Ensemble nonequivalence and Bose–Einstein condensation in weighted networks

The asymptotic (non)equivalence of canonical and microcanonical ensembles, describing systems with soft hard constraints respectively, is a central concept in statistical physics. Traditionally, the breakdown ensemble equivalence (EE) has been associated nonvanishing relative fluctuations thermodynamic limit. Recently, it reformulated terms entropy density between probabilities. earliest observations EE violation required phase transitions or long-range interactions. More recent research on binary networks found that an extensive number local can also break EE, even absence transitions. Here we study for first time nonequivalence weighted constraints. Unlike their counterparts, these undergo form Bose–Einstein condensation (BEC) producing core–periphery structure where finite fraction link weights concentrates core. This phenomenon creates unique setting coexist transition. We find surviving only condensed phase, as more traditional BEC settings. However, non-vanishing all temperatures, signalling due to presence constraints, irrespective BEC. Therefore, extensively many vanishing no longer guarantee EE.