Asymptotic behaviour for a charged particle transport problem with time-varying acceleration field

Abstract

Abstract In this paper we consider the one-dimensional linear Boltzmann equation which describes the time evolution of the spatially uniform distribution f(v, t) of charged particles moving with velocity v ∊ R within a host medium under the influence of a time-varying acceleration field a = a(t). The collision frequency v(v) is assumed to be integrable, and the acceleration field is supposed to be such that there exist a constant b > 0 and a number tr for which ∫tr +∞[a(ρ)−b]dρ exists and is finite. This condition on the long time behaviour of the acceleration field characterizes the fields giving the same asymptotics of the problem. In other words, we prove that under these conditions the asymptotic dynamics of the problem give rise to charged particles that are accelerated indefinitely, i.e. we are in presence of the so-called runaway phenomenon. This result is obtained in the framework of scattering theory, proving the existence of asymptotic wave operators which guarantee that the solution of the time...