## 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.

Original language | English |
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Pages (from-to) | 447-474 |

Number of pages | 28 |

Journal | Journal of Theoretical Probability |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - 1999 |

Externally published | Yes |

### Bibliographical note

Funding Information:The research of R. Durrett was partially supported by an NSF grant. The research of B. Granovsky and S. Gueron was partially supported by the Technion V.P.R. fund and by the Fund for the Promotion of Research at the Technion.

## Keywords

- Coagulation-fragmentation processes
- Interacting particle systems
- Reversibility; equilibrium

## ASJC Scopus subject areas

- Statistics and Probability
- Mathematics (all)
- Statistics, Probability and Uncertainty