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

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

Alessandro Campa, Thierry Dauxois, Stefano Ruffo 1. Complex Systems and Theoretical Physics Unit, Health and Technology Department, Istituto Superiore di Sanità, and INFN Roma1, Gruppo Collegato Sanità, Viale Regina Elena 299, 00161 Roma, Italy
2. Université de Lyon, Laboratoire de Physique de l’École Normale Supérieure de Lyon, CNRS, 46 allée d’Italie, 69364 Lyon cedex 07, France
3. Dipartimento di Energetica “S. Stecco” and CSDC, Università di Firenze, INFN, Via S. Marta, 3 I-50139, Firenze, Italy
July 9, 2019
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, 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.

pacs:

05.20.-y Classical statistical mechanics
05.20.Dd Kinetic theory
05.20.Gg Classical ensemble theory
64.60.Bd General theory of phase transitions
64.60.De Statistical mechanics of model systems

Keywords: Long-range interactions, ensemble inequivalence, negative specific heat, ergodicity breaking, Vlasov equation, quasi-stationary states.

I Introduction

A wide range of problems in physics concerns systems with long-range interactions. However, their statistical and dynamical properties are much less understood than those of short-range systems leshouches (); Assisi (). One finds examples of long-range interacting systems in astrophysics Padmanabhan90 (); Chavanisreview2006 (), plasma physics Escande (), hydrodynamics Robert90 (); Miller90 (), atomic physics Courteille (), nuclear physics chomazdd (). This ubiquitous presence in different physics disciplines alone would itself justify the need for a general and interdisciplinary understanding of the physical and mathematical problems raised by long-range interacting systems.

In this review, we are interested in systems with a large number of degrees of freedom, for which a statistical physics approach is mandatory, independently of the specific features of the interactions. Therefore, we will discuss in the following equilibrium and out–of–equilibrium properties of long-range systems relying on the tools of statistical mechanics Huang ().

Let us define which is the property of the interaction that makes it short or long-ranged. We consider systems where the interaction potential is given by the sum, over pairs of the elementary constituents, of a two-body translationally invariant potential. For sufficiently large distances , the absolute value of the two-body potential is bounded by . If the positive power is larger than the dimension of the space where the system is embedded, , we define the system to be short-range. Otherwise, if , the system is long-range. The reason for this definition is that, in the large limit all the mathematical and physical differences between short and long-range systems can be traced back to this property of the interaction potential. We should remark that this definition of the range of the interaction does not coincide with others, where the range is instead defined by a characteristic length appearing in the interaction potential. In this latter definition, any interaction decaying as a power law at large distances, thus without characteristic length, is considered as long range. However, for the physical and mathematical problems found in the statistical mechanics of many-body systems, the definition used throughout this review is more appropriate, since, as we have stressed, it is related to the interaction property that determines the behavior of such systems for large .

Our purpose will be to illustrate, especially through the use of simple models, the peculiar properties of long-range systems, and the tools and techniques that are employed to describe them. To give a flavour of the issues that will be considered, we would like to begin with a very simple description of the main problems that one faces in the study of these systems.

The aim of equilibrium statistical mechanics is to derive the thermodynamic properties of a macroscopic system from microscopic interactions Huang (). The connection between micro and macro is realized through the introduction of statistical ensembles. Different thermodynamic potentials describe situations in which different thermodynamic parameters are used in the characterization of the system, and, in the aforementioned connection with the microscopic interactions, different statistical ensembles are related to different thermodynamic potentials. However, it is usually stated (and experimentally verified for many physical systems) that, as far as macroscopic averages are concerned, i.e. in the thermodynamic limit (, with ), the predictions of statistical mechanics do not depend on the chosen ensemble. Ensemble equivalence is related to the fact that, given a sufficient number of macroscopic thermodynamic parameters (two in a one-component system), the others are fixed in the large volume limit, apart from vanishingly small relative fluctuations.

An important feature of long-range systems is that ensembles can be inequivalent Thirringdd (); Thirring (); KiesslingLebowitz97 (); BMR (); Chavanisreview2006 (), and therefore one of the main issues in the statistical mechanics of these systems is a careful examination of the relations between the different ensembles, in particular of the conditions that determine their equivalence or inequivalence. We emphasize that ensemble inequivalence is not merely a mathematical drawback, but it is the cause, as it will be shown in this review, of physical properties of these systems that could be experimentally verified. Probably, one of the most striking features of long-range systems is the possibility to display negative specific heat in the microcanonical ensemble Emden (); eddigton (); Antonov (); Lyndenwood68 (); Thirringdd (); Thirring (). Specific heat is always positive in the canonical ensemble, independently of the nature of the interactions, since it is given by the expectation value of a positive quantity. It turns out that microcanonical equilibrim contains all informations about canonical equilibrium, while the converse is wrong in case of ensemble inequivalence Dieter (); Touchette2003 (). The discrepancy between the two ensembles extends to other observables related to the response of the system to a change in a thermodynamic parameter: a concrete example will be given discussing magnetic susceptibility.

As far as out–of–equilibrium dynamical properties are concerned, many-body long-range systems again show peculiar behaviors. The approach to equilibrium of short-range systems is usually characterized by the time scales that govern the equations of motion of the elementary constituents Balian (). For systems without disorder, these time scales are typically small when compared to the observational time scales. Sometimes these systems can be trapped in metastable states that last for long times. These states are local extrema of thermodynamic potentials and, in practical cases, their realization requires a very careful preparation of the system (e.g. undercooled liquids and superheated solids). If perturbed, the system rapidly converges towards the equilibrium state. It can also happen that the relaxation time depends on the volume if hydrodynamics modes are present or in coarsening processes.

For long-range systems, dynamics can be extremely slow and the approach to equilibrium can take a very long time, that increases with the number of elementary constituents binneytremaine (). This feature is induced by the long-range nature of the interaction itself and is not a consequence of the existence of a collective phenomenon. The state of the system during this long transient is quasi-stationary Lyndenbell67 (); ChavanisSommeriaRobert (); lrr (); yoshi (), since its very slow time evolution allows us to define slowly varying macroscopic observables, like for local equilibrium or quasi-static transformations. It should be however remarked that quasi-stationary states are not thermodynamic metastable states, since they do not lie on local extrema of equilibrium thermodynamic potentials. The explanation of their widespread presence should rely only upon the dynamical properties of the systems. It must be stressed that the nature of quasi-stationary states can be strongly dependent on the initial condition, as it will become clear from the examples that we will give. In addition, a variety of macroscopic structures can form spontaneously in out–of–equilibrium conditions for isolated systems: a fact that should not be a surprise given that already the equilibrium states of long-range systems are usually inhomogeneous. In short-range systems, macroscopic structures can arise as an effect of an external forcing (Rayleigh-Bénard convection, Benjamin-Feir instability, Faraday waves) Fauve () or due to the nonlinearity of the governing equations of motion (solitons, breathers) dauxoispeyrard (), but are usually strongly selected by the specific dynamical properties and by the geometrical conditions. All this shows the great richness of the dynamics of long-range systems.

Summarizing, a satisfying theoretical framework concerning the behaviour of system with long-range interactions should necessarily address the following aspects:

• For what concerns equilibrium properties, the determination of the physical conditions that determine equivalence or inequivalence of the statistical ensembles and, for the latter case, the relation between macrostates in the different statistical ensembles.

• As for the out–of–equilibrium features, the development of a consistent kinetic theory able to explain the formation of quasi-stationary states and their final relaxation to equilibrium.

Although different aspects of systems with long-range interactions have been studied in the past in specific scientific communities, notably astrophysics and plasma physics, this has not constituted a seed for more general theoretical studies. In the last decade or so, it has become progressively clear that the ubiquitous presence of long-range forces needs an approach that integrates different methodologies leshouches (); Assisi (). This has induced a widespread interest in long-range systems throughout numerous research groups. The successive development has led to a better understanding of both the equilibrium and out–of–equilibrium properties of such systems. Time is ripe to summarize what is known on firm basis about the equilibrium statistical mechanics of systems with long-range interactions and to describe the preliminaries of a theory of non equilibrium.

In this review, we have chosen to present the main problems, tools and solutions by discussing paradigmatic examples, which are simple and general enough to be useful also for specific applications. Therefore, we will mostly emphasize:

• the equilibrium statistical mechanics solutions of simple mean-field toy models, which is a first step for understanding phase diagrams of more complex long-range systems;

• the basic ingredients of a kinetic theory of a model able to catch the essential properties of out–of–equilibrium dynamics.

