Abstract
We review and comment on styles of applied mathematics before exhibiting our own in regard to relating the theory of Lévy's stable distributions to dynamic processes in complex disordered materials. Lévy's probability distributions have long tails, infinite moments and elegant scaling properties. Our first example connects intermittant currents in certain xerographic films to a Lévy distribution of waiting times for the jumping of charges out of a distribution of deep traps. We then extend our analysis from transport to electron-hole recombination reactions in amorphous materials. A Lévy distribution of first passage times appears both in this recombination problem as well as in the dielectric relaxation phenomena described by the Williams-Watts formula. Lastly, the most famous scaling problem, "1/f noise", is shown to be related to a log-normal distribution of relaxation times. We derive the log-normal distribution in a generic fashion and show it to be a limiting form of a Lévy distribution.
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Montroll, E.W., Shlesinger, M.F. (1983). On the wedding of certain dynamical processes in disordered complex materials to the theory of stable (Lévy) distribution functions. In: Hughes, B.D., Ninham, B.W. (eds) The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation. Lecture Notes in Mathematics, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073256
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DOI: https://doi.org/10.1007/BFb0073256
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