Ensemble nonequivalence in system with local constraints

The equivalence between canonical and microcanonical ensembles (describing systems with soft and hard constraints respectively) is a basic assumption in statistical physics, traditionally verified through the vanishing of the relative fluctuations of the constraints in the thermodynamic limit. However, evidence has accumulated that, in presence of phase transitions or long-range interactions, this property will break down, a phenomenon known as ensemble nonequivalence. Normally, ensemble nonequivalence is ‘restricted’ to a certain region of parameter space, where the difference between canonical and microcanonical entropies is ‘strong’, i.e. of the same order as the entropies themselves. However, recent research on networks has shown that the presence of an extensive number of local constraints can also lead to ensemble nonequivalence, even in the absence of phase transitions. This new form of ensemble nonequivalence appears in the whole parameter space and is therefore ‘unrestricted’. One the other hand it is ‘weak’, i.e. it is characterized by a sub-leading entropy difference.
In this work, we focus on more general complex systems with local constraints (represented as generic matrices with constraints on their margins) and find that, surprisingly, ensemble nonequivalence can manifest itself in the whole parameter space. At the same time, when the degrees of freedom for each unit of the system remain finite in the thermodynamic limit, the entity of ensemble nonequivalence is of the strong type. This novel, simultaneously ‘strong and unrestricted’ form of ensemble nonequivalence is of the most robust type observed so far and imposes a principled choice of the ensemble in all real-world applications.
We also compare the traditional criterion for ensemble equivalence, based on vanishing relative fluctuations, with the more recent criterion based on the vanishing relative entropy density between microcanonical and canonical probabilities. In particular we consider weighted core-periphery networks with local constraints, for which the phenomenon of Bose-Einstein condensation can occur when a finite fraction of the link weights concentrates in the core. We find that the relative entropy density does not vanish, both in the condensed and in the non-condensed phase. By contrast, the relative fluctuations of the constraints do not vanish in the condensed phase, while they vanish in the non-condensed phase. To our knowledge, this model is the first one where local constraints coexist with a phase transition. Our results show that, in presence of local constraints, the traditional criterion based on vanishing relative fluctuations becomes misleading, while the one based on a vanishing relative entropy density remains correct.