• the analysis of the phase diagram of models with both short and long-range interactions, or with forces that weakly decay in space.

The structure of the review is the following. In section II, we introduce the subject presenting the definition of long-range interactions and discussing the non additivity property and its consequences. In section III, we briefly review the relevant physical systems with long-range interactions: gravitational systems, 2D hydrodynamics, 2D elasticity, charged systems, dipolar systems and small systems. Section IV constitutes the core of the review. We present there the equilibrium properties of several mean-field models for which one can analytically compute both the free energy and the entropy. The solution is obtained by different methods, including the powerful large deviation method, which allows to solve models with continuous variables. Dynamics is tackled in section V, where the main result reviewed is the widespread presence of quasi-stationary states that hinder relaxation to Boltzmann-Gibbs equilibrium. This phenomenon is strongly supported by numerical simulations and can be studied using kinetic equations explicitly devised for long-range interactions (Vlasov, Lenard-Balescu). Few analytical results exist for non mean-field models: we collect some of them in section VI. The message is that the introduction of short-range terms doesn’t spoil the features of mean-field models and that some weakly decaying interactions can be treated rigorously. Finally, we draw conclusions and we discuss some perspectives in section VII.

Ii The additivity property and the definition of long-range systems

ii.1 Extensivity vs. Additivity

In order to easily illustrate the issues of extensivity and additivity, it is useful to consider a concrete example and restrict ourselves to the energy as the thermodynamic extensive variable. We employ a very simple model that is used in the study of magnetic systems, namely the Curie-Weiss Hamiltonian

 HCW=−J2NN∑i,j=1SiSj=−J2N(N∑i=1Si)2, (1)

where the spins are attached to sites labeled by . In this example, the interaction does not decay at all with the distance: indeed, each spin interacts with equal strength with all the other spins. Such systems are usually referred to as mean-field systems. With the prefactor in (1), the total energy increases as , then the energy per spin converges to a finite value in the thermodynamic limit, which is a physically reasonable requirement. We recall that one can find rigorous definitions of the thermodynamic limit in Ruelle’s book ruelle (). Model (1) is extensive: for a given intensive magnetization

 m=∑iSiN=MN, (2)

where is the extensive magnetization, if one doubles the number of spins the energy doubles. On the other hand, Hamiltonian (1) is not additive, in spite of the presence of the regularizing factor . Indeed, let us divide the system, schematically pictured in Fig. 1, in two equal parts. In addition, let us consider the particular case in which all spins in the left part are equal to , whereas all spins in the right part are equal to . The energy of the two parts is . However, if one computes the total energy of the system, one gets . Since , such a system is not additive, at least for this configuration. One could easily generalize the argument to generic configurations. The problem is not solved by increasing system size , because the “interaction” energy increases with .

Extensivity in the Curie-Weiss model is provided by the prefactor in Hamiltonian (1), which makes the energy proportional to . This is the so-called Kac prescription KacUhlenbeck (). This energy rescaling guarantees a competition between energy and entropy , which is crucial for phase transitions. Indeed, introducing the free energy , where is temperature, Kac prescription implies that the two competing terms on the r.h.s. both scale as , since temperature is intensive and entropy scales like . This latter scaling deserves a further analysis which will be developed later on. Intuitively, the fact that the interaction is long-range does not alter the density of states, which usually grows factorially with . An alternative prescription would be to make temperature extensive , in such a way that energy and entropy term of the free energy both scale as . The two prescriptions give equivalent physical consequences. In systems with kinetic energy, being temperature the average kinetic energy per particle, rescaling temperature corresponds to a renormalization of velocities, and finally of the time scale.

After having defined the two distinct concepts of extensivity and additivity with reference to a specific model, let us clarify more generally these two notions.

Indeed, it is very important, as a first step, to give the general definitions of extensivity and additivity, and to clarify the distinction between these two concepts. It is convenient to first consider the situation that one encounters in short-range systems. We can imagine to divide a system at equilibrium in two parts occupying equal volumes. Some thermodynamic variables of each half of the system will be equal to the corresponding ones of the total system, others will be halved. Temperature and pressure are examples of the first kind of thermodynamic variables: they do not depend on the size of the system and are called intensive variables. Energy, entropy and free energy are variables of the second kind; their value is proportional to system size, i.e. to the number of elementary constituents (for given values of the intensive variables), and they are called extensive variables. The property that the size dependent thermodynamic variables are proportional to system size is called extensivity, and systems with this property are called extensive. The specific value of extensive variables (e.g., the energy per unit particle, or per unit mass, or per unit volume) give rise to new intensive quantities. Considering the energy of a system, we see that it has also the property of additivity, that consists in the following. Dividing the systems in two macroscopic parts, the total energy will be equal to , with the energy of the -th part, and the interaction energy between the two parts. In the thermodynamic limit the ratio tends to zero; therefore in this limit . This property is called additivity; systems with this property (for the energy as well as for other size dependent quantities) are called additive. It is evident that extensivity and additivity are related. Indeed, in the definition of extensivity just given, we could not have concluded that the energy of each part of the system is half the total energy, if the interaction energy would not be negligible. Additivity implies extensivity (thus non extensivity implies non additivity), but not the reverse, since the interaction energy might scale with , as in the Curie-Weiss model. This comment applies more generally to all long-range systems. The unusual properties of these systems derive from the lack of additivity.

ii.2 Definition of long-range systems

We have just shown that mean-field systems like the Curie-Weiss model can be made extensive using Kac’s trick. However, extensive systems could be non additive. Here, we will discuss the case of interactions that decay as a power law at large distances. We will show that the energy of a particle (excluding self-energy) diverges if the potential does not decay sufficiently fast, implying that the total energy grows superlinearly with volume at constant density, which violates extensivity. These interactions are called long-range or non integrable, just referring to this divergence of the energy. Energy convergence can be restored by an appropriate generalization of Kac’s trick.

Let us estimate the energy by considering a given particle placed at the center of a sphere of radius where the other particles are homogeneously distributed. We will exclude the contribution to coming from the particles located in a small neighborhood of radius (see Fig. 2). This is motivated by the necessity to regularize the divergence of the potential at small distances, which has nothing to do with its long-range nature.

If the other particles interact with the given one via a potential that at large distances decays like , we obtain in –dimensions

 ε=∫RδddrρJrα=ρJΩd∫Rδrd−1−αdr=ρJΩdd−α[Rd−α−δd−α],if α≠d , (3)

where is the generic density (e.g. mass, charge), is the coupling constant and is the angular volume in dimension ( in , in , etc.). When increasing the radius , the energy remains finite if and only if . This implies that the total energy will increase linearly with the volume , i.e. the system is extensive. Such interactions are the usual short-range ones. On the contrary, if the energy grows with volume as (logarithmically in the marginal case ). This implies that the total energy will increase superlinearly, , with volume. However, analogously to the Kac’s prescription, one can redefine the coupling constant and get a perfectly extensive system. Mean-field models, like Hamiltonian (1), correspond to the value , since the interaction does not depend on the distance, and one recovers the usual Kac’s rescaling. Cases where the energy grows superlinearly define the long-range nature of the interaction. However, as we have shown for mean-field systems, the fact that energy can be made extensive, does not imply that the system is additive. Implications of the lack of additivity for long-range systems will be discussed throughout the paper.

ii.3 Convexity in thermodynamic parameters

Let us consider the space of the extensive thermodynamic parameters. To be specific, as in Fig. 3, we consider the plane of energy and magnetization (2). The attainable region in this space is always convex when only short-range interactions are present. This property is a direct consequence of additivity. Consider two different subsystems with two different energies and , and two different magnetization values and . Introducing a parameter taking values between and , depending on the relative size of the subsystems, the system obtained by combining the two subsystems has an energy , and a magnetization . Any value of between and is realized thermodynamically, just varying the relative size of the two subsystems. This is exactly the convexity property of the attainable region of the space of thermodynamic parameters. In particular, one can consider two subsystems with the same energy but different magnetization. Varying the relative size of the two subsystems, the combined system will have the same energy and any possible magnetization between the two values of the two subsystems. It is important to stress that the convexity property is possible if additivity is satisfied, since the interaction energy between the two subsystems has been neglected. As already remarked, the additivity property is generically valid for large enough and short-range interacting systems. Moreover convexity implies that the space of thermodynamic parameters is connected.

