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 language | English (US) |
|---|---|
| Title of host publication | Understanding Complex Systems |
| Publisher | Springer Verlag |
| Pages | 125-161 |
| Number of pages | 37 |
| DOIs | |
| State | Published - Jan 1 2018 |
Publication series
| Name | Understanding Complex Systems |
|---|---|
| ISSN (Print) | 1860-0832 |
| ISSN (Electronic) | 1860-0840 |
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All Science Journal Classification (ASJC) codes
- Software
- Computational Mechanics
- Artificial Intelligence
Cite this
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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 proceeding › Chapter
TY - CHAP
T1 - Phase Transitions
T2 - The Giant Cluster
AU - Matsoukas, Themis
PY - 2018/1/1
Y1 - 2018/1/1
N2 - 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.
AB - 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.
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UR - http://www.scopus.com/inward/citedby.url?scp=85065838202&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-04149-6_5
DO - 10.1007/978-3-030-04149-6_5
M3 - Chapter
AN - SCOPUS:85065838202
T3 - Understanding Complex Systems
SP - 125
EP - 161
BT - Understanding Complex Systems
PB - Springer Verlag
ER -