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The electrodynamics and statistical mechanics of linear plasma response functions

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.

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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

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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

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Key words

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