The Boltzman Equation Theory of Charged Particle Transport

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.

Download Full-text

Boltzmann Equation Theory of Charged Particle Transport in Neutral Gases: Perturbation Treatment

Australian Journal of Physics ◽

10.1071/ph99053 ◽

1999 ◽

Vol 52 (6) ◽

pp. 999 ◽

Cited By ~ 10

Author(s):

Slobodan B. Vrhovac ◽

Zoran Lj. Petrovic

Keyword(s):

Distribution Function ◽

Perturbation Theory ◽

Charged Particle ◽

Boltzmann Equation ◽

Particle Transport ◽

Kinetic Equations ◽

Initial Distribution ◽

Formal Structure ◽

Time Dependent ◽

Charged Particle Transport

This paper examines the formal structure of the Boltzmann equation (BE) theory of charged particle transport in neutral gases. The initial value problem of the BE is studied by using perturbation theory generalised to non-Hermitian operators. The method developed by R�sibois was generalised in order to be applied for the derivation of the transport coecients of swarms of charged particles in gases. We reveal which intrinsic properties of the operators occurring in the kinetic equation are sucient for the generalised diffusion equation (GDE) and the density gradient expansion to be valid. Explicit expressions for transport coecients from the (asymmetric) eigenvalue problem are also deduced. We demonstrate the equivalence between these microscopic expressions and the hierarchy of kinetic equations. The establishment of the hydrodynamic regime is further analysed by using the time-dependent perturbation theory. We prove that for times t ? τ0 (τ0 is the relaxation time), the one-particle distribution function of swarm particles can be transformed into hydrodynamic form. Introducing time-dependent transport coecients ? *(p) (?q,t), which can be related to various Fourier components of the initial distribution function, we also show that for the long-time limit all ? *(p) (?q,t) become time and ?q independent in the same characteristic time and achieve their hydrodynamic values.

Download Full-text