Ensemble inequivalence and Maxwell construction in the self-gravitating ring model

•The paper revisits the well studied self-gravitating ring model by showing from first principles, i.e. by only solving Hamilton equations of motion for the first time in a system with long-range interactions, that mircocanonical and canonical ensembles are inequivalent. The thermal bath is also modeled in the same way. Although the latter is a know and stablished result, it is the approach used to illustrate this fact that is novel and helps to clarify some issues for researchers outside the long-range interactions community.•The paper illustrates the point that the Maxwell construction must be modified in the presence of long-range interactions by excluding as non-physical the whole region of p hase coexistence.•By giving actual examples from numerical solutions of the equations of motion, it shown that equilibrium states with negative heat capacity in the micro-canonical ensemble are stable and accessible by the dynamics of the system, and if the same states are coupled to a heath bath of the same temperature (canonical ensemble), they become unstable and evolve according to the particle dynamics to another states of same temperature minimizing the free energy. The statement that Gibbs equilibrium ensembles are equivalent is a base line in many approaches in the context of equilibrium statistical mechanics. However, as a known fact, for some physical systems this equivalence may not be true. In this paper we illustrate from first principles the inequivalence between the canonical and microcanonical ensembles for a system with long range interactions. We make use of molecular dynamics simulations and Monte Carlo simulations to explore the thermodynamics properties of the self-gravitating ring model and discuss on what conditions the Maxwell construction is applicable.