Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-30T05:06:13.684Z Has data issue: false hasContentIssue false

Some aspects of the Smoluchowski process

Published online by Cambridge University Press:  14 July 2016

Philip McDunnough*
Affiliation:
University of Toronto

Abstract

This article deals with the generalized Smoluchowski process, {n(t), t ≧ 0}, defined by the temporal fluctuating of the numbers of randomly moving particles contained in m < ∞ disjoint regions of space. The relationship of the Smoluchowski process {n(t), t ≧ 0} to the emigration–immigration process is discussed and conditions for their equivalence are presented.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1978 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aczél, J. (1966) Lectures on Functional Equations and their Applications. Academic Press, New York.Google Scholar
Bartlett, M. S. (1949) Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211229.Google Scholar
Bartlett, M. S. (1950) Recurrence times. Nature (London) 165, 727.Google Scholar
Bartlett, M. S. (1966) Introduction to Stochastic Processes , 2nd edn. Cambridge University Press.Google Scholar
Brillinger, D. R. (1975) Time Series, Data Analysis and Theory. Holt, Rinehart and Winston, New York.Google Scholar
Chandrasekar, S. (1943) Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 189. Republished in Selected Papers on Noise and Stochastic Processes (1954). Dover, New York.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Fürth, R. (1918) Statistik und Wahrscheinlichkeitsnachwirkung. Phys. Z. 19, 421426.Google Scholar
Fürth, R. (1919) Statistik und Wahrscheinlichkeitsnachwirking. Phys. Z. 20, 21.Google Scholar
Heyde, C. C. and Seneta, E. (1972) Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Prob. 9, 235256.Google Scholar
Heyde, C. C. and Seneta, E. (1974) Notes on ‘Estimation theory for growth and immigration rates in a multiplicative process’. J. Appl. Prob. 11, 572577.Google Scholar
Kac, M. (1959) Probability and Related Topics in Physical Sciences. Interscience, London.Google Scholar
Lewis, P. A. W., (Ed.) (1972) Stochastic Point Processes: Statistical Analysis, Theory and Applications. Wiley-Interscience, New York.Google Scholar
Lindley, D. V. (1954) The estimation of velocity distributions from counts. Proc. Int. Congress Mathematicians (Amsterdam) 3, 427444.Google Scholar
Mcdunnough, P. (1977) The Smoluchowski Process in Statistical Physics and Related Topics. Ph.D. Thesis, McGill University, Montreal.Google Scholar
Mahamunulu, D. M. (1967) A note on regression in the multivariate Poisson distribution. J. Amer. Statist. Assoc. 62, 251258.Google Scholar
Patil, V. T. (1957) The consistency and adequacy of the Poisson–Markoff model for density fluctuations. Biometrika 44, 4356.Google Scholar
Ruben, H. (1962) Some aspects of the emigration-immigration process. Ann. Math. Statist. 33, 111129.Google Scholar
Ruben, H. (1964) Generalized concentration fluctuations under diffusion equilibrium. J. Appl. Prob. 1, 4768.Google Scholar
Ruben, H. and Rothschild, Lord (1953) Estimation of mean speeds of organisms and particles by counting. Unpublished.Google Scholar
Smoluchowski, M. V. (1916) Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys. Z. 17, 557585.Google Scholar
Svedberg, T. (1912). Die Existenz der Molekule. Leipzig.Google Scholar
Westgren, A. (1916) Die Veränderungsgeschwindigkeit der lokalen Teilchenkonzentration in kolloiden Systemen. (Erste Mitteilung.) Ark. Mat. Astronom. Fys. Band 11, no. 14; (Zweite Mitteilung (1918) Band 13, No. 14.).Google Scholar