Abstract

Systems governed by short-range forces—such as gases or simple magnets—obey the property of additivity, where the total energy is simply the sum of the energies of its macroscopic components. In contrast, systems with long-range interactions (LRI), characterized by an inter-particle potential decaying as $V® \sim 1/r^\alpha$ (where $\alpha \le d$), are non-additive. A striking consequence of this non-additivity is ensemble inequivalence, in which the microcanonical (fixed energy $E$) and canonical (fixed temperature $T$) ensembles yield divergent physical predictions. Key phenomena associated with LRI include negative specific heat, temperature jumps at phase transitions, and the emergence of quasi-stationary states. I will demonstrate these features by discussing classical spin models, such as the Nagle-Kardar spin-1/2 model with competing mean-field and nearest-neighbor interactions, as well as the quantum Lipkin-Meshkov-Glick model augmented with multi-spin interactions.