Stochastic derivation of the switching function in the theory of microwave heating of ceramic materials

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Abstract
Stochastic arguments are presented to provide a fundamental derivation of a switching function introduced in a recent theory of microwave heating of ceramic materials. The phenomenon under study is thermal runaway wherein the temperature of a ceramic material increases suddenly after being heated under microwaves for a certain time. Observations in a wide variety of materials have been explained successfully via a recently constructed theory of the pheonomenon, which suggests that the mobility of absorbing entities, such as vacancies, bivacancies, or interstitials, has a temperature dependence that follows a switching behavior: The mobility, equivalently the absorption coefficient, starts at small values for small temperatures but rises quickly to high values as the temperature is increased. Phase-space considerations and static arguments have been given earlier to support the idea, and to calculate the detail, of the switching function. Here we provide a basic justification to the static arguments through a consideration of the dynamics. Although the evolution of the absorbing entities is governed by a nonlinear Langevin equation, which cannot be solved exactly, natural approximations are shown to lead to a Smoluchowski equation for the probability distribution, which can be solved. General results and specific calculational prescriptions are provided for a variety of potential shapes.

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