Phase Transitions: The Giant Cluster

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

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.

Original languageEnglish (US)
Title of host publicationUnderstanding Complex Systems
PublisherSpringer Verlag
Pages125-161
Number of pages37
DOIs
StatePublished - Jan 1 2018

Publication series

NameUnderstanding Complex Systems
ISSN (Print)1860-0832
ISSN (Electronic)1860-0840

Fingerprint

Phase transitions
Thermodynamics
Gelation
Polymers
Sols
Dynamical systems
Mathematical models
Neural networks
Kinetics

All Science Journal Classification (ASJC) codes

  • Software
  • Computational Mechanics
  • Artificial Intelligence

Cite this

Matsoukas, T. (2018). Phase Transitions: The Giant Cluster. In Understanding Complex Systems (pp. 125-161). (Understanding Complex Systems). Springer Verlag. https://doi.org/10.1007/978-3-030-04149-6_5
Matsoukas, Themis. / Phase Transitions : The Giant Cluster. Understanding Complex Systems. Springer Verlag, 2018. pp. 125-161 (Understanding Complex Systems).
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Matsoukas, T 2018, Phase Transitions: The Giant Cluster. in Understanding Complex Systems. Understanding Complex Systems, Springer Verlag, pp. 125-161. https://doi.org/10.1007/978-3-030-04149-6_5

Phase Transitions : The Giant Cluster. / Matsoukas, Themis.

Understanding Complex Systems. Springer Verlag, 2018. p. 125-161 (Understanding Complex Systems).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Matsoukas T. Phase Transitions: The Giant Cluster. In Understanding Complex Systems. Springer Verlag. 2018. p. 125-161. (Understanding Complex Systems). https://doi.org/10.1007/978-3-030-04149-6_5