Stochastic coagulation and fragmentation: incommensurability and first passage times

We develop a fully stochastic theory for coagulation and fragmentation in a finite system with a maximum cluster size constraint. The process is modeled using a high-dimensional Master equation for the probabilities of cluster configurations. For certain realizations of total mass and maximum cluster sizes, we are able to find exact analytical results for the expected equilibrium cluster distributions. If coagulation is fast relative to fragmentation, and if the total system mass is indivisible by the mass of the largest allowed cluster, we find a mean cluster size distribution that is strikingly broader than that predicted by the corresponding mass-action equations. We also find conditions under which equilibration is accelerated, eluding late-stage coarsening. Finally, we show first assembly times of a single maximal cluster subtly depend on whether the assembly process proceeds via coagulation-fragmentation or is restricted to only monomer binding and dissociation, and can be nonmonotonic in the fragmentation rate when kinetic traps arise. These results hold even in the thermodynamic limit and can be derived only from a discrete stochastic analysis of the process, highlighting how classical mass-action treatments of coagulation-fragmentation fail.