Breaking of Ensemble Equivalence in Networks
Tiziano Squartini1,2, Joey de Mol1,3, Frank den Hollander3
1Lorentz Institute for Theoretical Physics, University of Leiden, Leiden, The Netherlands.
Physical Review Letters
|January 15, 2016
Summary
In statistical mechanics, the microcanonical and canonical ensembles are usually equivalent. This study reveals ensemble nonequivalence in random graphs with topological constraints, challenging prior assumptions about its causes.
Area of Science:
- Statistical Mechanics
- Network Science
- Complex Systems
Background:
- The equivalence of microcanonical and canonical ensembles is a cornerstone of statistical mechanics in the thermodynamic limit.
- However, exceptions to this equivalence have been observed in specific systems, with a complete theoretical explanation still lacking.
- Ensemble nonequivalence has implications for understanding the behavior of diverse physical systems.
Purpose of the Study:
- To investigate ensemble nonequivalence in random graphs with topological constraints.
- To determine the conditions under which microcanonical and canonical descriptions diverge in network models.
- To provide a more comprehensive theoretical framework for understanding ensemble nonequivalence.
Main Methods:
- Analysis of random graphs with specified degree sequences and number of links.
- Application of large-deviation theory to compare microcanonical and canonical probabilities.
- Investigation of both unipartite and bipartite graph structures.
Main Results:
- Graphs with a fixed degree sequence exhibit ensemble nonequivalence, unlike those with a fixed number of links.
- This nonequivalence persists regardless of whether the graph's energy is additive or nonadditive.
- Ensemble nonequivalence can arise from an extensive number of local constraints, not solely from nonadditivity or long-range interactions.
Conclusions:
- Local constraints, rather than nonadditivity, can be a primary driver of ensemble nonequivalence in physical systems.
- The mathematical criterion for nonequivalence relates to the large-deviation behavior of individual microstates, offering a general and local test.
- These findings extend the understanding of ensemble nonequivalence beyond traditional physical systems to network structures.
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