Approach to Equilibrium of Electrons, Plasmons, and Phonons in Quantum and Classical Plasmas

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Abstract
The approach to equilibrium of electrons, plasmons, and phonons in finite-temperature plasmas is studied in the random phase approximation. It is first shown that for an electron plasma in equilibrium, the long-wavelength, well-defined, plasmons contribute a term to the free energy which is appropriate to a collection of independent bosons. In order to study nonequilibrium processes, an explicit plasmon distribution function is introduced. The matrix element for plasmon-electron coupling is calculated in the random phase approximation, and second-order perturbation theory is used to write down the equations which couple the electron and plasmon distribution functions. Equilibrium is shown to result from the competition between the spontaneous emission of plasma waves by single fast electrons and the Landau damping of the plasma waves due to the same group of particles. The equation for the time rate of change of the electron distribution function reduces, in the classical limit, to a Fokker-Planck equation in which there appear diffusion and friction terms associated with plasma waves, of the type first considered by Klimontovitch. When an initial arbitrary nonequilibrium electron distribution is considered, it is seen that a plasma wave instability corresponds to coherent excitation of plasma waves by the electrons in contrast to the incoherent excitation associated with spontaneous emission. The method is generalized to two-component electron-ion plasma in which well defined acoustic plasma waves exist, by introducing an explicit distribution function for the phonons (the acoustic plasmons). The approach to equilibrium and the two-stream instability are derived and discussed for the coupled electron-phonon system; results similar to those obtained for the electron-plasmon system are found.