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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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References

Bartlett, M. S. (1953) Recurrence and first passage times. Proc. Camb. Phil. Soc. 49, 263–275.CrossRefGoogle Scholar
Belayev, Yu. K. (1968) On the number of exits across the boundary of a region by a vector stochastic process. Theory Prob. Appl. 13, 320–324.CrossRefGoogle Scholar
Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89. Republished in Selected Papers on Noise and Stochastic Processes (1954) Dover, New York.CrossRefGoogle Scholar
Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Kac, M. (1959) Probability and Related Topics in Physical Sciences. Interscience, London.Google Scholar
Kac, M. and Slepian, D. (1959) Large excursions of Gaussian processes. Ann. Math. Statist. 30, 1215–1228.CrossRefGoogle Scholar
Khintchine, A. Y. (1960) Mathematical Methods in the Theory of Queueing. Griffin, London.Google Scholar
Lawrance, A. J. (1974) Theory of interval distributions for stationary point processes. Information and Control 25, 299–316.CrossRefGoogle Scholar
Leadbetter, M. R. (1966) On streams of events and mixtures of streams. J. R. Statist. Soc. B 28, 218–227.Google Scholar
Leadbetter, M. R. (1969) On the distributions of the times between events in a stationary stream of events. J. R. Statist. Soc. B 31, 295–302.Google Scholar
McDunnough, P. (1978) Some aspects of the Smoluchowski process. J. Appl. Prob. 15, 663–674.CrossRefGoogle Scholar
McDunnough, P. (1979) Estimating the law of randomly moving particles by counting. J. Appl. Prob. 16, 25–35.CrossRefGoogle Scholar
Ruben, H. (1964) Generalized concentration fluctuations under diffusion equilibrium. J. Appl. Prob. 1, 47–68.CrossRefGoogle Scholar
Ruben, H. and Rothschild, Lord (1953) Estimation of mean speeds of organisms and particles by counting. Unpublished.Google Scholar
Smoluchowski, M. V. (1915) Molekulartheoretische Studien über Umkehr thermodynamische irreversibler Vorgänge und über Wiederkehr abnormaler Zustände. S. B. Akad. Wiss. Wien (2a) 124, 339–368.Google Scholar
Smoluchowski, M. V. (1916) Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys. Z. 17, 557–571, 587–599.Google Scholar
Ylvisaker, N. D. (1965) The expected number of zeros of a stationary stochastic process. Ann. Math. Statist. 36, 1043–1046.CrossRefGoogle Scholar