Notes on the Statistical Mechanics of Systems with Long-Range Interactions

Abstract

Thermodynamic and dynamical properties of systems with long range pairwise interactions (LRI) which decay as 1/r^{d+σ} at large distances r in $d$ dimensions are reviewed in these Notes. Two broad classes of such systems are identified: (a) systems with a slow decay of the interactions, termed "strong" LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations and ergodicity breaking. And (b) systems with faster decay of the interaction potential where the energy is additive, thus resulting in less dramatic effects. These interaction affect the thermodynamic behavior of systems near phase transitions, where long range correlations are naturally present. Long range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems are considered within the framework outlined above.

Notes of lectures given at a Summer School in Les Houches (France), 4-29 August 2008

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