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A nucleation–growth process on the integers

Published online by Cambridge University Press:  01 July 2016

Bernhard Mellein
Bernhard Mellein*
Affiliation:
Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, La Plata, Argentina
*
∗ Present address: SANDOZ Ltd, Clinical Research, CH-4002 Basle, Switzerland.
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Abstract

We consider a Markov process with state space {0, 1}Z where Os become 1s irreversibly at rates which depend on whether none of a 0&s nearest neighbours (nucleation), its left-hand neighbour (right-hand growth), or its right-hand neighbour (left-hand growth) is in the 1-state. Furthermore, we assume that Os with both nearest neighbours in the 1-state remain in the 0-state forever and that at time 0 only the origin is in the 1-state. We determine the size distribution of the cluster (maximal sequence of 1s uninterrupted by Os) at the origin and the distribution of the time when its growth is stopped (by birth (nucleation) or competitive growth of neighbouring clusters). In the final state of the process, the spatial distribution of the (trapped) sites in the 0-state is considered. Some information on the size distribution of clusters well away from the cluster at the origin is obtained.

Keywords

IRREVERSIBLE REACTION PROCESSCLUSTER-SIZE DISTRIBUTIONTRAPPED SITESGENERATING FUNCTION

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 1 , March 1987 , pp. 26 - 43
Copyright
Copyright © Applied Probability Trust 1987 

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References

Abramowitz, M. and Stegun, I. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Barron, T. H. K., Bawden, R. J., and Boucher, E. A. (1974) Distribution of occupied sequences in one-dimensional arrays. J. Chem. Soc. Faraday Trans. II 70, 651–664.Google Scholar
Blaisdell, B. E. and Solomon, H. (1970) On random sequential packing in the plane and a conjecture of Palásti. J. Appl. Prob. 7, 667–698.Google Scholar
Boucher, E. A. (1972) Reaction kinetics of polymer substituents: Neighbouring-substituent effects in single-substituent reactions. J. Chem. Soc. Faraday Trans. I 68, 2295–2304.CrossRefGoogle Scholar
Downton, F. (1961) A note on vacancies on a line. J. R. Statist. Soc. B 23, 207–214.Google Scholar
Durrett, R. (1980) On the growth of one-dimensional contact processes. Ann. Prob. 8, 890–907.Google Scholar
Durrett, R. and Liggett, T. M. (1981) The shape of the limit set in Richardson&s growth model. Ann. Prob. 9, 186–193.Google Scholar
Evans, J. W. and Nord, R. S. (1985) Cluster-size distributions for irreversible cooperative filling of lattices. II. Exact one-dimensional results for noncoalescing clusters. Phys. Rev. A 31, 3831–3840.Google Scholar
Gonzalez, J. J., Hemmer, P. C., and Høye, J. S. (1974) Cooperative effects in random sequential polymer reactions. Chem. Phys. 3, 228–238.Google Scholar
Griffeath, D. (1981) The basic contact process. Stoch. Proc. Appl. 11, 151–185.Google Scholar
Harris, T. E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969–988.Google Scholar
Kamke, E. (1959) Differentialgleichungen, Lösungsmethoden und Lösungen, I. Chelsea, New York.Google Scholar
Mcquarrie, D. A. (1967) Stochastic approach to chemical kinetics. J. Appl. Prob. 4, 413–478.Google Scholar
Mellein, B. (1986) R-mer filling with general range R cooperative effects. J. Math. Phys. 27, 1839–1851.Google Scholar
Milchev, A. and Tsakova, V. (1985) Probabilistic aspects of mercury electrodeposition on a platinum single crystal cathode I. Electrochim. Acta 30, 133–142.Google Scholar
Nord, R. S., Hoffman, D. K., and Evans, J. W. (1985) Cluster-size distributions for irreversible cooperative filling of lattices. I. Exact one-dimensional results for coalescing clusters. Phys. Rev. A 31, 3820–3830.CrossRefGoogle ScholarPubMed
Olson, W. H. (1978) A Markov chain model for the kinetics of reactant isolation. J. Appl. Prob. 15, 835–841.Google Scholar
Page, E. S. (1959) The distribution of vacancies on a line. J. R. Statist. Soc. B 21, 364–374.Google Scholar
Parisi, G. and Zhang, Y-C., (1984) Eden model in many dimensions. Phys. Rev. Lett. 53, 1791–1794.Google Scholar
Platé, N. A., Litmanovich, A. D., Noah, O. V., Toom, A. L., and Vasilyev, M. V. (1974) Effect of neighboring groups in macromolecular reactions: distribution of units. J. Polymer Sci.: Polymer Chemistry Edition 12, 1265–1285.Google Scholar
Rawlings, P. K. and Schneider, F. W. (1974) Kinetics of helix-coil transitions for large perturbations according to the Zimm-Bragg model. Ber. Bunsenges. Physik. Chem. 78, 773–781.CrossRefGoogle Scholar
Schürger, K. (1981) A class of branching processes on a lattice with interactions. Adv. Appl. Prob. 13, 14–39.Google Scholar
Silberberg, A. and Simha, R. (1968) Kinetics of reversible reactions on linear lattices with neighbor effects. Biopolymers 6, 479–490.Google Scholar
Tsuchiya, T. and Szabo, A. (1982) Cooperative binding of n-mers with steric hindrance to finite and infinite one-dimensional lattices. Biopolymers 21, 979–994.Google Scholar
Williams, D. E., Westcott, C., and Fleischmann, M. (1985) Stochastic models of pitting corrosion of stainless steels. I. Modeling of the initiation and growth of pits at constant potential. J. Electrochem. Soc. 132, 1796–1804.Google Scholar