Statistical mechanics of systems with long range interactions

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1 Journal of Physics: Conference eries tatistical mechanics of systems with long range interactions To cite this article: Freddy Bouchet and Julien Barré 2006 J. Phys.: Conf. er View the article online for updates and enhancements. Related content - Lyapunov exponent and criticality in the Hamiltonian mean field model L H Miranda Filho, M A Amato and T M Rocha Filho - Continuous and discontinuous transitions in geophysical turbulence Marianne Corvellec and Freddy Bouchet - Long range interaction of antiprotonic helium with helium Quan-Long Tian, Zhen-Xiang Zhong and Vladimir I Korobovt Recent citations - Long-range orientational order in twodimensional microfluidic dipoles Itamar hani et al - An extension of the linear Luikov system equations of heat and mass transfer Fernanda R.G.B. ilva et al - Non-Debye relaxation in the dielectric response of nematic liquid crystals: urface and memory effects in the adsorption-desorption process of ionic impurities J. de Paula et al This content was downloaded from IP address on 06/07/2018 at 05:28

2 Institute of Physics Publishing Journal of Physics: Conference eries 31 (2006) doi: / /31/1/003 The Third 21CO ymposium: Astrophysics as Interdisciplinary cience tatistical mechanics of systems with long range interactions Freddy Bouchet 1, Julien Barré 2 1) INLN (Institut Non Linéaire de Nice) - CNR, UMR 6618 CNR / UNA, 1361 route des lucioles, Nice-ophia Antipolis, F06560 Valbonne, France 2) Laboratoire J.-A. Dieudonné, UMR 6621 du CNR, Université de Nice - ophia Antipolis, Parc Valrose, Nice Cedex 02, France -mail: Freddy.Bouchet@inln.cnrs.fr,jbarre@math.unice.fr Abstract. Many physical systems are governed by long range interactions, the main example being self-gravitating stars. Long range interaction implies a lack of additivity for the energy. As a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the statistical mechanics of such systems is a well understood subject. In this proceeding, we explain briefly the classical way to equilibrium and non equilibrium statistical mechanics, starting from first principles and emphasizing some new results. At equilibrium, we explain how the Boltzmann-Gibbs entropy can be proved to be the appropriate one, using large deviations tools. We explain the thermodynamics consequences of the lack of additivity, like the generic occurence of statistical ensemble inequivalence and negative specific heat. This well known behavior is not the only way inequivalence may occur, as emphasized by a recent new classification of phase transitions and ensemble inequivalence in systems with long range interaction. We note a number of generic situations that have not yet been observed in any physical systems. Out of equilibrium, we show that algebraic temporal correlations or anomalous diffusion may occur in these systems, and can be explained using usual statistical mechanics and kinetic theory. 1. Introduction In a large number of physical systems, any single particle feels a potential dominated by interactions with far away particles: this is our definition of long range interactions. In a system with algebraic decay of the inter-particle potential V (r) r r α, this occurs when α is less than the dimension of the system (these interactions are sometimes called non-integrable ). Then the energy is not additive, as the interaction of any subpart of the system with the whole is not negligible with respect to the internal energy of this given part. elf gravitating stars is the main example of systems with long range interaction. After the discovery of negative specific heat by Lynden Bell [39], this system has played a very important historical role, by emphasizing the peculiarities in the statistical mechanics of such systems. Besides astrophysical self gravitating systems [18, 21, 28, 29, 30, 33, 37, 41, 49, 53], the main physical examples of non-additive systems with long range interactions are two-dimensional or geophysical fluid dynamics [12, 13, 16, 34, 42, 50] and a large class of plasma effective models [22, 6, 26, 35]. pin systems [4] and toy models with long range interactions [1, 2, 20] have also 2006 IOP Publishing Ltd 18

