Statistical mechanics and dynamics of solvable models with long-range interactions

Abstract

The two-body potential of systems with long-range interactions decays at large distances as $V(r)\sim 1/r^{\alpha }$, with $\alpha \le d$, where $d$ is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties is at the origin of ensemble inequivalence, which implies that specific heat can be negative in the microcanonical ensemble and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity implies that ergodicity may be generically broken. We present here a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Relaxation towards thermodynamic equilibrium can be extremely slow and quasi-stationary states may be present. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation.

118 pages, review paper, added references, slight change of content

References in corpus (16)

• Phase transitions and configuration space topology
• Collisionless relaxation in gravitational systems: From violent relaxation to gravothermal collapse
• Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model
• Phase transitions and topology changes in configuration space
• Partial equivalence of statistical ensembles and kinetic energy
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Negative magnetic susceptibility and nonequivalent ensembles for the mean-field ${\varphi }^4$ spin model
• Controlled switching of intrinsic localized modes in a 1-D antiferromagnet
• Application of large deviation theory to the mean-field phi^4-model
• Phase transitions induced by saddle points of vanishing curvature
• On the mean-field spherical model
• Thermodynamic fluctuation relation for temperature and energy
• Topology, phase transitions and the spherical model
• Statistical mechanics of two-dimensional vortices and stellar systems
• Topological approach to phase transitions and inequivalence of statistical ensembles
• On the dynamical anomalies in the Hamiltonian Mean Field model

Cited by in corpus (1)

• The large deviation approach to statistical mechanics