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After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a "binary star" consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a "dilute" state, where the particles are well separated, and a "condensed" state, where the particles are bound together. We show that the evolution of the system is given by a Fokker–Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star ( N = 2 ), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N → + ∞ . [ABSTRACT FROM AUTHOR]

After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a “binary star” consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a “dilute” state, where the particles are well separated, and a “condensed” state, where the particles are bound together. We show that the evolution of the system is given by a Fokker–Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (N=2), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N→+∞.

In this paper, we propose a new generalized XY model with nearest-neighbor interaction which cannot be studied analytically. By adopting both the canonical Monte Carlo simulations and the microcanonical Monte Carlo simulations, the first-order phase transitions in both canonical ensemble and microcanonical ensemble are observed numerically, and the accurate phase-transition temperatures of the model with different parameters are given. From the distinct phase diagrams in two different ensembles, we conclude that the model we propose here exists as an ensemble equivalence. [ABSTRACT FROM AUTHOR]

We discuss a method to compute the microcanonical entropy at fixed magnetization without direct counting. Our approach is based on the evaluation of a saddle-point leading to an optimization problem. The method is applied to a benchmark Ising model with simultaneous presence of mean-field and nearest-neighbour interactions for which direct counting is indeed possible, thus allowing a comparison. Moreover, we apply the method to an Ising model with mean-field, nearest-neighbour and next-nearest-neighbour interactions, for which direct counting is not straightforward. For this model, we compare the solution obtained by our method with the one obtained from the formula for the entropy in terms of all correlation functions. This example shows that for general couplings our method is much more convenient than direct counting methods to compute the microcanonical entropy at fixed magnetization. [ABSTRACT FROM AUTHOR]

Abstract This review article takes stock of the progress made in understanding the phase transition in hot nuclei and highlights the coherence of observed signatures. [ABSTRACT FROM AUTHOR]

Broadly distributed random variables with a power-law distribution f (m) ~ m -(1+α) are known to generate condensation effects. This means that, when the exponent a lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean (0 < α < 1) one finds unconstrained condensation, whereas for α > 1 constrained condensation takes places fixing the total mass to a large enough value ... > Mc. In both cases, a standard indicator of the condensation phenomenon is the participation ratio ..., which takes a finite value for N → ∞ when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value M ~ N 1+ δ (δ > 0), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as M ~ N1/α for α < 1) and the extensive constrained mass. In particular we show that for exponents a < 1 a condensate phase for values δ > δc = 1/α - 1 is separated from a homogeneous phase at δ < δc from a transition line, δ = δc, where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass. [ABSTRACT FROM AUTHOR]

In recent years, studies of long-range interacting (LRI) systems have taken center stage in the arena of statistical mechanics and dynamical system studies, due to new theoretical developments involving tools from as diverse a field as kinetic theory, non-equilibrium statistical mechanics, and large deviation theory, but also due to new and exciting experimental realizations of LRI systems. In the first, introductory, Section 1, we discuss the general features of long-range interactions, emphasizing in particular the main physical phenomenon of non-additivity, which leads to a plethora of distinct effects, both thermodynamic and dynamic, that are not observed with short-range interactions: Ensemble inequivalence, slow relaxation, broken ergodicity. In Section 2, we discuss several physical systems with long-range interactions: mean-field spin systems, self-gravitating systems, Euler equations in two dimensions, Coulomb systems, one-component electron plasma, dipolar systems, free-electron lasers. In Section 3, we discuss the general scenario of dynamical evolution of generic LRI systems. In Section 4, we discuss an illustrative example of LRI systems, the Kardar-Nagel spin system, which involves discrete degrees of freedom, while in Section 5, we discuss a paradigmatic example involving continuous degrees of freedom, the so-called Hamiltonian mean-field (HMF) model. For the former, we demonstrate the effects of ensemble inequivalence and slow relaxation, while for the HMF model, we emphasize in particular the occurrence of the so-called quasistationary states (QSSs) during relaxation towards the Boltzmann-Gibbs equilibrium state. The QSSs are non-equilibrium states with lifetimes that diverge with the system size, so that in the thermodynamic limit, the systems remain trapped in the QSSs, thereby making the latter the effective stationary states. In Section 5, we also discuss an experimental system involving atoms trapped in optical cavities, which may be modelled by the HMF system. In Section 6, we address the issue of ubiquity of the quasistationary behavior by considering a variety of models and dynamics, discussing in each case the conditions to observe QSSs. In Section 7, we investigate the issue of what happens when a long-range system is driven out of thermal equilibrium. Conclusions are drawn in Section 8. [ABSTRACT FROM AUTHOR]

