Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T16:03:51.857Z Has data issue: false hasContentIssue false
  • English
  • Français

Coagulation Processes with Gibbsian Time Evolution

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Markov processes 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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48.CrossRefGoogle Scholar
Arratia, R., Barbour, A. D. and Tavaré, S. (2004). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.Google Scholar
Ball, J. M., Carr, J. and Penrose, O. (1986). The Becker–Döring cluster equations: basic properties and asymptotic behaviour of solutions. Commun. Math. Phys. 104, 657–692.CrossRefGoogle Scholar
Berestycki, N. and Pitman, J. (2007). Gibbs distributions for random partitions generated by a fragmentation process. J. Statist. Phys. 127, 381–418.Google Scholar
Buffet, E. and Pulé, J. V. (1990). On Lushnikov's model of gelation. J. Statist. Phys. 58, 1041–1058.Google Scholar
Buffet, E. and Pulé, J. V. (1991). Polymers and random graphs. J. Statist. Phys. 64, 87–110.CrossRefGoogle Scholar
Durrett, R., Granovsky, B. L. and Gueron, S. (1999). The equilibrium behavior of reversible coagulation-fragmentation processes. J. Theoret. Prob. 12, 447–474.Google Scholar
Freiman, G. and Granovsky, B. (2005). Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: asymptotic formulae and limiting laws. Trans. Amer. Math. Soc. 357, 2483–2507.Google Scholar
Gnedin, A. and Pitman, J. (2006). Exchangeable Gibbs partitions and Stirling triangles. J. Math. Sci. 138, 5674–5685.Google Scholar
Goldschmidt, C., Martin, J. B. and Spanó, D. (2008). Fragmenting random permutations. Electronic Commun. Prob. 13, 461–474.Google Scholar
Granovsky, B. (2012). Asymptotics of counts of small components in random structures and models of coagulation-fragmentation. To appear in ESAIM Prob. Statist. Google Scholar
Granovsky, B. L. and Erlihson, M. M. (2009). On time dynamics of coagulation-fragmentation processes. J. Statist. Phys. 134, 567–588.Google Scholar
Han, D., Zhang, X. S. and Zheng, W. A. (2008). Subcritical, critical and supercritical size distributions in random coagulation-fragmentation processes. Acta Math. Sinica (Engl. Ser.) 24, 121–138.CrossRefGoogle Scholar
Hendriks, E. M., Spouge, J. L., Elbi, M. and Schreckenberg, M. (1985). Exact solutions for random coagulation processes. Z. Phys. B 58, 219–227.Google Scholar
Hu, C., Han, D. and Hsu, C. (2007). Critical behavior of the heterogeneous random coagulation-fragmentation processes. J. Phys. A 40, 14649–14665.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.Google Scholar
Kolchin, V. F. (1999). Random Graphs (Encyclopedia Math. Appl. 53). Cambridge University Press.Google Scholar
Lushnikov, A. A. (1977). Coagulation in finite systems. J. Colloid Interface Sci. 65, 276–285.Google Scholar
Lushnikov, A. A. (1978). Certain new aspects of the coagulation theory. Atmos. Ocean. Phys. 14, 738–743.Google Scholar
Marcus, A. H. (1968). Stochastic coalescence. Technometrics 10, 133–143.Google Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin.Google Scholar
Van Dongen, P. G. J. and Ernst, M. H. (1984). Kinetics of reversible polymerization. J. Statist. Phys. 37, 307–324.CrossRefGoogle Scholar
Vershik, A. M. (1996). Statistical mechanics of combinatorial partitions, and their limit configurations. Funct. Analysis Appl. 30, 90–105.Google Scholar
Whittle, P. (1986). Systems in Stochastic Equilibrium. John Wiley, Chichester.Google Scholar