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

Abstract

In combinatorics, a Stirling number of the second kind S (n,k) 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” P i =0 . Accordingly, a new equation for partitions P (N, m) in a set of entities into both empty and nonempty subsets was derived. The indistinguishableness of particles (N identical atoms or molecules) makes only sense within a cluster (subset) with the size 0≤n i ≥N. The first-order phase transition is indeed the case of transitions, for example in the simplest interpretation, from completely liquid state type L = {n 1 = N , n 2 = 0} to the completely crystalline state type C = {n 1 =0, n 2 = N }. These partitions are well distinguished from the physical point of view, so they are ‘typed’ differently in the model. Finally, the present developments in the physics of complex systems, in particular the structural relaxation of super-cooled liquids and glasses, are discussed by using such stochastic cluster-based models.