Statistical theory of radiative processes in plasmas

Abstract The methods of quantum field theory have been shown to be adaptable to the study of equilibrium and nonequilibrium statistical mechanics. Especially noteworthy in this context is the work of Prigogine and his group (1), whose advances have led to a unification in statistical physics that is interesting and useful both in nonequilibrium and equilibrium situations, and can be developed in the framework of either classical or quantum mechanics. The present article is an extension of the plasma physics work of Prigogine and Balescu (2), which allows inclusion of radiation as well as particles. Our treatment will not meet the currently fashionable interest in fully ionized plasmas inasmuch as this initial account ignores the Coulomb interaction between electrons, focusing attention chiefly upon the behavior of photons and sparse electrons in the presence of ions and neutral molecules. It does allow for a transverse index of refraction, however, which results from an indirect interaction between electrons via the radiation field. The results of this paper are applicable to slightly ionized plasmas. As will be seen, however, the work permits certain conclusions which transcend this restriction. Although we shall develop our theory within a classical framework, this is not a significant limitation since most of the formalism may be directly transcribed into quantum mechanics. In fact, the parallelism is so close that we will frequently lapse into the suggestive language of quantum mechanics, even though the expressions we describe are strictly classical. The work presented here involves certain customary approximations which restrict its scope mainly to the recovery of known results. But these flow from a unique and integral approach which exhibits connections that are not apparent in a fragmented treatment dealing separately with such problems as plasma conductivity, bremsstrahlung, scattering of microwaves, and absorption. Where novel features appear they are noted in the text. Earlier work is confined to the study of atoms, electrons, and ions. It involves standard procedures of finding Liouville's equation via the Hamiltonian of the particles, and then derives irreversible master equations and kinetic equations in terms of matrix graphs by means of perturbation techniques. We have included, along with the particle Hamiltonian, additional terms representing the radiation and its interaction with the particles and examined the numerous additional features and matrix graphs which appear, both as to mathematical form and physical meaning. While this work was in progress, preliminary reports of similar approaches have been published (3–5). Insofar as they attack the same problems, these authors seem to reach results in agreement with ours. Other related work is found in refs. 6 through 12. Our study deals with homogeneous plasmas. The treatment is carried systematically to the fourth order of the Liouville perturbation operator. Conservation principles are derived and an over-all H-Theorem is proved for the radiating plasma. Concrete results are collected in the section entitled “Applications,” which shows how the general theory entails formulas for the scattering, absorption, and conductivity of microwaves, electron-atom and electron-ion bremsstrahlung, both spontaneous and stimulated, and other effects.