Phase Transitions: The Giant Cluster

A problem of great theoretical interest in thermodynamics is the possibility of phase transitions. Many dynamical systems exhibit behavior that is remarkably similar to phase transitions. Polymer gelation, shattering in fragmentation, the spread of epidemics, and the emergence of long-range connectivity in artificial and neural networks are examples of the emergence of a giant coherent structure, a behavior that is often discussed qualitatively in the language of phase transitions. If generic population obey thermodynamics, do they also undergo phase transitions? The answer is, yes. As in molecular thermodynamics, phase splitting in the cluster ensemble is associated with the violation of the stability conditions that guarantee the existence of a maximum in the microcanonical weight that defines the most probable distribution. In this chapter we formalize the stability conditions that ensure the existence of the most probable distribution and identify the giant cluster as a phase that is distinct from the sol, a stable population of dispersed clusters. We discuss two mathematical models that give rise to a giant cluster and solve them analytically. A kinetic model of gelation with a closer connection to a physical system of reacting polymers will be discussed in Chap. 9.