Recent progress in the study of long-range interactions

Transcription

1 Recent progress in the study of long-range interactions Thierry Dauxois 1 and Stefano Ruffo 1,2 1 Laboratoire de Physique de l École Normale Supérieure de Lyon, Université de Lyon, CNRS, 46 Allée d Italie, Lyon cédex 07, France 2 Dipartimento di Energetica S. Stecco, Università di Firenze and INFN, via s. Marta, 3, Firenze, Italy

2 Recent progress in the study of long-range interactions 2 1. General framework Systems with long-range interactions are characterized by a pair potential which decays at large distances as a power law, with an exponent smaller than the space dimension. Well-known examples are gravitational and Coulomb interactions, but one finds examples of long-range interacting systems also in two-dimensional hydrodynamics, atomic physics and condensed matter. This ubiquitous presence in different areas of physics alone justifies the need for a general and interdisciplinary understanding of the physical and mathematical problems raised by long-range interacting systems. Despite this, the thermodynamic and dynamical properties of such systems were poorly understood until a few years ago. Substantial progress has been made recently, since it has been realized that the lack of additivity induced by long-range interactions does not hinder the development of a fully consistent thermodynamic formalism, based on the mean-field limit. However, this has important consequences: entropy is no longer a convex function of macroscopic extensive parameters (energy, magnetization, etc) and the set of accessible macroscopic states does not form a convex region in the space of thermodynamic parameters. This is at the origin of ensemble inequivalence, which in turn determines curious thermodynamic properties, like negative specific heat in the microcanonical ensemble, first discussed in the context of astrophysics. As far as out-of-equilibrium dynamical properties are concerned, many-body longrange systems again show peculiar behaviours. Dynamics can be extremely slow and the approach to equilibrium can take a very long time, which increases with the number of elementary constituents. The state of the system during this long transient is quasi-stationary, since its very slow time evolution allows one to define slowly varying macroscopic observables, like for local equilibrium or quasi-static transformations. It should be remarked, however, 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 relies upon the dynamical properties of the system, by the development of kinetic theories adapted to long-range interactions. All this shows the great richness of the dynamics of long-range systems. 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. 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. All these aspects have been recently summarized in three books [1, 2, 3] and in review papers [4, 5]. However, this field is still very active, along many different lines of research and, following the invitation by the Editor-in-Chief of JSTAT, Henk van Beijeren, we present in this issue several contributions emphasizing different questions. 2. Different physical systems Historically, it is with the work of Emden and Chandrasekhar, and later Antonov, Hénon, Lynden-Bell and Thirring, in the context of astrophysics, that it has been realized that thermodynamic entropy might not have a global maximum, and therefore

3 Recent progress in the study of long-range interactions 3 thermodynamic equilibrium itself could not exist. The statistical mechanics of systems with gravitational interactions is nevertheless still an active field and several contributions belong to this category: Solvable model of a self-gravitating system by L Casetti and C Nardini, Cosmology in one dimension: fractal geometry, power spectra and correlation by B N Miller and J-L Rouet, Statistical mechanics of unbound two dimensional self-gravitating systems by T N Teles, Y Levin, R Pakter and F B Rizzato. Typicality analysis for the Newtonian N body problem on S 2 in the N limit by Michael K-H Kiessling. Coulomb systems is the second class of systems which have been extensively studied, and an interesting work is presented here: Binding and stability of Coulomb systems by W Thirring. The concept of negative temperature was first discussed in a seminal paper by Onsager on point vortices interacting via a long-range logarithmic potential in two dimensions. Two-dimensional hydrodynamics is an excellent, but much less wellknown, example of a system displaying characteristic features of systems with longrange interactions. One can cast in this line of research the paper Statistical mechanics of Beltrami flows in axisymmetric geometry by A Naso, S Thalabard, G Collette, P-H Chavanis and B Dubrulle. Pure electron plasmas, free electron lasers and charged particle beams exhibit the formation of multiple quasi-equilibria which persist over extremely long times. One can also induce long-range interaction in trapped atom and ion systems and observe striking self-organization phenomena. Related work is discussed in Experimental perspectives for systems based on long-range interactions by R Bachelard, T Manos, P de Buyl, F Staniscia, F S Cataliotti, G De Ninno, F Fanelli and N Piovella. 3. Simple toy models Some elements of the theory that might encompass long-range interactions (whose construction is still a challenge) have already emerged: (i) a thermodynamic formalism which must consistently include inequivalence of statistical ensembles; (ii) appropriate mathematical techniques to treat the thermodynamic limit of long-range interacting systems (large deviation theory as an example); (iii) a transport theory based on long-range (mean-field) effects rather than on collisional processes. Some contributions to this issue propose a bottom-up scenario, where simple toy models are investigated. These latter have the merit of encapsulating the main physical modalities while allowing for a dramatic reduction in complexity. The knowledge gained with reference to these preliminary applications can be subsequently transferred to other, hierarchically ordered, domains of investigations. Spin models are usually appropriate systems to test ideas. The first example Methods for calculating nonconcave entropies by H Touchette,