3 19 been widely studied. The links between these different subjects have been emphasized recently [20]. Long range interacting systems are known to display peculiar thermodynamic behaviors. As additivity is often seen as a cornerstone of usual statistical mechanics and thermodynamics, it is sometimes written in textbooks or articles that statistical mechanics or thermodynamics do not apply to systems with long range interactions. In this paper, we argue on the contrary that usual tools and ideas of statistical mechanics do apply to such systems, both at equilibrium and out of equilibrium. However, reviewing a variety of recent works, we will show that a careful application of these tools reveals truly unusual and fascinating behaviors, absent from the world of short range interacting systems. In section 2, we discuss the usual assumptions of equilibrium statistical mechanics and their interpretation in systems with long range interactions. For equilibrium, based on the assumption of equal probability of any configuration with given energy, we explain why the Boltzmann-Gibbs entropy actually measures the probability to observe a given distribution function. This relies on our ability to prove large deviations results for such systems. We extend the discussion to other examples where another macroscopic variable (not the distribution function) may be chosen [5]. The result of this analysis is that microcanonical and canonical ensembles of systems with long range interactions are described by two dual variational problems. We explain why such variational problems lead to possible generic ensemble inequivalence, and to a richer zoology of phase transitions than in usual systems. A natural question then arises: do we know all possible behaviors stemming from long range interactions, and, if not, what are the possible phenomenologies? In section 4, we answer this question by discussing a classification of all microcanonical and canonical phase transitions, in long range interacting systems, with emphasis on situations of ensemble inequivalence [9]. Because systems with long range interactions relax very slowly towards equilibrium, the study of out of equilibrium situations is physically essential. During the past century, there have been many attempts to find a general formalism for out of equilibrium statistical mechanics, which would give the equivalent of the Gibbs picture for out of equilibrium states. Unfortunately, as recognized by most of the statistical mechanics community, until now any such attempt failed. This is mainly due to the fact that our knowledge of out of equilibrium situation can not be parameterized by a small number of macroscopic quantities, playing the same role as dynamical invariants for the equilibrium theory. Then out of equilibrium statistical mechanics must be addressed by a case by case careful examination of dynamics, using some appropriate probabilistic description. tandard tools have been developed for this purpose, like kinetic theory. In section 5, we quickly discuss some recent results for the kinetic theory of the HMF model, a toy model with long range interactions. Using this model, we explain, how kinetic theory can explain long range temporal correlations and anomalous diffusions in such systems [10]. 2. quilibrium statistical mechanics of systems with long range interactions 2.1. Peculiarities of thermodynamics of systems with long range interactions For systems with long range interactions, the most intriguing thermodynamical property is the generic occurence of statistical ensemble inequivalence and negative specific heat. uch possibilities have first been recognized and studied in the context of self gravitating systems [39, 30, 31]. Afterwards, ensemble inequivalence and negative specific heat have been observed or predicted in a number of different physical systems : two dimensional turbulence [13, 51, 23], plasma physics [51, 35], spin systems or toy models [4, 2], or self gravitating systems in situations different from the simple initial case [18, 41, 53, 49, 28, 33, 52]. A detailed description of each

4 20 Figure 1. Caloric curve (temperature T = 1 as a function of the energy U) for the GR model, for ɛ =10 2. The energies corresponding to the dashed vertical lines are refered to, from left to right, as U low, U top, U c,andu high. The decreasing temperature between U top and U c characterizes a range of negative specific heat and thus of ensemble inequivalence. At U c, there is a microcanonical second order phase transition associated with the canonical first order phase transition (please see the text for detailed explanation). This contrasts with the usual ensemble inequivalence. uch a behavior is linked to the existence of tricritical points in both statistical ensembles (see figure 2). A classification of all possible routes to ensemble inequivalence is briefly described in section 4 or in [9]. of these cases is provided in [9]. To motivate the following development and let us give an example where ensemble inequivalence appears in an unsual way. We discuss the equilibrium properties of the elf Gravitating Ring (GR) model, a toy model for self gravitating systems. Whereas we mainly present here its equilibrium properties, we stress that this systems is very interesting also from a dynamical point of view, as it shows a number of out of equilibrium quasi-stationary states [32, 46]. The Hamiltonian of the GR model is : H = 1 Ni=1 2 p 2 i 1 Ni,j= Particles 1 cos(θi θ j )+ɛ are constrained on a ring (0 θ i 2π). The angles θ i are conjugate to the momenta p i. ɛ is a small scale softening of the gravitational interaction. We study the phase transition of this system and how they evolve when ɛ is varied. Please see [52] for a detailed discussion. Figure 1 shows the caloric curve T (U) forɛ =10 2, where T is the temperature and U is the energy. For U top <U<U c the temperature is decreasing. The specific heat C = du/dt is thus negative in this area, showing that statistical ensembles are not equivalent (in the canonical ensemble, the specific heat is always positive). The horizontal dashed line is the Maxwell construction, which links microcanonical and canonical ensembles. From it one sees that in the canonical ensemble, when is varied, there is a first order phase transition characterized by an energy jump between the values U low and U high. This is a common feature in case of ensemble inequivalence. What is less common is the concomitant existence of a second order phase transition in the microcanonical ensemble, at the energy U c. At this point, the temperature T is continuous, whereas its derivative is discontinuous as it is clear from the curve. In figure 2, we show that this type of ensemble inequivalence, with the coexistence of a first order canonical first order phase transition and of a microcanonical second order phase transition, is linked to the existence of a tricritical point in both ensembles. The GR model displays one possible route to ensemble inequivalence, out of several others. However, there are some constraints on the possible phenomenologies. For instance, at a

