# Statistical mechanics and dynamics of solvable models with long-range interactions - Condensed Matter > Statistical Mechanics

Abstract: The two-body potential of systems with long-range interactions decays atlarge distances as $Vr\sim 1-r^\alpha$, with $\alpha\leq d$, where $d$ is thespace dimension. Examples are: gravitational systems, two-dimensionalhydrodynamics, two-dimensional elasticity, charged and dipolar systems.Although such systems can be made extensive, they are intrinsically nonadditive. Moreover, the space of accessible macroscopic thermodynamicparameters might be non convex. The violation of these two basic properties isat the origin of ensemble inequivalence, which implies that specific heat canbe negative in the microcanonical ensemble and temperature jumps can appear atmicrocanonical first order phase transitions. The lack of convexity impliesthat ergodicity may be generically broken. We present here a comprehensivereview of the recent advances on the statistical mechanics andout-of-equilibrium dynamics of systems with long-range interactions. The coreof the review consists in the detailed presentation of the concept of ensembleinequivalence, as exemplified by the exact solution, in the microcanonical andcanonical ensembles, of mean-field type models. Relaxation towardsthermodynamic equilibrium can be extremely slow and quasi-stationary states maybe present. The understanding of such unusual relaxation process is obtained bythe introduction of an appropriate kinetic theory based on the Vlasov equation.

Author: A. Campa 1, T. Dauxois 2, S. Ruffo 3 1 Complex Systems and Theoretical Physics Unit, ISS and INFN, Rome, Italy 2 Laboratoire de P

Source: https://arxiv.org/