The Equilibrium Behavior of Reversible Coagulation-Fragmentation Processes

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 ΩN, the set of all partitions of N. We obtain the stationary distribution (invariant measure) on ΩN 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.

Duke Scholars

Author Richard Timothy Durrett Mathematics