5 21 Figure 2. Phase diagrams for the GR model. Left: for each value of ɛ, the critical energies where a microcanonical phase transition occurs are represented by black points or circles. When the softening parameter is varied, at the tricritical point ɛ = ɛ µ T, first order phase transitions (black points) change to second order phase transitions (circles). Right: the bottom dashed line and black line represent respectively U low (see fig. 1) and U high. The difference U high U low is the energy jump associated to a first order phase transition in the canonical ensemble. This jump decreases to zero at canonical tricritical point ɛ c T where the canonical first order phase transition changes to a canonical second order phase transition. The tricritical points have different energies and ɛ values in both ensembles U 0 U oftening parameter oftening parameter second order phase transition, the negative specific heat jump must be positive. By contrast, the temperature jumps at a discontinuity associated with a first order microcanonical phase transition must be negative (this means that when energy is increased the system has a negative temperature jump). ummarizing all these constraints yields a classification [9] of all possible ensemble inequivalences and their links with phase transitions. To prepare the discussion of this classification in section 4, we recall now the main hypothesis and definitions of statistical equilibrium, in the context of systems with long range interactions Additivity and extensivity of systems with long range interactions Let us consider N particles which dynamics is described by the Hamiltonian H = 1 N p 2 i + c N V (r i r j ) (1) 2 2 i=1 i,j=1 where c is a coupling parameter. As discussed in the introduction, for systems with long range interactions the potential V is not integrable. As a consequence the system is not additive, in the sense that the potential at a given point is dominated by the contributions of far away particles. For self gravitating stars, V (r) =1/r and c = Gm 2 (equal mass stars), it is well known that a proper choice of space and time variable allow one to rescale the coupling parameter. This type of rescaling is actually possible for any system with long range interaction; it is convenient to choose the variables so that c = ±1/N : H N = 1 2 N i=1 p 2 i ± 1 2N N i,j=1 V (r i r j ) (2) This classical scaling of the coupling parameter is called the Kac s presciption [36] or sometimes the mean field scaling (see for instance [40]). Within this scaling, taking the limit N with

6 22 all other parameters fixed (fixed volume for instance), the sum over i and j is clearly of order N 2, and the energy H N is extensive H N N N. This scaling is also the relevant one in order to obtain the collisionless Boltzmann equation, for the dynamics, in the large N limit. From a physical point of view, such a scaling is relevant as soon as the energy per particle remains finite with the new variables. This is indeed the case for galaxies, globular clusters, hot plasmas and for many other physical problems. Thus, the long range thermodynamic limit N with all other parameters fixed (fixed volume) is the relevant one for systems with long range interactions; this is at variance with short range interacting systems where the usual thermodynamic limit must be considered (in this case, the volume is not kept fixed) The microcanonical and canonical ensembles In the sections 3 and 4, we will consider equilibrium properties of systems with long range interactions. We suppose that the energy of our system is known. We consider the microcanonical ensemble. In this statistical ensemble all phase space configurations with energy have the same probability. We thus consider the microcanonical measure µ N = 1 Ω N () N dr i dp i δ (H N ({r i,p i }) ), i=1 where Ω N () is the volume of the energy shell in the phase space Ω N () dri dp i δ (H N ({r i,p i }) ). We consider here the energy as the only parameter, however generalization of the following discussion to other quantities conserved by the dynamics is straightforward. The only hypothesis of equilibrium statistical mechanics is that averages with respect to µ N will correctly describe the macroscopic behavior of our system. This hypothesis is usually verified after a sufficiently long time, when the systems has relaxed to equilibrium. The Boltzmann entropy per particle is defined as N () 1 N log Ω N () In the following, we will justify that in the long range thermodynamic limit, the entropy per particle N () has a limit: N () N () The canonical ensemble is defined similarly, using the canonical measure µ c,n = 1 Z N () N dr i dp i exp [ H N ({r i,p i })], i=1 with the associated partition function Z N () dr i dp i exp [ H N ({r i,p i })] and free energies F N () 1 N log Z N () andf N () N F () 3. Large deviation results 3.1. Justification of the Boltzmann-Gibbs entropy Let us consider the particle distribution on µ space f (r, p) (f (r, p) drdp is the probability to observe a particle with position r and momentum p). f defines a macrostate as many microscopic states corresponds to a given f. As explained in section 2, the hypothesis of usual statistical mechanics is that all microscopic states with a given energy are equiprobable. Given this uniformity in phase space, we address the question: what is the number of microscopic states having the distribution f?

