Skip to main content

The electrodynamics and statistical mechanics of linear plasma response functions

Journal of Statistical Physics volume 1, pages 415–466 (1969)Cite this article

Abstract

The purpose of this paper is to review and to extend, wherever possible, the Kramers-Kronig relations, sum rules, and symmetry properties for the electrodynamic transport tensors of a linear plasma medium. For complete generality, we consider both nonrelativistic and relativistic plasmas with and without external magnetic fields. Our study is carried out first within the framework of classical electrodynamics. We then exploit the statistical-mechanical fluctuation-dissipation theorem to further obtain the Onsager symmetry relations and Kubo sum-rule frequency moments. Of special significance is the emergence of a variety of new Kramers-Kronig formulae andf-sum rules for the inverse dispersion tensor.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

Abbreviations

E(k,Ω):

electric field intensity

Ê(k,Ω):

electric field in absence of plasma particles,

Ě(k,Ω):

electric field due to the plasma particles (=E-Ê)

B(k,Ω):

magnetic induction

D(k,Ω):

electric induction

H(k,Ω):

magnetic field strength

B 0 :

constant external magnetic field

A 0 :

vector potential corresponding toB 0

ρ(k,Ω),j(k, co):

charge and current densities due to the plasma particles

σ(k,Ω),J(k,Ω):

charge and current densities of the external agency

ɛ(k,Ω,B 0):

dielectric tensor of the plasma medium in the presence of B0

Ν(k,Ω,B 0):

diamagnetic tensor

σ(k, co,B 0):

ε(k,Ω,B 0) − 1, electric polarizability tensor

ξ(k,Ω,B 0):

magnetic polarizability tensor

σ(k,Ω,B 0):

ordinary conductivity tensor

σ(k,Ω,B 0):

external conductivity tensor

D(k,Ω,B 0):

n2T−ε(k,Ω,B 0), dispersion tensor, where T=1-kk is the transverse projection tensor (k being the unit vector in the direction ofk) andn = kc/Ω the index of refraction

δ:

n2T − 1, = vacuum wave operator (value of D in vacuum)

σ :

1/2(σ + σ ‡), Hermitian part of σ

σ^ :

1/2(σ − σ‡), Anti-Hermitian part of a

σ′, σ″:

real and imaginary parts of a

R(r,t):

dissipated power per unit volume of plasma

U :

total energy absorbed by the plasma

R(k,Ω):

E*(k,Ω) · σ(k,Ω,b 0) ·E(k,Ω) corresponding spectral energy density

W(r,t):

1/2ε0E2(r, 0 + (l/2Μ0) B2(r,t), field energy density

W(k,Ω):

1/2ε0E*k,Ω) ·E(k,Ω) + (l/2Μ0)B *(k,Ω) · B(k,Ω), energy content in a certain domain of (k,Ω)-space for a single mode

x i,p i,v i :

coordinate, momentum, and velocity of ith electron

γ i :

[1−(Νi 2/c2)]−1/2

X j,P j,V j :

coordinate, momentum, and velocity of jth ion

{A q}, {Eq}:

field coordinates and momenta

jk′(t),J k′(t):

perturbations in the microscopic electron and ion current densities due to the presence of the small external vector potential agencyâ(r,t) = (1/L3) âk(t) expi k ·r

Ω :

Liouville distribution function = Ω0 + Ω′

Ω0 :

macrocanonical distribution function characterizing the equilibrium state of the system in the infinite past

Ω′ :

small perturbation due toA

H0 :

Hamiltonian of equilibrium system which includes interaction

H′ :

Hamiltonian for the interaction between the system and the small external perturbing agencyA

〈⋯〉0 = ∫ dγR(⋯)Ω0 :

expectation value of any quantity over the equilibrium ensemble (dγR is an element of hypervolume in γ-phase space)

G(12):

two-particle distribution function

F(1):

one-particle distribution function

g(¦x2 − x1 ¦):

[G(12)/F(1)F(2)] − 1, pair correlation function

N :

total number of electron in volume L3

n 0 :

equilibrium density (of electrons)

