Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-20T20:22:30.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Modified Vlasov equation and dispersion relations for a relativistic radiating electron plasma

Part of: Focus on Plasma Astrophysics

Published online by Cambridge University Press:  20 June 2019

Juan F. García-Camacho ,
Gonzalo Ares de Parga Open the ORCID record for Gonzalo Ares de Parga [Opens in a new window]  and
Dionisio Tun
Juan F. García-Camacho
Affiliation:
Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, U. P. Adolfo López Mateos, Zacatenco, C.P. 07738, México City, México
Gonzalo Ares de Parga*
Affiliation:
Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, U. P. Adolfo López Mateos, Zacatenco, C.P. 07738, México City, México
Dionisio Tun
Affiliation:
Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, U. P. Adolfo López Mateos, Zacatenco, C.P. 07738, México City, México
*
†Email address for correspondence: gadpau@hotmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A modified Vlasov equation is obtained by developing a covariant statistical mechanics for a system of electrons without considering the effects of the ions and including the Landau–Lifshitz equation of motion. General dispersion relations for the transverse and longitudinal modes for any temperature are expressed. The results are similar to those found by Hakim & Mangeney (Phys. Fluids, vol. 14, 1971, pp. 2751–2781) for both the modified Vlasov equation and the dispersion relations. However, for the longitudinal mode, unlike the development of Hakim and Mangeney, correct expansions are done in order to give a numerical approach to obtain the longitudinal relativistic dispersion relations for any value of the wavenumber. Accordingly, new loop solutions, with turning points, crossing the super-luminous region and the super-thermal region are found. Although the expressions for the Landau damping and the damping due to the radiation reaction force coincide with the Hakim and Mangeney results for some particular cases, in general they are different. A Landau anti-damping appears in the second branch of the loop in a small region between the cutoff point and the intersection with the super-thermal line. The analysis of this effect leads us to a kind of wave pulse. We will call them bipolar waves. The treatment contains the relativistic interactions between all the electrons in the system with retarded effects. This explain the differences with Zhang’s recent work (Phys. Plasmas, vol. 20, 2013, 092112–092132). It is shown that for low densities, the cutoff of the wave is due to the dispersion relations and not due to the radiation reaction force damping. While for both high densities and temperatures, the damping due to the radiation reaction force is important.

