Abstract
It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that (1)physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by long-range interactions or nonadditivity, (2)mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in general.
Highlights
In contrast with previous expectations, our results show that (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by longrange interactions or nonadditivity, (2) mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates
Background.—In statistical physics, calculating the equilibrium properties of a system with a given energy requires averaging over the so-called microcanonical ensemble [1,2], i.e., the uniform distribution on the set of all particle configurations having a prescribed energy
Starting with the work of Gibbs [2], the microcanonical and canonical ensembles have been shown to be equivalent in the thermodynamic limit for physical systems with short-range interactions
Summary
Tiziano Squartini, Joey de Mol, Frank den Hollander, and Diego Garlaschelli1,*. In contrast with previous expectations, our results show that (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by longrange interactions or nonadditivity, (2) mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in general. Large deviation theory [17] shows that the ensembles are nonequivalent on all three levels when the microcanonical specific entropy is nonconcave
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