On the contrary, systems with long-range interactions are not additive, and thus intermediate values of extensive parameters are not necessarily accessible (see Fig. 3). This feature has profound consequences on the dynamics of systems with long-range interactions. Gaps may open up in the space of extensive variables. Since the space of thermodynamic parameters is no more connected, ergodicity breaking might appear when considering continuous microcanonical dynamics of such a system. We will discuss again this question in detail in Sec. IV.5.4 by emphasizing simple examples.

ii.4 Lattice systems

When defining the long-range nature of the interaction, care must be taken of the specific nature of the microscopic variables. These could be divided in two classes: the coordinates related to the translational degrees of freedom (e.g. cartesian coordinates), and those giving the internal state of each particle (e.g. spin variables). Both could be either continuous or discrete. When coordinates take fixed discrete values, one speaks of lattice systems. As for the internal degrees of freedom the variables could be discrete (e.g. spin systems) or continuous (e.g. systems of rotators).

For systems with continuous translational degrees of freedom that do not possess internal degrees of freedom, the potential energy can be written in the general form

 U(→r1,…,→rN)=∑1≤i

where are cartesian coordinates in -dimensional space and we assume that the translationally invariant pair potential depends only on the modulus of the distance between two particles. Systems of gravitational point masses or Coulomb point charges fall into this category.

In a lattice system, for which each site of a -dimensional lattice, located by the position vector , hosts a particle possessing one or more internal degrees of freedom, collectively denoted by the vector (the dimensionality of this vector is independent of ), the potential energy can be written as

 U(q1,…,qN)=∑1≤i

where the coupling constants are translationally and rotationally invariant (i.e., they depend only on ). We allow also the presence of an external field that couples to the particles via the function with a strength . As pointed out above, the variables may take continuous or discrete values.

We have already discussed in the previous Subsection that long-range systems of type (4) are those for which, at large distance, with . Similarly, long-range lattice systems (5) can be characterized by a slow decay of the coupling constants . If these latter behave at large distance like with , the system is long-range. The energy grows also in this case superlinearly with the volume and one will need to introduce a Kac rescaling factor. At variance with systems of type (4) we need not worry about the behavior at short distances, because the lattice regularizes any possible divergence.

A kinetic energy term can be added to the potential energy one for systems of type (4) and for lattice systems (5) when the variables are continuous.

In this review, we will mostly concentrate our attention on lattice systems. Since they cannot display any short distance singularity, their thermodynamic and dynamical behaviour highlights the essential features of long-range interactions. Therefore, these systems are more suitable to present an overview of the main results and of the tools used to deal with long-range interactions.

ii.5 Non additivity and the canonical ensemble

Let us recall that, in , neglecting the dependence on the number of particles and on the volume , the microcanonical partition function (proportional to the number of microstates with a given energy ) is defined as

 Ω(E)∼∫d3Nqd3Np δ(E−H(p,q)), (6)

where are the phase space coordinates, is the Hamiltonian and we forget for the moment about multiplicative -dependent factors and dimensional constants (for a more precise definition see Section IV). The entropy is defined via the classical Boltzmann formula

 S(E)=lnΩ(E) , (7)

where we adopt units for which the Boltzmann constant is equal to .

The non additivity has strong consequences on the construction of the canonical ensemble from the microcanonical. The reasoning usually goes as follows. One considers an isolated system with energy , that we divide into a “small” part with energy (the subsystem of interest) and a “large” part with energy which plays the role of the bath. The additivity of the energy implies that the probability distribution that the “small” system has an energy , let us call it , is given by

 p(E1) = ∫Ω2(E2) δ(E1+E2−E)dE2 (8) = Ω2(E−E1). (9)

Using the entropy to express and expanding the term , one gets

 p(E1) = exp[S2(E−E1)] (10) ≈ exp[S2(E)−E1∂S2∂E∣∣∣E+⋯] (11) ∝ Ω2(E)e−βE1, (12)

where

 β=∂S2∂E∣∣∣E. (13)

One would end up with the usual canonical distribution for the system of interest after performing the thermodynamic limit ruelle (). Let us however remark that this derivation is valid also before taking the thermodynamic limit. This has led many authors to develop a thermodynamic formalism for finite systems chomazdd ().

It is clear that additivity is crucial to justify the factorization hypothesis implied in (8), and hence the existence of the canonical distribution. This hypothesis is clearly violated when the system size is finite, because of a contribution to the entropy coming from the surface that separates the two subsystems. However, after performing the thermodynamic limit, this contribution becomes negligible when the interactions are short-range. This is not true for long-range interactions which are non additive also in the thermodynamic limit. This has led the community to split into two different attitudes. On one side are those who think that the canonical ensemble cannot be appropriately defined for long-range interactions and claim that all the analyses should be performed using the microcanonical ensemble Dieter (); grossdd (). Although clearly appropriate for isolated system, this approach cannot describe open systems. On the other side are those who stress that the canonical ensemble could be still formally defined and used kiesling89 (); Padmanabhan90 (); paddy (); ChavanisHouches (). Among the two sides are those who attempt operative definitions of a heat bath. For instance, one has imagined that the heat bath is a short-range system that interacts with the system of interest with short-range interactions, so that the non additivity is due only to the system BaldovinOrlandini (); BaldovinOrlandinibis (). The failure of the usual derivation of the canonical ensemble suggests that non additive systems might have a very peculiar behavior if they are in contact with a thermal reservoir (see Refs. PoschThirringpre2006 (); Lyndenbell_epl2008 (); RamirezHernandezPRL (); RamirezHernandezPRE (); VelazquezCurilef () for recent literature on this topic). A third position was recently initiated by Bouchet and Barré who proposed that when considering systems with long-range interaction, the canonical ensemble does not describe fluctuations of a small part of the whole system. However, they argued julienfreddyjstat (); Villain () that it may describe fluctuations of the whole system when coupled to a thermostat via a negligibly small coupling. This interesting line of thought needs to be pursued theoretically and confirmed by numerical simulations.

In the following Section, we will discuss some physical examples before proceeding to the core of the review.

Iii Physical examples of long-range interacting systems

There are many systems in nature where particles interact with a pair potential that decays at large distances as with . Although these systems are deeply studied in their own sake (e.g. gravitational many body systems, Coulomb systems, systems of vortices, etc.), they rarely appear in books of statistical mechanics. This is of course due to the difficulty to deal with systems that are non extensive and non additive. We give here a brief sketch of the physics involved in some of these systems.

In Fig. 4, we draw in the plane some of the physical systems with long-range interactions that we will discuss afterwards.

iii.1 Gravitational systems

Gravitational systems, which correspond to in dimension , clearly belong to the category of long-range interacting systems. The gravitational problem is particularly difficult because, in addition to the non additivity due to the long-range character of the interaction, one also needs a careful regularization of the potential at short distances to avoid collapse. To be more specific, let us consider the canonical partition function of a system of self-gravitating particles of the same mass moving inside a volume

 ZN=1(2πλ2)3N/2N!∫VN∏i=1d→ri exp[−βU(→r1,…,→rN)], (14)

where

 U(→r1,…,→rN)=−Gm2N∑i

with

 V(r)=1r , (16)

where is the inverse temperature, the gravitational constant, and the De Broglie wavelength. In the following, we will not introduce dimensional constants, i.e. , in the definition of partition functions for a matter of convenience (see e.g. Eq. (55)).

From the shape of the potential represented in Fig. 5 by the solid line, one clearly sees that will diverge if at least two particles collapse towards the same point. This difficulty arises because the potential is not bounded from below as for the Lennard-Jones or Morse potential. In quantum mechanics, the collapse of self-gravitating fermions HertelThirringcommmathphys (); Chavanisreview2006 (), a system which is physically relevant for dwarfs and neutron stars, is forbidden by the Pauli exclusion principle, that introduces a natural effective small scale cut-off. However, to avoid the use of quantum concepts and stick to a classical model, the usual trick is to introduce an ad-hoc cut-off binneytremaine (). One way of regularizing the potential is shown in Fig. 5 by the dotted line. One can imagine that the potential has a hard core, which represents the particles’ size. As a consequence, the inequality allows to easily determine a finite upper bound of the configurational partition function

 ZN≤VN(2πλ2)3N/2N!exp[βC′N(N−1)/2] . (17)

As far as the microcanonical ensemble is concerned, a standard argument tells us that the entropy of a self-gravitating system in a finite volume might have convergence problems if the potential is not regularized at short distances. Indeed, let us consider gravitating particles grouped together in a finite volume. A strong decrease of the potential energy of a pair of particles is obtained when tends to . Since the total energy is a conserved quantity, the kinetic energy would correspondingly increase. This process leads to an increase of the accessible phase volume in the direction of momentum, and hence to an entropy increase. Since the process extends to the limit of zero distance among the particles, it might induce a divergence of the density of states, and then of Boltzmann entropy. However, one can show that this does not happens for and a more subtle derivation Padmanabhan90 (); Chabanol () reveals that the entropy integral diverges only when . It’s remarkable that this also corresponds to the transition from an integrable () to a non integrable () gravitational system. As a consequence of this discussion, “equilibrium” states can exist only in association with local entropy maxima Antonov (); Lyndenbell67 ().

