We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non- random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit self-organised criticality in the cluster size distribution, with universal exponent 3 2.
The critical behaviour of the stationary cluster size distribution, found in Theorem , has also been seen in a different form of fragmentation-coalescence process studied by Bressaud and Fournier @cite_2 . This was a mean-field approximation to a one-dimensional forest-fire model on @math . Edges (each of weight one) are glued together if the site between them is occupied, which happens at rate 1, and clusters of @math edges have all sites removed at rate @math . A mean-field approximation to this model, ignoring correlations, follows Then, they found that the stationary cluster size distribution converges weakly to @math as @math ( fragmentation' rate converges to zero) where which matches the distribution in Theorem in the case where @math .
@cite_2
1585432259
We report a remarkable universality in the patterns of violence arising in three high-profile ongoing wars, and in global terrorism. Our results suggest that these quite different conflict arenas currently feature a common type of enemy, i.e. the various insurgent forces are beginning to operate in a similar way regardless of their underlying ideologies, motivations and the terrain in which they operate. We provide a microscopic theory to explain our main observations. This theory treats the insurgent force as a generic, self-organizing system which is dynamically evolving through the continual coalescence and fragmentation of its constituent groups.