In nonadditive systems, like small systems or like long-range interacting systems even in the thermodynamic limit, ensemble inequivalence can be related to the occurrence of negative response functions, this in turn being connected with anomalous concavity properties of the thermodynamic potentials associated with the various ensembles. We show how the type and number of negative response functions depend on which of the quantities E, V and N (energy, volume and number of particles) are constrained in the ensemble. In particular, we consider the unconstrained ensemble in which E, V and N fluctuate, which is physically meaningful only for nonadditive systems. In fact, its partition function is associated with the replica energy, a thermodynamic function that identically vanishes when additivity holds, but that contains relevant information in nonadditive systems.

Broadly distributed random variables with a power-law distribution f ( m ) ∼ m - ( 1 + α ) are known to generate condensation effects. This means that, when the exponent α lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean ( 0 < α < 1 ) one finds unconstrained condensation, whereas for α > 1 constrained condensation takes places fixing the total mass to a large enough value M = ∑ i = 1 N m i > M c . In both cases, a standard indicator of the condensation phenomenon is the participation ratio Y k = 〈 ∑ i m i k / ( ∑ i m i ) k 〉 ( k > 1 ), which takes a finite value for N → ∞ when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value M ∼ N 1 + δ ( δ > 0 ), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as M ∼ N 1 / α for α < 1 ) and the extensive constrained mass. In particular we show that for exponents α < 1 a condensate phase for values δ > δ c = 1 / α - 1 is separated from a homogeneous phase at δ < δ c from a transition line, δ = δ c , where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass.

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called 'exactness of the mean-field theory'. It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the α-Potts model with annealed vacancies and the α-Potts model with invisible states. [ABSTRACT FROM AUTHOR]

Abstract: We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as at large distances in dimensions. Two broad classes of such systems are discussed. (i) 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, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions 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 with local dynamics are considered within the framework outlined above. [ABSTRACT FROM AUTHOR]

Abstract: For systems with long-range interactions, the two-body potential decays at large distances as , with , where 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, and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameter 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. [Copyright &y& Elsevier]

Abstract: The Robert–Sommeria–Miller equilibrium statistical mechanics predicts the final organization of two dimensional flows. This powerful theory is difficult to handle practically, due to the complexity associated with an infinite number of constraints. Several alternative simpler variational problems, based on Casimir’s or stream function functionals, have been considered recently. We establish the relations between all these variational problems, justifying the use of simpler formulations. [Copyright &y& Elsevier]

Abstract: We present a complete analytical solution of a system of Potts spins on a random k-regular graph in both the canonical and microcanonical ensembles, using the Large Deviation Cavity Method (LDCM). The solution is shown to be composed of three different branches, resulting in a non-concave entropy function. The analytical solution is confirmed with numerical Metropolis and Creutz simulations and our results clearly demonstrate the presence of a region with negative specific heat and, consequently, ensemble inequivalence between the canonical and microcanonical ensembles. [Copyright &y& Elsevier]

