica Scripta. 30 (1): 73–77. Bibcode:1984PhyS...30...73T. doi:10.1088/0031-8949/30/1/015. ISSN 1402-4896. S2CID 250769529.
  10. Fox, W.; Porkolab, M.; Egedal, J.; Katz, N.; Le, A. (2008). "Laboratory Observation of Electron Phase-Space Holes during Magnetic Reconnection". Physical Review Letters. 101 (25): 255003. Bibcode:2008PhRvL.101y5003F. doi:10.1103/PhysRevLett.101.255003. PMID 19113719.
  11. Vasko, I. Y.; Kuzichev, I. V.; Agapitov, O. V.; Mozer, F. S.; Artemyev, A. V.; Roth, I. (2017). "Evolution of electron phase space holes in inhomogeneous plasmas". Physics of Plasmas. 24 (6): 062311. Bibcode:2017PhPl...24f2311V. doi:10.1063/1.4989717. ISSN 1070-664X.
  12. Quon, B. H.; Wong, A. Y. (1976). "Formation of Potential Double Layers in Plasmas". Physical Review Letters. 37 (21): 1393–1396. Bibcode:1976PhRvL..37.1393Q. doi:10.1103/physrevlett.37.1393. ISSN 0031-9007.
  13. Montgomery, D. S.; Focia, R. J.; Rose, H. A.; Russell, D. A.; Cobble, J. A.; Fernández, J. C.; Johnson, R. P. (2001). "Observation of Stimulated Electron-Acoustic-Wave Scattering". Physical Review Letters. 87 (15): 155001. Bibcode:2001PhRvL..87o5001M. doi:10.1103/PhysRevLett.87.155001. PMID 11580704.
  14. Chen, Francis F. (1984). Introduction to plasma physics and controlled fusion (2nd ed.). New York: Plenum Press. pp. 261–262. ISBN 0306413329. OCLC 9852700.
  15. Van Kampen, N. G. (1955). "On the theory of stationary waves in plasmas". Physica. 21 (6–10): 949–963. Bibcode:1955Phy....21..949V. doi:10.1016/S0031-8914(55)93068-8. ISSN 0031-8914.
  16. H. Schamel, Stationary solitary,snoidal and sinusoidal ion acoustic waves, Plasma Phys. 14,905 (1972)
  17. H. Schamel, Cnoidal electron hole propagation: trapping, the forgotten nonlinearity in plasma and fluid dynamics, Phys. Plasmas 19,020501 (2012).
  18. H. Schamel, Particle trapping: A key requisite of structure formation and stability of Vlasov-Poisson plasmas, Phys. Plasmas 22,042302 (2015).
  19. H. Schamel, D. Mandal, and D. Sharma, Evidence of a new class of cnoidal electron holes exhibiting intrinsic substructures,its impact on linear and (nonlinear) Vlasov theories and role in anomalous transport,Phys. Scr.95, 055601 (2020)).
  20. H. Schamel, D. Mandal, and D. Sharma, Diversity of solitary electron holes operating with non-perturbative trapping, Phys. Plasmas 27, 062302 (2020).
  21. H. Schamel, Unconditionally marginal stability of harmonic electron hole equilibria in current-driven plasmas, Phys. Plasmas 25, 062115 (2018)).
  22. H.Schamel, Pattern formation in Vlasov-Poisson plasmas beyond Landau caused by the continuous spectra of electron and ion hole equilibria, arXiv:2110.01433v2 [physics.plasm-ph].
  23. Demeio, L. (2007). "Quantum Corrections to Classical BGK Modes in Phase Space". Transport Theory and Statistical Physics. 36 (1–3): 137–158. Bibcode:2007TTSP...36..137D. doi:10.1080/00411450701456857. ISSN 0041-1450. S2CID 122915619.
  24. Lange, Horst; Toomire, Bruce; Zweifel, P. F. (1996). "Quantum BGK modes for the Wigner-poisson system". Transport Theory and Statistical Physics. 25 (6): 713–722. Bibcode:1996TTSP...25..713L. doi:10.1080/00411459608222920. ISSN 0041-1450.
  25. Haas, F.; Manfredi, G.; Feix, M. (2000). "Multistream model for quantum plasmas". Physical Review E. 62 (2): 2763–2772. arXiv:cond-mat/0203405. Bibcode:2000PhRvE..62.2763H. doi:10.1103/PhysRevE.62.2763. PMID 11088757. S2CID 42012068.


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