Time-dependent solutions of transport equations

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Abstract
In transport problems (e.g. conduction in metals) involving a distribution function P, it is usual to assume that when, say, an electric field is applied, a steady state is soon established, and correspondingly we seek solutions of a steady-state transport equation in which we set ∂P/∂t = 0. However, it has already been suggested in the past (e.g. the work of Fröhlich) that dielectric breakdown can arise because electrons may find themselves in an unstable situation, gaining energy from the applied field faster than it can be dissipated through collisions with the lattice. Evidently such possibly unstable situations must depend on the functional behaviour of the relaxation time and the magnitude of the applied field. We felt it would be interesting to see whether, on an elementary basis with a simple and explicitly time-dependent transport equation, unstable [runaway] solutions could be found. This is indeed so and we have determined such solutions for a number of models. We propose a simple criterion for stability (probably too naïve for physical stability) which depends only on the assumption that scattering from any given state is a Markovian process. We notice in particular that there is a limiting form of velocity dependence of the relaxation time which in terms of our model should in itself maintain stability at all applied fields.

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