7 23 It is a classical result to show that the logarithm of number of microscopic states corresponding to a distribution f is given by s [f] = drdp f log f where s is sometimes called the Boltzmann-Gibbs entropy. It is the Boltzmann entropy associated to the macrostate f, in the sense that it counts the number of microstates corresponding to f. We stress that no other functional has this probabilistic meaning, and that this property is independent of the Hamiltonian. Thanks to the long range nature of the interaction, for most configurations, the energy can be expressed in term of the distribution function f, using h [f] N dr 1 dp 1 dr 2 dp 2 f(r 1, p 1 )f(r 2, p 2 )V (r 1 r 2 ). (3) This mean field approximation for the energy allows to conclude that the equilibrium entropy is given by N () =log(ω N ()) N N() with () =sup{s(f) s(f) =} (4) m In the limit of a large number of particles, the mean field approximation q. (3) and its consequence the variational problem (4) have been justified rigorously for many systems with long range interactions. The first result assumes a smooth potential V and has been proved by Messer and pohn [40], see also the works by Hertel and Thirring on the self gravitating fermions [31] Large deviations We explained why the Boltzmann-Gibbs entropy is the correct one to describe the probability of a given f. Large deviations provide a modern tool to obtain similar results in a wider context. The above picture then generalizes. We refer to [5] for a simple detailed explanation of many large deviations results in the context of long range interacting systems. We also refer to the book of llis and the very interesting contributions of llis and coworkers [7, 23, 25, 24]. In a first step one describes the system at hand by a macroscopic variable; this may be a coarse-grained density profile f, a density of charges in plasma physics, a magnetization profile for a magnetic model. In the following, we will generically call this macroscopic variable m; it may be a scalar, a finite or infinite dimensional variable. One then associates a probability to each macrostate m. Large deviation theory comes into play to estimate Ω(m), the number of microstates corresponding to the macrostate m: log (Ω N (m)) N Ns(m). This defines the entropy s(m). In a second step, one has to express the constraints (energy or other dynamical invariants) as functions of the macroscopic variable m. In general, it is not possible to express exactly H; however, for long range interacting systems, one can define a suitable approximating mean field functional h(m), as in q. (3). Having now at hand the entropy and energy functionals, one can compute the microcanonical density of states Ω() [23]: the microcanonical solution is simply given by the variational problem log (Ω N ()) N N() with () =sup{s(m) h(m) =} (5) m

