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] - Gubceac, Gennadii; Guțu, Roman; Paladi, Florentin (SCRIP, 2013)
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.
Copyright © 2013 Ghennadii Gubceac et al . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distri bution, and reproduction in any medium, pr ovided the original work is properly cit ed.
Keywords: partitions; agent-based models; stochastic processes; complex systems