A New Formula for Partitions in a Set of Entities into Empty and Nonempty Subsets, and Its Application to Stochastic and Agent-Based Computational Models [Articol]

Abstract

In combinatorics, a Stirling number of the second kind (n,k),Snk is the number of ways to partition a set of n objects into k nonempty subsets. The empty subsets are also added in the models presented in the article in order to describe properly the absence of the corresponding type i of state in the system, i.e. when its “share”Pi 0 . Accordingly, a new equation for partitions 0ip,PNmtype in a set of entities into both empty and nonempty subsets was derived. The indis- tinguishableness of particles (N identical atoms or molecules) makes only sense within a cluster (subset) with the size . The first-order phase transition is indeed the case of transitions, for example in the simplest interpretation, from completely liquid state to the completely crystalline state . These partitions are well distinguished from the physical point of view, so they are ‘typed’ differently in the model. Fi- nally, the present developments in the physics of complex systems, in particular the structural relaxation of supercooled liquids and glasses, are discussed by using such stochastic cluster-based models.