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The lifetimes of configuration states in statistical physics

Published online by Cambridge University Press:  01 July 2016

Harold Ruben
Harold Ruben*
Affiliation:
McGill University
*
∗Postal address: Department of Mathematics, Burnside Hall, McGill University, 805 Sherbrooke St. W., Montreal, Canada H3A 2K6.
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Abstract

The moments and probability distribution of the lifetime of a configuration state relative to m disjoint regions in ℝd for particles under stochastic motion are expressed in terms of the derivatives at the origin of the probability after-effects for the m regions and for the union of the regions, together with the single integrated and the m randomized first-passage-time distributions, relative to the union of the m regions and to the separate complements of the m regions, respectively. The lifetime, suitably normed in terms of the mean lifetime, is shown to have a limiting standard exponential distribution. Finally, the distributions of lifetime when the motion of the particles is either Brownian or such as to generate a persistent generalized Smoluchowski process are discussed; in the first case, the distribution of lifetime reduces to a standard problem in heat conduction, and in the second case the distribution is expressed in terms of an exponential function and the m probability after-effects for the m regions.

Keywords

GENERALIZED SMOLUCHOWSKI PROCESSLIFETIMES AND RESIDUAL LIFETIMES OF CONFIGURATION STATESPROBABILITY AFTER-EFFECTSINTEGRATED AND RANDOMIZED FIRST-PASSAGE-TIME DISTRIBUTIONSPALM–KHINTCHINE RELATIONS FOR ALTERNATING POINT PROCESSESLIMITING EXPONENTIAL DISTRIBUTIONBROWNIAN–SMOLUCHOWSKI AND PERSISTENT PROCESSESHEAT CONDUCTION
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 704 - 719
Copyright
Copyright © Applied Probability Trust 1981 

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