Abstract: We calculate the thermodynamic entropy of the mean-field spin model in the microcanonical ensemble as a function of the energy and magnetization of the model. The entropy and its derivatives are obtained from the theory of large deviations, as well as from Rugh''s microcanonical formalism, which is implemented by computing averages of suitable observables in microcanonical molecular dynamics simulations. Our main finding is that the entropy is a concave function of the energy for all values of the magnetization, but is nonconcave as a function of the magnetization for some values of the energy. This last property suggests that the magnetic susceptibility of the model can be negative when calculated microcanonically for fixed values of the energy and magnetization. This provides a magnetization analogue of negative heat capacities, which are well-known to be associated in general with the inequivalence of the microcanonical and canonical ensembles. Here, the two ensembles that are nonequivalent are the microcanonical ensemble in which the energy and magnetization are held fixed and the canonical ensemble in which the energy and magnetization are fixed only on average by fixing the temperature and magnetic field. [Copyright &y& Elsevier]

Abstract: We shortly review recent progress in the field of long-range interactions, analyzed from the viewpoint of statistical mechanics and of dynamical systems theory. Long-range interaction potentials decay at large distance with a power that is smaller than space dimension. This feature causes a lack of additivity, which in turn implies inequivalence between microcanonical and canonical ensemble for what concerns predictions near first order canonical phase transitions. Inequivalence manifests itself physically by the presence of curious effects such as negative specific heat and temperature jumps. We also briefly mention dynamical effects which we believe are generic for long-range interactions: (i) quasi-stationary states whose lifetime increases with system size; (ii) metastable states that live proportionally to the exponential of systems size times the entropy barrier; (iii) broken ergodicity. [Copyright &y& Elsevier]

Abstract: The effect of nearest-neighbor coupling on the thermodynamic and dynamical properties of the ferromagnetic Hamiltonian mean field model (HMF) is studied. For a range of antiferromagnetic nearest-neighbor coupling, a canonical first-order transition is observed, and the canonical and microcanonical ensembles are non-equivalent. In studying the relaxation time of non-equilibrium states it is found that as in the HMF model, a class of non-magnetic states is quasi-stationary, with an algebraic divergence of their lifetime with the number of degrees of freedom N. The lifetime of metastable states is found to increase exponentially with N as expected. [Copyright &y& Elsevier]

Abstract: The relation between thermodynamic phase transitions in classical systems and topology changes in their state space is discussed for systems in which equivalence of statistical ensembles does not hold. As an example, the spherical model with mean field-type interactions is considered. Exact results for microcanonical and canonical quantities are compared with topological properties of a certain family of submanifolds of the state space. Due to the observed ensemble inequivalence, a close relation is expected to exist only between the topological approach and one of the statistical ensembles. It is found that the observed topology changes can be interpreted meaningfully when compared to microcanonical quantities. [Copyright &y& Elsevier]

We consider systems in which the canonical partition function can be expressed as the integral in an n-dimensional space (the order parameter space) of a function that also depends parametrically on the number N of degrees of freedom and on the inverse temperature β. We show how to compute, together with the canonical entropy and specific heat, also the corresponding microcanonical quantities, generalizing in this way some results already in the literature for the case n=1. From the expressions that are obtained it is possible to derive the necessary conditions for the equivalence of the canonical and of the microcanonical ensembles. We finally study a simple model, with a two-dimensional order parameter space, in which ensemble inequivalence is realized. [Copyright &y& Elsevier]

Ensemble inequivalence has been observed in several systems. In particular, it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field-like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble. [Copyright &y& Elsevier]

In nonadditive systems, like small systems or like long-range interacting systems even in the thermodynamic limit, ensemble inequivalence can be related to the occurrence of negative response functions, this in turn being connected with anomalous concavity properties of the thermodynamic potentials associated with the various ensembles. We show how the type and number of negative response functions depend on which of the quantities E, V and N (energy, volume and number of particles) are constrained in the ensemble. In particular, we consider the unconstrained ensemble in which E, V and N fluctuate, which is physically meaningful only for nonadditive systems. In fact, its partition function is associated with the replica energy, a thermodynamic function that identically vanishes when additivity holds, but that contains relevant information in nonadditive systems. [ABSTRACT FROM AUTHOR]