Β −1 :

temperature (in energy units)

Ω0 :

(n0e2/mε0)1/2, equilibrium electron plasma frequency

Ωc :

¦e ¦−B 0/m, electron frequency

κ −1 :

(ε 0/Βn0e2)1/2, Debye length

Ω 0 :

(n0Ze2/Mε0)1/2, equilibrium ion plasma frequency

Ω c :

ZeB0/M, ion cyclotron frequency

References

  1. P. Nozières and D. Pines,Nuovo Cimento 9, 470 (1958)

    Google Scholar 

  2. A. A. Rukhadze and V. P. Silin,Usp. Fiz. Nauk 74, 223 (1961); [Soviet Phys.-Usp.4, 459 (1961)];

    Google Scholar 

  3. F. Englert and R. Brout,Phys. Rev. 120, 1085 (1960).

    Google Scholar 

  4. H. Nyquist,Phys. Rev. 32, 110 (1928);

    Google Scholar 

  5. H. B. Callen and T. A. Welton,Phys. Rev. 83, 34 (1951);

    Google Scholar 

  6. R. Kubo,J. Phys. Soc. Japan 12, 570 (1957).

    Google Scholar 

  7. M. N. Rosenbluth and N. Rostoker,Phys. Fluids 5, 776 (1962).

    Google Scholar 

  8. Yu. L. Klimontovich and V. P. Silin,Soviet Phys.—JETP 15, 199 (1962).

    Google Scholar 

  9. L. D. Landau and E. M. Lifshitz,Electrodynamics of Continuous Media (Pergamon Press, London, 1960), pp. 247–250, 256–262, 347;

    Google Scholar 

  10. L. D. Landau and E. M. Lifshitz,Statistical Physics (Pergamon Press, London, 1958), pp. 391–398.

    Google Scholar 

  11. S. Ichimaru,Ann. Phys. (N.Y.) 20, 78 (1962).

    Google Scholar 

  12. R. Balescu,Statistical Mechanics of Charged Particles (Interscience Publishers, John Wiley and Sons, Ltd., London, 1963), Vol. IV, pp. 91, 98.

    Google Scholar 

  13. D. Pines,Elementary Excitations in Solids (W. A. Benjamin, Inc., New York, 1964), pp. 288–292;

    Google Scholar 

  14. D. Pines and P. Nozières,The Theory of Quantum Liquids (W. A. Benjamin, Inc., New York, 1966).

    Google Scholar 

  15. G. Kalman,The Dielectric Constant of a Plasma, Lecture Notes (Université de Paris, Faculté des Sciences, Centre d'Orsay, Laboratoire des Hautes Enérgies, 1963) TN-5, pp. 1–10.

  16. P. C. Martin,Phys. Rev. 161, 143 (1967).

    Google Scholar 

  17. J. Lindhard,Kgl. Danske Videnskab Selskab, Mat.-Fys. Medd. 28, No. 8 (1954).

  18. J. Neufeld,Phys. Rev. 123, 1 (1961);J. Appl. Phys. 34, 2549 (1963).

    Google Scholar 

  19. A. G. Sitenko,Electromagnetic Fluctuations in Plasma (Academic Press, New York, 1967).

    Google Scholar 

  20. R. Kubo, inLectures in Theoretical Physics, edited by W. E. Brittin and L. G. Dunham (Interscience Publishers, New York, 1959), Vol. I, pp. 134–143, 147, 148, 155, 156;

    Google Scholar 

  21. R. Kubo,Rep. Prog. Phys. (Institute of Physics and the Physical Society, London)29, Part I, 263–283 (1966).

    Google Scholar 

  22. E. W. Montroll and J. C. Ward,Physica 25, 423 (1959); E. W. Montroll, inTheory of Neutral and Ionized Gases, edited by C. DeWitt and J. F. Detoeuff (Université de Grenoble, Ecole d'été de physique théorique, Les Houches, 1960), pp. 95–100.