Besides these unusual properties of the canonical and microcanonical partition functions, self-gravitating systems were historically the first physical system for which ensemble inequivalence was discovered through the phenomenon of negative specific heat. The possibility of finding a negative specific heat in gravitational systems was already emphasized by Emden Emden () and Eddington eddigton (). The divergence of the phase space volume for gravitational systems was proved by Antonov Antonov (), and the fact that this implies negative specific heats was later stressed by Lynden-Bell Lyndenwood68 (). An early remark on the possibility of having a negative specific heat can be found in the seminal review paper on statistical mechanics by Maxwell Maxwell (). This was for long time considered a paradox, until Thirring Thirringdd (); Thirring () finally clarified the controversial point by showing that the paradox disappears if one realizes that the microcanonical specific heat can be negative only in the microcanonical ensemble. Therefore, one can attribute to Thirring the discovery of ensemble inequivalence.

Let’s rephrase Thirring’s argument. In the canonical ensemble, the mean value of the energy is computed from the partition function (14) as

 ⟨E⟩=−∂lnZ∂β. (18)

It is then straightforward to compute the heat capacity at constant volume

 CV=∂⟨E⟩∂T=β2⟨(E−⟨E⟩)2⟩>0. (19)

Let’s remind that . This clearly shows that the canonical specific heat is always positive. Notice also that this condition is true for systems of any size, regardless of whether a proper thermodynamic limit exists or not.

For self-gravitating systems at constant energy (i.e., in the microcanonical ensemble) a simple physical argument which justifies the presence of a negative specific heat has been given by Lynden-Bell LyndenPhysA (). It is based on the virial theorem, which, for the gravitational potential, states that

 2⟨K⟩+⟨U⟩=0, (20)

where and are the kinetic and potential energy, respectively. Recalling that the total energy is constant

 E=⟨K⟩+⟨U⟩=−⟨K⟩, (21)

where in the second identity we have used the virial theorem (20), and since the kinetic energy defines the temperature, one gets

 CV=∂E∂T∝∂E∂K<0. (22)

Loosing its energy, the system becomes hotter.

There is a further difficulty in the case of gravitational interactions: the system is open, i.e. without boundary, strictly speaking. Therefore, the microcanonical partition function (6) will diverge. This divergence is actually not peculiar of self-gravitating systems since it would also occur for a perfect gas. Although for gases it is natural to confine them in a box, this is completely unjustified for gravitational systems. A way out from this pitfall is to consider an expanding universe and reconsider the problem in a wider context paddy ().

A detailed discussion on phase transitions in self-gravitating systems, in both the canonical and microcanonical ensemble, is not the aim of this review, and can be found in Refs. Padmanabhan90 (); Chavanisreview2006 ().

iii.2 Two-dimensional hydrodynamics

Two-dimensional incompressible hydrodynamics is another important case where long-range interactions appear. Although high Reynolds number flows have a very large number of degrees of freedom, one can often identify structures in the flow. This suggests that one could use a much smaller number of effective degrees of freedom to characterize the flow. This remark is particularly valid for two-dimensional flows, where the inverse energy cascade leads to the irreversible formation of large coherent structures (e.g. vortices). A system with a large number of degrees of freedom which can be characterized by a small number of effective parameters, is reminiscent of what happens in thermodynamics Castaing (), where a few macroscopic variables describe the behavior of systems composed of many particles. Statistical mechanics for turbulence is nowadays a very active field of research EyingSreenivasan (), initiated long ago by Lars Onsager Onsager49 (). We will discuss in the following the long-range character of two-dimensional hydrodynamics.

The velocity of a two-dimensional flow can be expressed in terms of the stream function , where is the coordinate on the plane

 vx = +∂ψ∂y (23) vy = −∂ψ∂x . (24)

The vorticity is related to the velocity field

 ω=∂vy∂x−∂vx∂y (25)

and, hence, to the stream function by the Poisson equation

 ω=−Δψ . (26)

Using the Green’s function , one easily finds the solution of the Poisson equation in a given domain

 ψ(→r)=∫Dd→r′ω(→r′)G(→r,→r′), (27)

plus surface terms Alastueybook (). In an infinite domain,

 G(→r,→r′)=−12πln|→r−→r′|. (28)

The energy is conserved for the Euler equation and is given by

 H = ∫Dd→r12(v2x+v2y) (29) = ∫Dd→r12(∇ψ)2 (30) = 12∫Dd→r ω(→r)ψ(→r) (31) = −14π∫D∫Dd→r% d→r′ω(→r′)ω(→r)ln|→r−→r′| . (32)

This emphasizes that one gets a logarithmic interaction between vorticies at distant locations, which corresponds to a decay with an effective exponent , well within the case of long-range interactions (see Fig. 4). For a finite domain the Green’s function contains additional surface terms Alastueybook (), which however gives no contribution to the energy (29) if the velocity field is tangent to the boundary of the domain (no outflow or inflow).

Another important conserved quantity is enstrophy, defined as

 A=12∫Dd→r[ω(→r)]2. (33)

The long-range character of the interaction is even more evident if one approximates the vorticity field by point vortices located at , with a given circulation

 ω(→r)=N∑i=1Γiδ(→r−→ri) . (34)

The energy of the system reads now

 H=−14π∑i≠jΓiΓjln|→ri−→rj| , (35)

where we have dropped the self-energy term because, although singular, it would not induce any motion MarchioroPulvirenti (). Considering the two coordinates of the point vortex on the plane, the equations of motion are

 Γidxidt = +∂H∂yi (36) Γidyidt = −∂H∂xi. (37)

The phase-space volume contained inside the energy shell can be written as

 Φ(E)=∫N∏i=1d→riθ(E−H(→r1,...,→rN)), (38)

being the Heaviside step function. The total phase space volume is , where is the area of the domain . One immediately realizes that is a non-negative increasing function of the energy with limits and . Therefore, its derivative, which is nothing but the microcanonical partition function (6), is given by

 Ω(E)=Φ′(E)=∫N∏i=1d→riδ(E−H(→r1,...,→rN)), (39)

and is a non-negative function going to zero at both extremes . Thus the function must achieve at least one maximum at some finite value where . For energies , will then be negative. Using the entropy , one thus gets that the inverse temperature is negative for . This argument for the existence of negative temperatures was proposed by Onsager Onsager49 () two years before the experiment on nuclear spin systems by Purcell and Pound PurcellPound51 () reported the presence of negative “spin temperatures”.

Onsager also pointed out that negative temperatures could lead to the formation of large-scale vortices by clustering of smaller ones. Although, as anticipated, the canonical distribution has to be used with cautions for long-range interacting systems, the statistical tendency of vortices of the same circulation sign to cluster in the negative temperature regime can be justified using the canonical distribution . Changing the sign of the inverse temperatures would be equivalent to reversing the sign of the interaction between vortices, making them repel (resp. attract) if they are of opposite (resp. same) circulation sign.

After a long period in which Onsager’s statistical theory was not further explored, this domain of research has made an impressive progress recently. One might in particular cite the work of Joyce and Montgomery Joyce73 (), who have considered a system of vortices with total zero circulation. Maximizing the entropy at fixed energy, they obtain an equation for the stream function which gives exact stable stationary solutions of the 2D Euler equations, able to describe the macroscopic vortex formation proposed by Onsager for negative temperatures. Later, Lundgren and Pointin Lundgren () studied the effect of far vorticity field on the motion of a single vortex, showing that it produces a positive eddy viscosity term leading to an increase of cluster size. Subsequently, Robert Robert90 () and Miller Miller90 () have elaborated an equilibrium statistical mechanical theory directly for the continuum 2D Euler equation. Nice prolongations along these lines are Refs. RobertSommeria91 (); MichelRobert94 (); ChavanisSommeriaRobert (), together with applications to the Great Red Spot of Jupiter Turkington (); Bouchet01 (); Bouchet02 (); chavanise ().

