The Boltzman Equation Theory of Charged Particle Transport

Abstract

It is shown how a formally exact Kubo-Iike response theory equivalent to the Boltzmann equation theory of charged particle transport can be constructed. Our response theory gives· the general wavevector and time-dependent velocity distribution at any time in terms of an initial distribution function, to which is added the 'response' induced by a generalized 'perturbation' over the intervening time. The usual Kubo linear response result for the distribution function is recovered by choosing the initial velocity distribution to be Maxwellian. For completeness the response theory introduces an exponential convergence function into the 'response' time integral. This is equivalent to using a modified Boltzmann equation but the general form of the transport theory is not changed. The modified transport theory can be used to advantage where possible convergence difficulties occur in numerical solutions of the Boltzmann equation. This paper gives a systematic development of the modified transport theory and shows how our response theory fits into the broader scheme of solving the Boltzmann equation. Our discussion extends both the work of Kumar et al. (1980), where the distribution function is expanded out in terms of tensor functions pj), and the propagator description where the non-hydrodynamic time development of the distribution function is related to the wavevector dependent Green function of the Boltzmann equation.