Abstract
The coagulation-fragmentation process models the stochastic evolution of a population of N particles distributed into groups of different sizes that coagulate and fragment at given rates. The process arises in a variety of contexts and has been intensively studied for a long time. As a result, different approximations to the model were suggested. Our paper deals with the exact model which is viewed as a time-homogeneous interacting particle system on the state space Ω N, the set of all partitions of N. We obtain the stationary distribution (invariant measure) on Ω N for the whole class of reversible coagulation-fragmentation processes, and derive explicit expressions for important functionals of this measure, in particular, the expected numbers of groups of all sizes at the steady state. We also establish a characterization of the transition rates that guarantee the reversibility of the process. Finally, we make a comparative study of our exact solution and the approximation given by the steady-state solution of the coagulation-fragmentation integral equation, which is known in the literature. We show that in some cases the latter approximation can considerably deviate from the exact solution.
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REFERENCES
Aizenman, M., and Bak, T. A. (1979). Convergence to equilibrium in a system of reacting polymers. Commun. Math. Phys. 65, 203–230.
Aldous, D. (1997). Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists. Preprint.
Aldous, D. (1998). Stochastic coalescence. Documenta Mathematica, Proc. Int'l. Congress of Mathematicians, Vol. III, Invited Lectures, Berlin, pp. 205–211.
Anderson, W. (1990). Continuous-Time Markov Chains, Springer-Verlag, New York.
Ball, J. M., Carr, J., and Penrose, O. (1986). The Becker-Döring cluster equations: Basic properties and asymptotic behavior of solutions. Commun. Math. Phys. 104, 657–692.
Barrow, J. D. (1981). Coagulation with fragmentation. J. Phys. A: Math. Gen. 14, 729–733.
Beguin, M., Gray, L., and Ycart, B. (1998). The load transfer model. Am. Appl. Prob. 8(2), 337–353.
Bender, C. M., and Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York.
Drake, R. L. (1972). A general mathematical survey of the coagulation equation. Topics in Current Aerosol Res., Part 2, 3, 201–376.
Dubovskii, P. B., and Stewart, I. W. (1995). The order of singularity of solutions for the stationary coagulation equation. Appl. Math. lett. 8, 17–20.
Durrett, R. (1995). Ten lectures on particle systems. Springer, New York, Lecture Notes in Math. 1608, 97–201.
Ernst, M. H. (1986). Kinetics of clustering in irreversible aggregation. Fractals in Physics, pp. 289–302.
Ewens, W. J. (1995). Remarks on the law of succession. Lecture Notes in Statist. 114, 229–244.
Gueron, S. The steady-state distributions of coagulation-fragmentation processes. J. Math. Biol. (in press).
Gueron, S., and Levin, S. A. (1995). The dynamics of group formation. Math. Biosci. 128, 243–264.
Hendriks, E. M., Ernst, M. H., and Ziff, R. M. (1983). Coagulation equations with gelation. J. Statist. Phys. 31, 519–563.
Hidy, G. M., and Brock, J. R. (eds.) (1972). Topics in current aerosol research. Int. Rev. in Aerosol Phys. and Chem., Pergamon Press, Oxford.
Jeon, I. (1997). Gelation phenomena. Preprint.
Kelly, F. (1979). Reversibility and Stochastic Networks, Wiley.
McLeod, J. H. (1962). On an infinite set of nonlinear differential equations I. Quart. J. Math. Oxford 13(2), 119–128.
Liggett, T. (1985). Interacting Particle Systems, New York.
Morgan, B. J. T. (1976). Stochastic models of grouping changes. Adv. Appl. Prob. 8, 30–57.
Okubo, A. (1986). Dynamical aspects of minimal grouping; swarms, schools, flocks and herds. Adv. Biophys. 22, 1–94.
Penrose, O. (1989). Metastable states for the Becker-Döring cluste equations. Commun. Math. Phys. 124, 515–541.
Penrose, O. (1997). The Becker-Döring equations at large times and their connection with the LSW theory of coarsening. J. Statist. Phys. 89(1/2), 305–320.
Pommerenke, C. (1975). Univalent Functions, Göttingen.
Spouge, J. L. (1984). An existence theorem for the discrete coagulation-fragmentation equations. Math. Proc. Camb. Phil. Soc. 96, 351–356.
Stanley, R. (1986). Enumerative Combinatorics, Wadsworth and Brooks, California.
Stewart, I. W. (1989). A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11, 627–648.
Stewart, I. W. A uniqueness theorem for the coagulation-fragmentation equation. Math. Proc. Camb. Phil. Soc. 107, 573–578.
Stewart, I. W. (1991). Density conservation for a coagulation equation. J. Appl. Math. Phys. 42, 746–756.
Stewart, I. W., and Dubovskii, P. B. (1996). Approach to equilibrium for the coagulation fragmentation equation via a Lyapunov functional. Math. Methods. Appl. Sci. 19, 171–184.
Thompson, C. J. (1988). Classical Equilibrium Statistical Mechanics, Oxford University Press, Oxford.
van Dongen, P. G. J., and Ernst, M. H. (1984). Kinetics of reversible polymerization. J. Stat. Phys. 37, 301–324.
van Dongen, P. G. V., and Ernst, M. H. (1985). Cluster size distribution in irreversible aggregation at large times. J. Phys. A: Math. Gen. 18, 2779–2793.
White, W. H. (1980). A global existence theorem for Smoluchowski's coagulation equations. Proc. Amer. Math. Soc. 80, 273–276.
Whittle, P. (1986). Systems in Stochastic Equilibrium, Wiley.
Ziff, R. M., and McGrady, A. A. (1985). The kinetics of cluster fragmentation and depolymerization. J. Phys. A 18, 3027–3037.
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Durrett, R., Granovsky, B.L. & Gueron, S. The Equilibrium Behavior of Reversible Coagulation-Fragmentation Processes. Journal of Theoretical Probability 12, 447–474 (1999). https://doi.org/10.1023/A:1021682212351
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DOI: https://doi.org/10.1023/A:1021682212351