The canonical ensemble can also be defined for both the Euler equation in and the Onsager point vortex model. It turns out that in both cases, canonical and microcanical ensembles may be inequivalent Caglioti_95 (); KiesslingLebowitz97 (); ellisdd (); Ellis02 (). In particular for the Euler equation in the region of negative temperature (where vorticity tends to accumulate at the center of the domain) some hybrid states are shown to be realized in the microcanonical ensemble but not in the canonical ellisdd (). As far as the point vortex model is concerned, microcanonical stable states, that are unstable in the canonical ensemble, are found for specific geometries KiesslingLebowitz97 (). In both cases, microcanonical specific heat is negative. Reviews on 2D turbulence can be found in Refs chavanisf (); chavanisg (); chavanish (); ChavanisHouches (). Another related interesting case of ensemble inequivalence has been recently reported in the context of physical oceanography venaillebouchet ().

iii.3 Two-dimensional elasticity

Let us discuss the planar stress and displacement fields around the tip of a slit-like plane crack in an ideal Hookean continuum solid. The classical approach to a linear elasticity problem of this sort involves the search for a suitable “stress function” that satisfies the so-called biharmonic equation

 ∇2(∇2ψ)=0 (40)

where is the Airy stress function, which has to satisfy appropriate boundary conditions. The deformation energy density is then defined as where is the fracture stress field around the tip, whereas is the deformation field. Considering a crack-width and using the exact Muskhelishvili’s solution muskhelishvili (), one obtains the elastic potential energy due to the crack

 U≃σ2∞(1−ν)2Ea2r2, (41)

where is the Young modulus, the stress field at infinity, the Poisson coefficient and the distance to the tip. The elasticity equation in the bulk of two-dimensional materials leads therefore to a marginal case of long-range interaction, since in . Looking at the important engineering applications, the dynamics of this non conservative system should be better studied: the difficulty lies again in the long-range nature of the interaction. In addition, in such a two dimensional material, the presence of several fractures could exhibit a very interesting new type of screening effects.

iii.4 Charged systems

The partition function of system of charges is basically the same as the one for gravitational systems displayed in formula (14) with the potential given by

 U(→r1,…,→rN)=14πε0N∑i

where is the charge of the particle and is given by formula (16). If the total charge is non zero, the excess charge is expelled to the boundary of the domain and the bulk is neutral. Hence, neglecting boundary contributions (which are non extensive), one usually considers an infinite medium with total zero charge LebowitzLieb69 (); LiebLebowitz72 (). Similarly to the gravitational case, the partition function would diverge if not properly regularized at short distances. This is usually done by supposing that additional forces are present at short distances, either hard core of radius or smoothed Coulombic singularities of the type

 Vsmooth∼1−exp(−r/λ)r. (43)

The regularized partition function allows one to perform the thermodynamic limit and derive physical quantities, like the pressure.

As far as the large distance behavior is concerned, several rigorous results exist Martin () that prove, under appropriate hypotheses (low density and high temperature), that the effective two-body potential is Debye-Hückel screened

 Veff∝exp(−r/ℓD)r, (44)

where is the Debye length, with the density. Among the hypotheses, the most important one for its possible physical consequences is the one of low density or high temperature. This suggests that all pathologies related to ensemble inequivalence (e.g. negative specific heat), that is a consequence of the long-range nature of the interaction, will be indeed absent for charged systems.

An interesting different situation is the one of plasmas consisting exclusively of single charged particles (pure electron or pure ion plasmas) DubinOneil99 (). Charged particles are confined by external electric and magnetic fields. The thermodynamics of such systems is not affected by short-distance effects because particles repel each other. On the contrary, the behavior at large distances is different from the one of globally neutral low density plasmas. Nevertheless, by studying the correlation at equilibrium, a Debye length emerges DubinOneil99 () (this behavior is also present in self-gravitating systems). Experimentally, relaxation to thermal equilibrium has been observed only in some specific conditions. The system has been shown to relax into quasi-stationary states, including minimum enstrophy states (see Eq. (33)) and vortex crystal states HuangDriscoll (); Fine (). Indeed, in the particular case of a pure electron plasma in a cylindrical container with a strong magnetic field applied along the axis of the cylinder, it can be shown that the electron motion in the plane perpendicular to the magnetic field obeys equations that are the same as the Onsager point vortex model (36) and (37). Therefore, all what has been written above about two-dimensional hydrodynamics applies to this system, including the existence of negative specific heat which has been explicitly demonstrated in a magnetically self-confined plasma torus Kiessling03 ().

iii.5 Dipolar systems

Systems of electric and magnetic dipoles share many similarities, but also have important differences (force in non uniform external fields, the symmetry axial/polar of the dipolar vector). However, for what concerns statistical mechanics, the two systems are equivalent. Magnetic dipolar systems are easier to realize in nature; let us then concentrate on them. The interaction energy between two magnetic dipoles is

 Eij=μ04π⎡⎢⎣→μi⋅→μj|→rij|3−3(→μi⋅→rij)(→μj⋅→rij)|→rij|5⎤⎥⎦ , (45)

where is the magnetic moment, is the distance between the two dipoles and is the magnetic permeability of the vacuum. Dipolar interaction energy is strongly anisotropic: on a lattice, magnetic moments parallel to a bond interact ferromagnetically, while when they are perpendicular to the bond they interact antiferromagnetically. When magnetic moments are placed on a triangular or a square lattice the interaction is frustrated. Dipolar forces are long-range only in , because the energy per spin scales as . They are therefore marginally long-range in , while they are short-range in . It should be remarked that elasticity is instead marginally long-range in .

It can be shown in general that, for samples of ellipsoidal shape and when the magnetization lies along the longest principal axis of the ellipsoid, the energy per volume of a system of dipoles on a lattice can be written as

 HV=12V∑i,jEij=E0+12μ0|M|2D , (46)

where is an energy that depends on the crystal structure, and is the so called demagnetizing factor, which is equal to for spherically shaped samples, tends to for needle shape samples and to for disk shaped ones. The well-known shape dependence of dipolar energy is hence accounted for by the highly frustrating antiferromagnetic demagnetizing term in formula (46) LandauLifshitz (). We will see examples of energies of the form (46) in subsection VI.1, while in subsection VI.2.4, we will work out in detail a specific example corresponding to a realistic system.

Using methods similar to those introduced to prove the existence of a thermodynamic limit for short-range forces ruelle (), it can be shown Griffiths68 (); Griffiths00 () that a system of dipolar spins posses a well defined bulk free energy, independent of sample shape, only in the case of zero applied field. The key to the existence of this thermodynamic limit is the reduction in demagnetization energy when uniformly magnetized regions break into ferromagnetically ordered domains Kittel51 (). Technically, the proof is performed by showing that under the hypothesis of zero field the free energy is additive, using invariance under time reversal of the dipole energy, which is a consequence of its bilinearity in .

In the future, magnetic dipolar systems might constitute a field where the results on the statistical mechanics of systems with long-range will find a fruitful application. Besides the example discussed in subsection VI.2.4, holmium titanate materials Bramwell01 (), where dipolar interactions dominate over Heisenberg exchange energy, deserve some attention. The possibility to perform experiments with single domain needle shaped dipolar materials has been also stressed Barbara ().

iii.6 Small systems

As we have seen, the presence of long-range interactions causes the lack of additivity in macroscopic systems. However, even if the interactions are short-range, systems of a linear size comparable to the range of the interaction are nonadditive. Then, we should expect that some of the peculiar features found for macroscopic systems with long-range interactions are present also in microscopic or mesoscopic systems. Examples of this sort are: atomic clusters, quantum fluids, large nuclei, dense hadronic matter.

As it will be shown in the following section, the study of phase transitions is very important for the characterization of the properties of macroscopic systems with long-range interactions, especially in relation to the issue of ensemble inequivalence. However, phase transitions do occur also in atomic clusters (liquid-gas and solid-liquid transitions), quantum fluids (Bose-Einstein condensation or super-fluid transition), large nuclei (liquid-gas transition), dense hadronic matter (formation of quark-gluon plasma). Extensive studies have been devoted to phase transitions in the thermodynamic limit. This limit introduces simplifications in the analytical treatment, also in the case of long-range interactions. On the contrary, a consistent theory of phase transitions for small systems has not yet been developed. Signatures of phase transitions in finite systems are, however, often found both in numerical and laboratory experiments.

Gross has focused his attention on the theoretical treatment of “Small” systems grossphysrep (); Dieter (); grossdd (). His approach privileged the use of the microcanonical ensemble. Therefore, he immediately realized the possibility that specific heat could be negative and pointed out the feasibility of experiments with heavy nuclei.

