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On the wedding of certain dynamical processes in disordered complex materials to the theory of stable (Lévy) distribution functions

  • Elliott W. Montroll
  • Michael F. Shlesinger
Conference Lectures
Part of the Lecture Notes in Mathematics book series (LNM, volume 1035)

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.

Keywords

Dielectric Relaxation Stable Distribution Deep Trap Dispersive Transport Xerox Corporation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Elliott W. Montroll
    • 1
    • 2
  • Michael F. Shlesinger
    • 1
    • 2
  1. 1.Institute for Physical Science and TechnologyUniversity of MarylandCollege Park
  2. 2.La Jolla InstituteLa Jolla

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