Stochastic Theory of Discrete Binary Fragmentation-Kinetics and Thermodynamics.

We formulate binary fragmentation as a discrete stochastic process in which an integer mass k splits into two integer fragments j , k-j, with rate proportional to the fragmentation kernel Fj,k-j. We construct the ensemble of all distributions that can form in fixed number of steps from initial mass M and obtain their probabilities in terms of the fragmentation kernel. We obtain its partition function, the mean distribution and its evolution in time, and determine its stability using standard thermodynamic tools. We show that shattering is a phase transition that takes place when the stability conditions of the partition function are violated. We further discuss the close analogy between shattering and gelation, and between fragmentation and aggregation in general.