According to Chomaz and collaborators, the key point is to begin with the general definition of entropy in the framework of information theory (see Refs. chonucphys1999 (); chopre2001 (); choepj2006 (); chomazassisi (); chomazdd (); chopre2002 ()). In order to treat on the same ground classical and quantum systems, Chomaz introduces the density matrix

 ^D=∑n|Ψn⟩pn⟨Ψn|, (47)

where are the states of the system and their probabilities. The entropy is thus defined by

 S[^D]=−Tr^Dln^D. (48)

Different Gibbs ensembles are obtained by maximizing with respect to the probabilities under some constraints. Each set of constraints defines an ensemble. The different number and functional forms of the constraints are related to the different physical situations. Within this framework, the finiteness of small systems is described by the introduction of specific constraints. It has been shown that phase transitions in finite systems can be equivalently signalled by three different effects. The first one is the bimodality of the density of states as a function of energy LabastieWhetten (), with the distance between maxima corresponding to different phases scaling as the number of particles. The second one is a negative slope of the microcanonical caloric curve, i.e. a negative specific heat LyndenMrMME (). The third one is the presence of anomalously large fluctuations in the energy partition between potential energy and kinetic energy. When the interactions are short-range, all these features give rise in the thermodynamic limit to the usual phase transitions, with the disappearance of the negative slope of the caloric curve. It is also possible to find a further signature of a phase transition, that makes a connection with Yang-Lee theory of phase transitions. In this latter theory, all zeroes of the partition function lie in the complex plane of the temperature for the canonical ensemble or of the fugacity for the grand-canonical ensemble, with an imaginary part different from zero as long as the system is finite. Phase transitions in the infinite system are associated to the approach of some of these zeros to the real axis, as system size increases. It has been stressed that the way in which zeroes approach the real axis may serve as a classification of phase transitions in finite systems chomazassisi ().

Some of the previously mentioned signatures of phase transitions in finite systems have been also experimentally reported. We mention here experiments on atomic clusters haberland (); gobet (), and experiments on nuclear fragmentation dagostino (). In all experiments, the microcanonical caloric curve is compatible with the presence of an energy range where specific heat is negative.

The first set of experiments is realized using atomic sodium clusters Na and hydrogen cluster ions H(H). In the first case, the negative specific heat has been found in correspondence to a solid-liquid phase transition, while in the second case in the vicinity of a liquid-gas transition. Sodium clusters haberland () are produced in a gas aggregation source and then thermalized with Helium gas of controlled temperature and selected to a single cluster size by a first mass spectrometer. Energy of the clusters is then increased by laser irradiation leading finally to evaporation. A second mass spectrometer allows the reconstruction of the size distribution, and correspondingly of their energies. Performing this experiment at different temperatures of the Helium gas, a caloric curve is constructed. The procedure assumes that, after leaving the source, a microcanonical temperature can be assigned to the clusters. Conceptually, this is probably the most delicate point of the experiment. A region of negative specific heat, corresponding to the solid-liquid transition, has been reported.

In the second set of experiments, performed with hydrogen cluster ions gobet (), the energy and the temperature are determined from the size distribution of the fragments after collision of the cluster with a Helium projectile. This is done using a method introduced in Ref. bonasera (). The reported caloric curve gobet () shows a plateau. Work along this line is in progress and seems to show a negative specific heat region farizonassisi (), corresponding to a liquid-gas transition.

In the third set of experiments on nuclear fragmentation dagostino (), the presence of negative specific heat is inferred from the event by event study of energy fluctuations in excited Au nuclei resulting from Au + Au collisions. The data seem to indicate a negative specific heat at an excitation energy around MeV/u. However, the signature corresponds to indirect measurements, and the authors cautiously use the word “indication” of negative specific heat.

Iv Equilibrium statistical mechanics: ensemble inequivalence

Our purpose in this section is to propose a fully consistent statistical mechanics treatment of systems with long-range interactions. An overview of the different methods (saddle-point techniques, large deviations, etc.) will be presented. Simple mean-field models will be used both for illustrative purposes and as concrete examples. Most of the features of long-range systems that we will discuss are valid also for non mean-field Hamiltonians as will be discussed in Sec. VI.

iv.1 Equivalence and inequivalence of statistical ensembles: physical and mathematical aspects

iv.1.1 Ensemble equivalence in short-range systems

In short-range systems, statistical ensembles are equivalent. A thorough proof of this result can be found in the book by Ruelle ruelle (). Let us first discuss the physical meaning of ensemble equivalence in order to clarify why for long-range interacting systems equivalence does not always hold.

The three main statistical ensembles are associated to the following different physical situations:

• a completely isolated system at a given energy : microcanonical ensemble;

• a system that can exchange energy with a large thermal reservoir characterized by the temperature : canonical ensemble;

• a system that can exchange energy and particles with a reservoir characterized by the temperature and the chemical potential : grand canonical ensemble.

Equivalence of the ensembles relies upon two important physical properties:

• in the thermodynamic limit, excluding critical points, the relative fluctuations of the thermodynamic parameters that are not held fixed (e.g. energy in the canonical ensemble) vanish;

• a macroscopic physical state that is realizable in one ensemble can be realized also in another (equivalence at the level of macrostates).

Let us concentrate our attention on the second item and let’s refer to the equivalence between microcanonical and canonical ensemble: an isolated system with a given energy has an average temperature, i.e., an average kinetic energy. If instead we put the system in contact with a thermal bath at that temperature, we have an average energy equal to the energy of the isolated system. Therefore there is a one-to-one correspondence between energy values and temperature values. Actually, in the presence of phase transitions, this statement has to be made more precise, as we will comment in subsection IV.1.2.

The practical consequence of ensemble equivalence is that, for computational purposes, one has the freedom to choose the ensemble where calculations are easier, and typically this is not the microcanonical ensemble (it is easier to integrate Boltzmann factors than -functions). Thus, in spite of its fundamental importance in the construction of statistical mechanics, the microcanonical ensemble is practically never used to perform analytical calculations. On the contrary it is very much used in numerical simulations, since it constitutes the fundamental ingredient of molecular dynamics FrenkelSmith ().

Ensemble equivalence is mathematically based on certain properties of the partition functions. To illustrate this point, we again consider the microcanonical and canonical ensembles, referring the reader to Ruelle ruelle () for a complete and rigorous discussion. A more precise definition of the microcanonical partition function (see formula (6)) of a system in with particles confined in a volume is given by

 Ω(E,V,N)=1N!∫Γdq3Ndp3Nδ(E−H(p,q)), (49)

where the domain of integration is the accessible phase space .

For lattice systems the definition of is slightly different. There is no explicit volume dependence (because volume is fixed once the lattice constant and are given) and no term due to the distinguishability of the lattice sites (for more details on the Gibbs paradox see  Huang ()). The microcanonical partition function is in this case

 Ωlattice(E,N)=∫N∏idqiN∏id% piδ(E−K({pi})−U({qi})), (50)

where and are the conjugate variables attached to site (see formula (5)) and the kinetic energy.

The entropy is defined by

 S(E,V,N)=lnΩ(E,V,N). (51)

The thermodynamic limit corresponds to , and such that and , where the density and the energy per particle are finite. The limit

 s(ε,n)=limN→∞1NS(E,V,N) (52)

exists and gives the entropy per particle. The function is continuous, increasing in at fixed , so that the temperature

 T=(∂s/∂ε)−1 (53)

is positive. Measuring the temperature by this formula seems hardly feasible. However, for Hamiltonians with kinetic energy, it can be shown that coincides with the average kinetic energy, which is accessible experimentally RUGH (). For short-range systems, is a concave function of at fixed , i.e.

 s(cε1+(1−c)ε2,n)≥cs(ε1,n)+(1−c)s(ε2,n) (54)

for any choice of and , with (in lattice systems without kinetic energy the energy can be bounded from above, and in turn this implies that can be negative; however, concavity is still guaranteed if the interactions are short-range). This property is important in connection with the partition function of the canonical ensemble, given by

 Z(β,V,N)=1N!∫Γdq3Ndp3Nexp[−βH(p,q)], (55)

with the inverse temperature. A similar definition of a canonical lattice partition function can be given, as done for the microcanonical lattice partition function (50). In the thermodynamic limit, the free energy per particle is

 f(β,n)=−1βlimN→∞1NlnZ(β,V,N) . (56)

Moreover, at fixed , the function is concave in . In the following will be called the rescaled free energy.

