For systems with long-range interactions, the two-body potential decays at large distances as V (r) ∼ 1/r α , with α ≤ 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: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameters space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable 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. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.

This book deals with an important class of many-body systems: those where the interaction potential decays slowly for large inter-particle distance. In particular, systems where the decay is slower than the inverse inter-particle distance raised to the dimension of the embedding space. Gravitational and Coulomb interactions are the most prominent examples. However, it has become clear that long-range interactions are more common than previously thought. This has stimulated a growing interest in the study of long-range interacting systems, which has led to a much better understanding of the many peculiarities in their behaviour. The seed of all particular features of these systems, both at equilibrium and out-of-equilibrium, is the lack of additivity. It is now well understood that this does not prevent a statistical mechanics treatment. However, it does require a more in-depth study of the thermodynamic limit and of all related theoretical concepts. A satisfactory understanding of properties generally considered as oddities only a couple of decades ago has now been reached: ensemble inequivalence, negative specific heat, negative susceptibility, ergodicity breaking, out-of-equilibrium quasi-stationary-states, anomalous diffusion, etc. The first two parts describe the theoretical and computational instruments needed for addressing the study of both equilibrium and dynamical properties of systems subject to long-range forces. The third part of the book is devoted to discussing the applications of such techniques to the most relevant examples of long-range systems. The only prerequisite is a basic course in statistical mechanics.

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We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in N -particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated N → ∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite N , dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via non stationary states: we characterize numerically this dynamical instability in the finite N system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite N system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N 1.7 . Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the q-exponential distributions derived from Tsallis statistics.

We review theoretical results obtained recently in the framework of statistical mechanics to study systems with long range forces. This fundamental and methodological study leads us to consider the different domains of applications in a trans-disciplinary perspective (astrophysics, nuclear physics, plasmas physics, metallic clusters, hydrodynamics,...) with a special emphasis on Bose-Einstein condensates.

Many thermodynamic instabilities in one dimension (e.g. DNA thermal denaturation, wetting of interfaces) can be described in terms of simple models involving harmonic coupling between nearest neighbors and an asymmetric on-site potential with a repulsive core, a stable minimum and a flat top. The paper deals with the case of the Morse on-site potential, which can be treated exactly in the continuum limit. Analytical expressions for correlation functions are derived; they are shown to obey scaling; numerical transfer-integral values obtained for a discrete version of the model exhibit the same critical behavior. Furthermore, it is shown in detail that the onset of the transition can be characterized by an entropic stabilization of an -otherwise unstable-, nonlinear field configuration, a soliton-like domain wall (DW) with macroscopic energy content. The statistical mechanics of the DW provides an exact estimate of the critical temperature for a wide range of the discretization parameter; this suggests that the transition can be accurately viewed as being "driven" by a nonlinear entity.

We study a model for the dynamical stretching of DNA which uses the stretching of the hydrogen bonds in a basepair as its main variable. We present a statistical mechanical analysis of the denaturation and specific heat curves, obtained with the transfer integral method; discreteness effects are treated exactly by a numerical solution of the transfer integral operator. Second order self-consistent phonon theory agrees with the exact transfer integral results in the low and intermediate temperature range and explains the phonon softening observed in the molecular dynamics simulations. When the temperature approaches the denaturation temperature, the second order self-consistent phonon results deviate significantly from the exact ones, pointing to the fundamental role of nonlinear processes in DNA denaturation.

We explain the ubiquity and extremely slow evolution of non-Gaussian out-of-equilibrium distributions for the Hamiltonian mean-field model, by means of traditional kinetic theory. Deriving the Fokker-Planck equation for a test particle, one also unambiguously explains and predicts striking slow algebraic relaxation of the momenta autocorrelation, previously found in numerical simulations. Finally, angular anomalous diffusion are predicted for a large class of initial distributions. Non-extensive statistical mechanics is shown to be unnecessary for the interpretation of these phenomena.