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Coagulation Processes with Gibbsian Time Evolution

Part of: Markov processes Time-dependent statistical mechanics (dynamic and nonequilibrium) Combinatorics

Published online by Cambridge University Press:  04 February 2016

Boris L. Granovsky  and
Alexander V. Kryvoshaev
Boris L. Granovsky*
Affiliation:
Technion - Israel Institute of Technology
Alexander V. Kryvoshaev*
Affiliation:
Technion - Israel Institute of Technology
*
∗ Postal address: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel.
∗ Postal address: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel.
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Abstract

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We prove that a stochastic process of pure coagulation has at any time t ≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ(i, j) of single coagulations are of the form ψ(i; j) = if(j) + jf(i), where f is an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the function f. For the three corresponding models, we study the probability of coagulation into one giant cluster by time t > 0.

Keywords

Stochastic process of coagulationtime dynamicsGibbs distribution

MSC classification

Primary: 82C23: Exactly solvable dynamic models
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 05A18: Partitions of sets
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 612 - 626
Copyright
© Applied Probability Trust 

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