The equivalence between microcanonical and canonical ensemble is a consequence of the concavity of and and of the relation between these two functions given by the Legendre-Fenchel Transform (LFT). Indeed, one can prove that is the LFT of

 ϕ(β,n)=βf(β,n)=infε[βε−s(ε,n)] . (57)

A brief sketch of the proof goes as follows

 exp(−βNf(β,n)) = Z(β,V,N) (58) = 1N!∫dE∫dq3Ndp3Nδ(H(p,q)−E)exp(−βE), (59) = ∫dEΩ(E,V,N)exp(−βE), (60) = ∫dEexp(−N[βε−s(ε,n)]), (61)

where the last equality is valid for large . The saddle point of the last integral gives formula (57). Let us remark that it would not have been necessary to hypothesize the concavity of in from the beginning, because it follows from the fact that the LFT of a generic function is a concave function. Also the inverse LFT holds, since is concave in

 s(ε,n)=infβ[βε−ϕ(β,n)]. (62)

This indeed proves ensemble equivalence, because for each value of there is a value of that satisfies Eq. (57), and, conversely, for each value of there is a value of satisfying Eq. (62). Fig. 6 provides a visual explanation of the relation between and and of the correspondence between and .

Relations similar to those described in this subsection explain the equivalence of other ensembles. For instance Van Hove vanhove () proved that the equation of state for a short-range classical system is the same in the canonical and grand-canonical ensemble. This implies that isothermal compressibility is positive in both ensembles.

iv.1.2 Phase separation and Maxwell construction in short-range systems

It is important to discuss ensemble equivalence in the presence of phase transitions. We will see that some interesting features arise. Phase transitions are associated to singularities of thermodynamic functions Huang (). Therefore, in the microcanonical and canonical ensemble, they will be signaled by discontinuities in a derivative of some order of the entropy or the rescaled free energy .

Let us concentrate on the dependence of and on and , respectively. Consider for example a model in which, for some choice of parameters, has a zero curvature in some energy range . At both extremes of this interval the second derivative of has a discontinuity (see Fig. 7). Within the range , the function is not strictly concave (i.e., Eq. (54) is satisfied with an equality). All energy values in this range correspond to the same value of the inverse temperature , the slope of the straight segment of in Fig. 7. For each energy in this range, the system separates in two phases of different energies and . Therefore, the energy will be given by

 ε=cε1+(1−c)ε2 (63)

where is the fraction of phase and, of course, the fraction of phase . We emphasize that this relation is a direct consequence of additivity. It is easy to show that the Legendre-Fenchel transform of , the rescaled free energy (shown in Fig. 7) has a discontinuity in the first derivative with respect to at (). Following Ehrenfest’s classification, this is a first order phase transition.

Let us remark that in this example there is no one-to-one correspondence between and : several microcanonical macroscopic states are represented by a single canonical state. This shows that, in the presence of first-order phase transitions the relation between the ensembles must be considered with care. We would not say that the ensembles are inequivalent in this case, which is a marginal one; therefore we do not adopt the term used in mathematical physics in this case: “partial equivalence” TouchettePhysRep ().

Tuning the parameters of the model in such a way that the straight segment in Fig. 7 reduces to one point, one recovers a strictly convex entropy function and a one-to-one correspondence between and . Consequently, the discontinuity in the first derivative of the rescaled free energy is removed. This is the case of a second order phase transition.

It could also happen that the entropy, instead of showing the straight segment of Fig. 7, has a convex region (see the full line in Fig. 8a). For short-range interactions this is what is observed for finite systems near a phase transition. It has been shown numerically Dieter (); chomazdd (); chomazassisi () and for simple models MadameLyndenBell (); MadameLyndenBellFirst () that by increasing system size the entropy approaches the “concave envelope” which is constructed by replacing the full line in the energy range by the straight thick dashed line in Fig. 8a. In statistical mechanics, this procedure goes under the name of Maxwell’s construction and is mostly known in connection with the Van der Waals theory of liquid-gas transition Huang (). The relation between the construction of the “concave envelope” and the Maxwell construction can be easily established by looking at the following relation

 s(ε2)=s(ε1)+∫ε2ε1dεβ(ε), (64)

which derives from the definition of the inverse temperature , which is plotted in Fig. 8b as a function of energy. Another way of obtaining is by integrating along the thick dashed line in Fig. 8a.

 s(ε2)=s(ε1)+(ε2−ε1)βt, (65)

where . This implies that

 ∫ε2ε1dεβ(ε)=(ε2−ε1)βt. (66)

Splitting the integral in two intervals and one gets

 (ε1−ε3)βt+∫ε3ε1dββ(ε)=(ε2−ε3)βt+∫ε3ε2dββ(ε) . (67)

The value is the one obtained from the entropy by looking where, in the convex region, the entropy has slope . Condition (67) is equivalent to the equality of the areas and in Fig. 8. Introducing the generalized free energy, which is a function of both energy and inverse temperature,

 ˆf(β,ε)=ε−1βs(ε), (68)

one obtains from Eq. (67) that

 ˆf(βt,ε1)=ˆf(βt,ε2) . (69)

This shows that the requirement that the entropy is concave is equivalent to Maxwell’s equal areas construction and, in turn, equivalent to demand that the generalized free energies, computed at the transition inverse temperature and at the two energies and which delimit the coexistence region, are equal (and looking at Fig. 8a, also equal to .)

The Maxwell construction is related to the application of a maximum entropy principle for additive systems. Indeed, for all energies in the range , the entropy corresponding to the full line in Fig. 8a is smaller than the entropy corresponding to the dashed line at the same energy. This latter entropy is related to a system which has performed phase separation and is therefore obtained as a mixture composed of a certain fraction of a state with energy and the remaining fraction with energy , as in formula (63). Having this latter system a larger entropy, the natural tendency will be to phase separate. Hence the “concave envelope” recovers maximum entropy states.

It should be remarked that the truly “locally” convex part of the entropy is the one in the range , while the range is “globally” convex. One should therefore expect a difference in the properties of the physical states in the various ranges. Indeed, states in the range are unstable (dotted line in Fig. 8b), while states in the ranges and are metastable (dashed lines in Fig. 8b): at a solid-liquid phase transition they would correspond to superheated solids and supercooled liquids, respectively. While the unstable states cannot be observed, the metastable states are observable but are not true equilibrium states, because higher entropy phase separated states are accessible.

Considering again small systems, the nucleation of a bubble might lead to creation of a convex entropy region. Indeed, once a bubble of phase is nucleated in phase , the energetic cost of the interface is proportional to the surface, while the energetic gain is proportional to the volume. If the system is small enough, these two energies might be comparable, implying that the additivity property is not satisfied Binder03 (); Binder04 ().

iv.1.3 Ensemble inequivalence in long-range systems: negative specific heat

As already anticipated in the Introduction, an important physical property of systems with long-range interactions is that ensembles can be inequivalent. This means that experiments realized in isolated systems, described by the microcanonical ensemble, may give different results from similar experiments performed with well thermalized systems, for which the canonical ensemble is the appropriate one. For instance, while the specific heat will turn out to be always positive for a system in contact with a heat bath, it might be negative for an isolated long-range system.

When the interactions are long-range, an entropy function with a convex “intruder” Dieter () like the one shown by the solid line in Fig. 8a can represent truly stable equilibrium states. In Sec. IV.2, we will give a concrete example to illustrate this important property. Here, we will develop some general considerations which are not specific to a given model.

The construction which has led to the “concave envelope” for short-range systems cannot be realized for long-range systems. On the one hand the same notion of phase is ill defined for long-range systems (which are inherently inhomogeneous). On the other hand, even if a definition of phase would be possible, the lack of additivity of long-range systems would not allow to obtain a mixed state and, in particular, to derive relations like (63).

The starting point for the construction of a consistent thermodynamics of long-range systems is the calculation of microcanonical entropy associated to a given macrostate. A microstate is defined by the phase space variables of the system, and thus it refers to a precise microscopic state, while a macrostate is described in terms of a few macroscopic or coarse-grained variables, and then it generally defines a large set of microscopic states, all of them giving rise to the same values of the macroscopic variables. The derivation of free energy from microcanonical entropy using the Legendre-Fenchel transform (57) is still valid for long-range systems, ensuring that the function is concave also for these systems. However, when the entropy has a convex region, the inversion of the Legendre-Fenchel transform, Eq. (62) does not give the correct microcanonical entropy, but rather its “concave envelope” TouchettePhysRep (). Physically, this implies a lack of equivalence of ensembles at the level of macrostates, i.e. all microcanonical macrostates with energies between and do not have a corresponding macrostate in the canonical ensemble Touchette2003 ().