8 Figure 3. At some critical values of its energy, a system may have phase transitions. We study all the possible changes in such phase transitions when one external parameter is varied (codim 1). This figure summarizes bifurcations in the microcanonical ensemble (first collumn) and the associated changes in the canonical ensemble (third collumn) (in order to study the links between these two ensembles, one has to consider the convexification of the entropy curves, which is also summarized ; please see [9] for a detailed explanation). One recover the usual phase transitions (triple point, azeotropy, critical point) in both ensembles. What is new is the list of the possible links between the behaviors in each ensembles, and the associated appearance of ensemble inequivalence. ingularities Concavity Concavification () () Critical point CCC CVC Maximization singularities Triple point CCV CVV VCV VVV CC Azeotropy Azeotropy VV VC Convexity modification C V Critical point Convexity First order and cano. destab. CV C CV V VC C VC V Concavification Triple point In the canonical ensemble, similar considerations lead to the conclusion that the free energy and the canonical equilibrium are given by the variational problem log (Z N ()) N NF() with F () =inf m { s(m)+h(m)} (6) We insist that this reduction of the microcanonical and canonical calculations to the variational problems (5) and (6) is in many cases rigorously justified. 4. Classification of phase transitions in systems with long range interactions The thermodynamics of long range interacting systems in the microcanonical and canonical ensembles is now reduced to the dual variational problems (5) and (6). The precise form of the solution obviously depends on the problem at hand through the functions s(m) and h(m). However, many qualitative features of the thermodynamics depend only on the structure of the variational problem. Thus, it is possible to classify all the different possible phenomenologies that one may find in the study of a particular long range interacting system. This is done in [9], with a particular emphasis on the situations of ensemble inequivalence, and the way these may arise. All types of phase transitions and ensemble inequivalences found in the literature so far are reproduced in the classification. In addition, the classification predicts the possibility of new phenomenologies, and new routes to ensemble inequivalence, that have not so far been observed in any specific model.

9 25 Figure 4. Angles diffusion ( < (θ(t) θ(0)) 2 > as a function of time) in the HMF model, for a quasi-stationnary state. Points are from a N body numerical simulation, the straight line is the analytic prediction by the kinetic theory [10]. For large times < (θ(t) θ(0)) 2 > t t ν with ν 1. uch an unexpected anomalous diffusion is also observed at equilibrium (see [10] for more details) 1e+11 1e+10 x**(1.5)*5e+8 "average.m-n50000" u (1/50000):(4) 1e+09 1e+08 1e+07 1e Out of equilibrium statistical mechanics and kinetic theory After a rapid violent relaxation, systems with long range interactions relax towards quasistationnary (virialized) states [55]. These states are often out of equilibrium. They then relax very slowly towards equilibrium on a time scale t R diverging with the number of particle (for instance t R log(n)/n for self gravitating systems or t R N for smooth potentials or t R >> N for one dimensionnal systems [10]). In this relaxation regime, thanks to the separation of time scale between the evolution of fluctuations and the evolution of the quasi-stationnary density f, a kinetic approach of the relaxation towards equilibrium is justified. This gives the classical Lenard-Balescu equation for plasma physics, or the kinetic equations for self-gravitating systems [29, 47]. We note a peculiarity for one dimensionnal systems where the Lenard-Balescu operator vanishes [10]. We also stress, that the Boltzmann-Gibbs entropy is still the correct entropy to measure the relative probability of an out of equilibrium state, in such systems 1. We want to stress some recent results, in the old and long history of kinetic theory for systems with long range interactions. Let us consider the Hamiltonian Mean Field model [1] with Hamiltonian (2), with a cosine interaction. Figure 4 shows the numerically computed diffusion of angles compared with an asymptotic behavior predicted using kinetic theory. This figure illustrates some unexpexted behavior of systems with long range interactions. Because of a very small diffusion in the momentum space, some very long temporal correlations do exist. This is illustrated by the algebraic tails in the autocorrelation function [45]. This behavior can be predicted using kinetic theory [10] and is directly linked to the anomalous diffusion. This shows that ususal kinetic theory can explain unusual behavior of the dynamics of systems with long range interactions (we note that we disagree with an other interpretation using Tsallis statistics. We refer to [10] and references therehein for a complete discussion). ome recent related results have also been reported in [19]. Thanks to kinetic theory, the index for anomalous diffusion can thus be predicted from first principles. Acknowledgments We thank P.H. Chavanis, T. Dauxois,. Ruffo, T. Tatekawa and Y. Yamaguchi for the interesting works and discussion during last years. Freddy Bouchet thanks K. Maeda and his group, for the kind hospitality in Tokyo, and the very interesting discussions held there. 1 This property is not valid in general, out of equilibrium. It may be valid when the two-point correlations between particule can be neglected. This is the case for systems with long range interaction in the kinetic regime. This is also the case for a dilute gas, as shown by the celebrated works of Boltzmann more than one century ago.

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