The steady-state distributions of coagulation-fragmentation processes

Coagulation-fragmentation processes arise in many scientific applications and have been studied extensively by means of deterministic approximations: integral equations and discrete systems of differential equations. These equations are motivated by physical intuition but from a mathematical point of view they are not derived rigorously from a stochastic model. This paper proposes a stochastic model formulation for coagulation-fragmentation processes. We consider a population of N particles distributed into groups that coagulate and fragment at given rates, and seek the resulting stationary group size distribution f(i), i = 1, 2,..., N. The coagulation-fragmentation process is defined as an ergodic Markov chain whose finite state space Omega(N) is the set of all partitions of N. We obtain the related transition rates matrix B, compute the stationary measure pi(N) on Omega(N), as an eigenvector of B, and then obtain f(i). Practical use of this method is limited to small N because the dimensions of B grow exponentially with N. To overcome this restriction we propose a feasible Monte Carlo simulation method which enables us to compute f(i) for large N. We also derive a new and exact version of the discrete coagulation-fragmentation equations, which are a deterministic description. These become a syst:em of N quadratic equations for f(i) only after we eliminate terms that represent high order correlations, which depend on pi(N) and cannot be expressed in terms f(i) alone. We investigate the consistency and accuracy of this approximation by several examples. For the special case where only pairs and singles interact, we prove that the results of the deterministic approximation approach the exact steady state of the stochastic process as N --> infinity. We compare the results of the stochastic model with large N to the stationary solutions of the coagulation-fragmentation integral equation. The agreement we obtain for some test cases suggests that the integral equation may indeed be a good deterministic approximation to the stochastic process, though only when N is large. However, some inconsistencies that we encounter indicate that this conclusion may not always hold.