The existence of a convex “intruder” in the entropy-energy curve, as in Fig. 8a, is associated to the presence of negative specific heat. Indeed,

 ∂2S∂E2=−1CVT2. (70)

where the heat capacity at fixed volume is . Hence, in the energy range , the convexity of the entropy, , implies that the heat capacity is negative . This, in turn, implies that the conveniently normalized specific heat is also negative.

In the canonical ensemble, the specific heat is always positive, even if the interactions are long-range. This is a straightforward consequence of the concavity of the function , which is given by Eq. (57) also for long-range systems. Indeed,

 ∂2ϕ∂β2=−cVT2<0 , (71)

implying that . There is a subtlety related to the calculation of at . For a first order phase transition, since there is a discontinuity of the first derivative of at , the specific heat is not well defined and one rather speaks of latent heat, related to the jump of the energy as shown in Fig. 8b. At second order phase transitions, the second derivative of is instead well defined and is discontinuous at .

As explained above, the presence of a convex “intruder” like in Fig.8a in the entropy of long-range systems does not imply the appearance of singularities in the entropy, and therefore it could be doubted that this behavior signals a true phase transition in the microcanonical ensemble. Since this feature has been found first for gravitational systems, it is sometimes called “gravitational phase transition”. We will better clarify this issue analyzing what happens in the canonical ensemble. The rescaled free energy is again expressed by Eq. (57) and the mean value of the energy is

 ε(β)=∂ϕ∂β . (72)

The plot of is obtained from the curve in Fig. 8b by considering the ordinate as the control variable. The Maxwell construction is realized by the horizontal dashed line at . If, in the canonical ensemble, we start from a value of such that the energy of the system is less than and we gradually decrease , the system will reach the energy and then will jump to the energy , and after the jump will decrease continuously. Therefore, while in the microcanonical ensemble there is no singularity of the entropy, in the canonical ensemble there is a discontinuity of the derivative of the rescaled free energy , corresponding to a jump in the energy (associated to a latent heat). In the canonical ensemble the system has therefore a first order phase transition. Equilibrium macroscopic states with energies in the range do not exist, since the lack of additivity, as we noted, does not allow, contrary to short-range systems, to have mixtures of states as in Eq. (63). The temperature of the phase transition in the canonical ensemble is obtained by the Maxwell construction. Correspondingly, microcanonical microstates exist in the energy range and phase separation is not thermodynamically favoured in this ensemble.

The fact that the presence of a canonical first order phase transition is necessary to obtain ensemble inequivalence was conjectured in Ref. BMR (). This statement has been put on a more rigorous basis in Refs. Touchette2003 (); julienfreddyjstat (), analyzing the convexity properties of the entropy . In fact, it has been shown Touchette2003 () that if the rescaled free energy is differentiable, then the entropy can be obtained by its Legendre-Fenchel transform. This applies also for second order phase transitions, when the second derivative of is discontinuous. Therefore, in the presence of a second order phase transition in the canonical ensemble, the microcanonical and canonical ensembles are equivalent.

In this subsection we have discussed in detail the case where no singularity are present in the entropy. Although already showing all the features of ensemble inequivalence, this case is not generic and we’ll discuss in the next subsection a model that has a second order phase transition in the microcanonical ensemble and still a first order transition in the canonical ensemble.

Let’s conclude this subsection with a remark. We have remarked that energies between and correspond to the same value of in the canonical ensemble. It is interesting to figure out what happens if an initially isolated system with negative specific heat and with an energy between and , is put in contact with a heat bath that has its inverse temperature . Looking at Fig. 9 can be of help to understand the argument. We consider the case where the energy of the system lies in the range in which the specific heat is negative when the system is put in contact with the bath. Let us take for instance point in Fig. 9 as an initial point. We are interested to study the behavior of the system subjected to small perturbations, so that it can still be considered to be initially close to a microcanonical system. We see immediately that the system becomes unstable. In fact, if it gets a small amount of energy from the bath, its temperature lowers (negative specific heat!), and therefore further energy will flow from the bath to the system, inducing a lowering of the system’s temperature and then creating an instability. If, on the contrary, the initial energy fluctuation decreases the system’s energy, its temperature rises, inducing a further energy flow towards the bath, and, hence, an increase of system’s temperature. Thus, in contact with a heat bath, the system does not maintain energies in which its microcanonical specific heat is negative. The flow of energy started by the initial energy fluctuations stops when the system reaches again the same temperature of the bath, but at an energy for which its specific heat is positive. Looking at Fig. 9, it is clear that this could be either outside the range , i.e. point , or inside this range, point . This feature is valid for all points inside . Once in , the system will be in a thermodynamically metastable state and a sufficiently large fluctuation in the energy exchange with the bath will make it leave this metastable state, ending up again in a state with energy outside , i.e. point , which has the same inverse temperature of the bath . If the system instead jumps directly from to , it will stay there because this point lies on a thermodynamically stable branch.

iv.2 An analytical solvable example: the mean-field Blume-Emery-Griffiths model

We have presented above the main physical and mathematical aspects related to ensemble equivalence or inequivalence in the study of long-range systems. Other mathematical approaches and tools, that exist, will be presented in connection with concrete examples. Actually, this subsection is dedicated to a toy model that exhibits all features that have been discussed so far, in particular ensemble inequivalence and negative specific heat in the microcanonical ensemble. Historically, the relation between first order phase transition and negative specific heat for long-range systems in the thermodynamic limit was first pointed out in Refs. Antoni1 (); Antoni2 (). The phenomenology we are going to discuss in this section has been heuristically described in Ref. Antoni4 ().

iv.2.1 Qualitative remarks

The Blume-Emery-Griffiths (BEG) model is a lattice spin model with infinite range, mean-field like interactions whose phase diagram can be obtained analytically both within the canonical and the microcanonical ensembles. This study enables one to compare the two resulting phase diagrams and get a better understanding of the effect of the non-additivity on the thermodynamic behavior of the model.

The model we consider is a simplified version of the Blume-Emery-Griffiths model Blume (), known as the Blume-Capel model, where the quadrupole-quadrupole interaction is absent. The model is intended to reproduce the relevant features of superfluidity in He-He mixtures. Recently, it has also been proposed as a realistic model for metallic ferromagnetism Ayuela (). It is a lattice system (5), and each lattice point is occupied by a spin-1 variable, i.e., a variable assuming the values . We will consider the mean-field version of this model, for which all lattice points are coupled with the same strength. The Hamiltonian is given by

 H=ΔN∑i=1S2i−J2N(N∑i=1Si)2 , (73)

where is a ferromagnetic coupling constant and controls the energy difference between the ferromagnetic , or , and the paramagnetic, , states. In the following we will set , without loss of generality since we consider only ferromagnetic couplings. The paramagnetic configuration has zero energy, while the uniform ferromagnetic configurations have an energy . In the canonical ensemble, the minimization of the free energy at zero temperature is equivalent to the minimization of the energy. One thus finds that the paramagnetic state is the more favorable from the thermodynamic point of view if , which corresponds to . At the point , there is therefore a phase transition; it is a first order phase transition since, it corresponds to a sudden jump of magnetization from the ferromagnetic state to the paramagnetic state.

For vanishingly small ratio , the first term of Hamiltonian (73) can be safely neglected so that one recovers the Curie-Weiss Hamiltonian (1) with spin , usually introduced to solve the Ising model within the mean-field approximation. It is well known that such a system has a second order phase transition when (we remind that we are adopting units for which ). Since one has phase transitions of different orders on the and axis (see Fig. 10), one expects that the phase diagram displays a transition line separating the low temperature ferromagnetic phase from the high temperature paramagnetic phase. The transition line is indeed found to be first order at large values, while it is second order at small ’s.

iv.2.2 The solution in the canonical ensemble

The canonical phase diagram of this model in the is known since long time Blume0 (); Capel (); Blume (). The partition function reads

 Z(β,N)=∑{S1,…,SN}exp⎛⎝−βΔN∑i=1S2i+βJ2N(N∑i=1Si)2⎞⎠. (74)

Using the Gaussian identity

 exp(bm2)=√bπ∫+∞−∞dxexp(−bx2+2mbx), (75)

(often called the Hubbard-Stratonovich transformation) with and , one obtains

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 Z(β,N)=∑{S1,…,SN}exp(−βΔN∑i=1S2i)√Nβ2π∫+∞−∞dxexp(−Nβ2x2+mNβx). (76)