Keywords

plasma waves
Type
Research Article
Information
Journal of Plasma Physics , Volume 85 , Issue 3 , June 2019 , 905850307
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ares de Parga, G. 2006 A physical deduction of an equivalent Landau–Lifshitz equation of motion in classical electrodynamics. A new expression for the large distance radiation rate of energy. Foundations Phys. 36, 101474–101510.Google Scholar
Ares de Parga, G. & López-Carrera, B. 2011 Relativistic statistical mechanics vs relativistic thermodynamics. Entropy 13, 1664–1693.Google Scholar
Ares de Parga, G., López-Carrera, B. & Angulo-Brown, F. 2005a A proposal for relativistic transformations in thermodynamics. J. Phys. A 38, 2821–2834.Google Scholar
Ares de Parga, G., Mares, R. & Domínguez-Hernández, S. 2005b Landau–Lifshitz equation of motion for a charged particle revisited. Annales de la Fondation Louis de Broglie 30, 283–289.Google Scholar
Bogolyubov, N. N. 1946 Problems of a Dynamical Theory in Statistical Physics. State Technical Press.Google Scholar
Brown, G.2005 Some notes on the ideal fermion gas. https://home.strw.leidenuniv.nl/∼nefs/ FermionGas.pdf.Google Scholar
Connor, J. W., Hastie, R. J., Marchetto, C. & Roach, C. M. 2018 A one-dimensional tearing mode equation for pedestal stability studies in Tokamaks. J. Plasma Phys. 84, 725840301–725840321.Google Scholar
Cubero, D., Casado-Pascual, J., Dunkel, J., Talkner, P. & Hänggi, P. 2007 Thermal equilibrium and statistical thermometers in special relativity. Phys. Rev. Lett. 99, 170601–170621.Google Scholar
Di Piazza, A. 2008 Exact solution of the Landau–Lifshitz equation in a plane wave. Lett. Math. Phys. 83, 305–335.Google Scholar
Di Piazza, A., Müller, C., Hatsagortsyan, K. Z. & Keitel, C. H. 2012 Extremely high-intensity laser interactions with fundamental quantum systems. Rev. Mod. Phys. 84, 1177–1200.Google Scholar
Dirac, P. A. M. 1938 Classical theory of radiating electrons. Proc. R. Soc. Lond. A 167, 148–169.Google Scholar
Eliezer, C. J. 1948 On the classical theory of particles. Proc. R. Soc. Lond. A 194, 543–555.Google Scholar
Ford, G. W. & O’Connell, R. F. 1991 Radiation reaction in Electrodynamics and the elimination of runaway solutions. Phys. Lett. A 157, 217–232.Google Scholar
Ford, G. W. & O’Connell, R. F. 1993 Relativistic form of radiation reaction. Phys. Lett. A 174, 182–202.Google Scholar
Gamba, A. 1967 Physical quantities in different reference systems according to relativity. Am. J. Phys. 35, 83–88.Google Scholar
García-Camacho, J. F., Salinas, E., Avalos-Vargas, A. & Ares de Parga, G. 2015 Mathematical differences and physical similarities between Eliezer–Ford–O’Connell equation and Landau–Lifshitz equation. Rev. Mex. Fís. 61, 363–371.Google Scholar
Hakim, R. & Mangeney, A. 1968 Relativistic kinetic equations including radiations effects. I. Vlasov approximation. J. Math. Phys. 9, 116–148.Google Scholar
Hakim, R. & Mangeney, A. 1971 Collective oscillations of a relativistic radiating electron plasma. Phys. Fluids 14, 2751–2781.Google Scholar
Hakim, R. I 1967 Remarks on relativistic statistical mechanics. I. J. Math. Phys. 8, 1315–1340.Google Scholar
Hakim, R. II 1967 Remarks on relativistic statistical mechanics. II. Hierarchies for the reduced densities. J. Math. Phys. 18, 1379–1399.Google Scholar
Hammond, R. T. 2010a Relativistic particle motion and radiation reaction in electrodynamics. Electron. J. Theor. Phys. 7, 221–258.Google Scholar
Hammond, R. T. 2010b Radiation reaction at ultrahigh intensities. Phys. Rev. A 81, 062104–062124.Google Scholar
Jackson, J. D. 1998 Classical Electrodynamics, 3rd edn. John Wiley and Sons.Google Scholar
Jovanović, M. S. & Dimitrijević, D. R. 2013 Nonlinear dispersion relation of ultra-relativistic plasma waves. Plasma Phys. 10, 1514–1536.Google Scholar
Jüttner, F. 1911 Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie. Annalen der Physik 34, 856–889.Google Scholar
Klimontovich, Y. L. 1960 Relativistic transport equations for a plasma. I. Sov. Phys. JETP 37, 524–544.Google Scholar
Krall, N. A. & Trivelpiece, A. W. 1973 Principles of Plasma Physics. McGraw-Hill.Google Scholar
Kravets, Y., Noble, A. & Jaroszynski, D. 2013 Radiation reaction effects on the interaction of an electron with an intense laser pulse. Phys. Rev. E 88, 011201–011232.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1962 The Classical Theory of Fields, 3rd revised english edn. Pergamon Press.Google Scholar
López-Carrera, B. & Ares de Parga, G. 2008 Relativistic transformation of the canonical distribution function in relativistic statistical mechanics. Physica A 387, 1099–1109.Google Scholar
Magneville, A. 1990 Plasma waves in hot relativistic beam-plasma systems. Part 1. Dispersion relations. J. Plasma Phys. 44, 191–211.Google Scholar
Nakamura, T. K. 2006 Relativistic energy-momentum of a body with a finite volume. Space Sci. Rev. 122, 271–278.Google Scholar
Pesch, T. C. & Kull, H. 2010 Dispersion relations of nonlinearly coupled electromagnetic and electrostatic waves in relativistic plasmas. Phys. Plasmas 17, 012305–012335.Google Scholar
Phillips, A. C. 1998 The physics of stars. In The Manchester Physics Series (ed. Ellison, R. J., Mandl, F. & Sandiford, D. J.). John Wiley & Sons.Google Scholar
Prentice, A. J. R. 1968 Dispersion relations in relativistic Vlasov plasmas. Phys. Fluids 11, 1036–1058.Google Scholar
Riviere, A. C. 1971 Penetration of fast hydrogen atoms into a fusion reactor. Nucl. Fusion 11, 363–369.Google Scholar
Rohrlich, F. 2000 The self-force and radiation reaction. Am. J. Phys. 68, 1109–1128.Google Scholar
Rohrlich, F. 2001 The correct equation of motion of a classical point charge. Phys. Lett. A 283, 276–298.Google Scholar
Shen, C. S. 1972 Magnetic Bremsstrahlung in an intense magnetic field. Phys. Rev. D 6, 2736–2771.Google Scholar
Spohn, H. 2000 The critical manifold of the Lorentz–Dirac equation. Europhys. Lett. 50, 287–315.Google Scholar
Synge, J. L. 1956 The Relativistic Gas. North-Holland Publishing Company.Google Scholar
Zhang, H., Wu, S. Z., Zhou, C. T., Zhu, S. P. & He, X. T. 2013 Study on longitudinal dispersion relation in one-dimensional relativistic plasma: linear theory and Vlasov simulation. Phys. Plasmas 20, 092112–092132.Google Scholar