Geometric Models of the Statistical Theory of Fragmentation

We propose an approach to describing a medium fragmentation process based on studying the stochastic geometry of the medium states. This approach allows accounting for the interrelation of the produced fragments relative to their positions and, in particular, allows taking the size of the fragmenting object into account. We use this approach to analyze a one-dimensional model – a stochastic process with discrete time and a phase space consisting of partitions into fragments of the real axis. We derive the driving equation for the partition function with respect to sizes and prove the existence of a limit distribution.