The Cluster Ensemble

Abstract

The generic population that is the subject of our study consists of M indistinguishable members assembled into N groups, or clusters, such that no group is empty. The “member” is the fundamental unit of the population and plays the same role as “monomer” in a polymeric system, or “primary particle” in granular materials. The cluster is characterized by the number of members it contains. We will refer to the number of members in the cluster as the size or mass of the cluster and will use the terms interchangeably. The goal in this chapter is to define a sample space of distributions of clusters and assign a probability measure over it. This probability space of distributions will be called cluster ensemble and forms the basis for the development of generalized thermodynamics. In Chap. 7 we will reformulate the theory on the basis of a more abstract space of distributions.