    Google Scholar 

  23. G. V. Chester,Rep. Progr. Phys. (Institute of Physics and the Physical Society, London)26, 411 (1963).

    Google Scholar 

  24. D. Bohm and D. Pines,Phys. Rev. 82, 625 (1951); Iu. L. Klimontovich,Soviet Phys.-JETP 34, 119 (1958); A. Simon,Phys. Fluids 3, 245 (1960); N. Rostoker, R. Aamodt, and O. Eldridge,Ann. Phys. (N.Y.) 31, 243 (1965); P. J. Mallozzi and H. Margenau,Ann. Phys. (N.Y.) 38, 177 (1966).

    Google Scholar 

  25. L. Onsager,Phys. Rev. 37, 405 (1931);38, 2265 (1931).

    Google Scholar 

  26. W. P. Allis, S. J. Buchsbaum, and A. Berhs,Waves in Anistropic Plasmas (MIT Press, Cambridge, Mass., 1963).

    Google Scholar 

  27. A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko, and K. N. Stepanov,Collective Oscillations in a Plasma (MIT Press, Cambridge, Mass., 1967).

    Google Scholar 

  28. A. A. Rukhadze and V. P. Silin,Usp. Fiz. Nauk 76, 79 (1962) [Soviet Phys.—Usp.5, 37 (1962)].

    Google Scholar 

  29. A. G. Sitenko and K. N. Stepanov,Soviet Phys.—JETP 4, 512 (1957).

    Google Scholar 

  30. J. H. Piddington,Monthly Notices of the Royal Astronomical Society 114, 638 and 651 (1955); T. G. Cowling,Monthly Notices of the Royal Astronomical Society 116, 114 (1956); H. K. Sen and A. A. Wyller,Radio Science 69D, 95 (1965); C. Dubs, B. Prasad, and R. Finn, “Generalization of Cowling's Conductivity for Wave Propagation in a Partially Ionized Gas,” presented at the Symposium on Electromagnetic Waves, Stresa, Italy, June 24–29, 1968.

    Google Scholar 

  31. D. Pines,The Many Body Problem (W. A. Benjamin, Inc., New York, 1963).

    Google Scholar 

  32. P. Nozières and D. Pines,Phys. Rev. 190, 741 (1958).

    Google Scholar 

  33. B. A. Trubnikov,Electromagnetic Waves in a Relativistic Plasma in a Magnetic Feld, Plasma Physics and the Problem of Controlled Thermonuclear Reactions (Pergamon Press, London, 1959), Vol. III.

    Google Scholar 

  34. B. R. A. Nijboer and L. Van Hove,Phys. Rev. 85, 777 (1952).

    Google Scholar 

  35. N. Davidson,Statistical Mechanics (McGraw-Hill Book Co., New York, 1962).

    Google Scholar 

  36. S. Ichimaru, D. Pines, and N. Rostoker,Phys. Rev. Letters 8, 231 (1962).

    Google Scholar 

  37. J. L. Synge,The Relativistic Gas (North Holland Publishing Company, Amsterdam, 1957).

    Google Scholar 

  38. G. Kaiman,Phys. Rev. 161, 156 (1967).

    Google Scholar 

  39. A. G. R. Prentice,Plasma Physics 9, 433 (1967).

    Google Scholar 

  40. J. Dawson and C. Oberman,Phys. Fluids 5, 517 (1962).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Astrophysics Institute, Brandeis University, Waltham, Massachusetts

    K. I. Golden & G. Kalman

Authors
  1. K. I. Golden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Kalman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Golden, K.I., Kalman, G. The electrodynamics and statistical mechanics of linear plasma response functions. J Stat Phys 1, 415–466 (1969). https://doi.org/10.1007/BF01106580

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01106580

Key words

  • plasmas
  • linear response functions
  • response functions
  • dielectric function
  • electrodynamics
  • Kramer-Kronig relations
  • sum rules
  • Onsager relation
  • fluctuation-dissipation theorem
  • transverse interaction