4 Recent progress in the study of long-range interactions 4 presents different methods which can be used to analytically calculate entropies that are nonconcave as functions of the energy in the thermodynamic limit. A second example is given by the contribution Models with short and long-range interactions: phase diagram and reentrant phase by T Dauxois, P de Buyl, L Lori and S Ruffo, in which the authors discuss ensemble inequivalence together with the possibility of observing reentrant phases. In addition, this contribution discuss competing effects of short- and long-range interactions, a largely open question. Finally, the authors of the paper Ensemble inequivalence in the ferromagnetic p-spin model in random fields by Z Bertalan, T Kuma and H Nishimori analyze the phase diagram of a ferromagnetic p-spin model with long-range interactions and a random magnetic field. While similar phase diagrams have been observed before in other models, the new feature here is the fact the system has a quenched randomness. Quantum aspects of long-range interacting systems are addressed in two different contributions again for simple spins systems, namely Nonequivalence of ensembles in the anisotropic quantum Heisenberg model by M Kastner, Dynamics of random dipoles: chaos vs ferromagnetism by F Borgonovi and G L Celardo. Quasi-stationary states. It has been recognized that systems with long-range interactions display long-lived states, in which the system remains trapped for a long time before relaxing towards thermodynamic equilibrium. These states, which have been later called Quasi Stationary States (QSS), arise in several different physical contexts of broad applied and theoretical interest. The ubiquity of QSS has been accepted as an important general feature. This question is considered in Relaxation to thermal equilibrium in the self-gravitating sheet model by M Joyce and T Worrakitpoonpon, while the description of the kinetic theory of long-range interacting systems is presented in Kinetic equations for systems with long-range interactions: a unified description by P-H Chavanis. The robustness of QSS with respect to stochastic processes beyond deterministic dynamics within a microcanonical ensemble is considered in Relaxation dynamics of stochastic long-range interacting systems by S Gupta and D Mukamel. Inhomogeneous states. It must be stressed that the nature of quasi-stationary states can be strongly dependent on the initial condition. 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. However, very few papers have addressed this question of inhomogeneity so far. The two following papers are devoted to this important question:

5 Recent progress in the study of long-range interactions 5 Dynamics of perturbations around inhomogeneous backgrounds in the HMF model by J Barré, A Olivetti and YYYamaguchi, Dynamical stability criterion for inhomogeneous quasi-stationary states in longrange systems by A Campa and P-H Chavanis. 4. Mathematical aspects Long-range interacting systems are also very interesting from the mathematical point of view: a few questions are discussed in this issue. It has been established by Neunzert, Braun, Hepp, Spohn and others how to treat the stability in finite times of the trajectories of interacting particles. The Vlasov equation has thus been shown to be of paramount importance for systems with long-range interactions. However, several questions are still pending. For potentials less singular than the classical electrostatic kernel, the paper Stability of trajectories for N-particles dynamics with singular potential by J Barré, M Hauray and P E Jabin discusses these issues for initial positions/velocities distributed according to the Gibbs equilibrium of the system. Invariant measures of partial differential equations, such as the Euler or the Vlasov equations, are discussed in Invariant measures of the 2D Euler and Vlasov equations by F Bouchet and M Corvellec. We hope that through this selection of problems the reader will discover interesting new topics. This domain of research is very promising and still challenging since basic properties that are common for systems with short-range interactions (like additivity and extensivity) are here violated. Key features are inequivalence of ensembles for equilibrium properties, and extremely slow relaxation for what concerns the dynamical properties, but more subtle effects are still to be discovered. References [1] Dauxois T, Ruffo S, Arimondo E and Wilkens M (eds), 2002 Dynamics and Thermodynamics of Systems with Long-Range Interactions (Lecture Notes in Physics vol. 602) (Berlin: Springer) [2] Campa A, Giansanti A, Morigi G and Sylos Labini F (eds), 2008 Dynamics and Thermodynamics of Systems with Long-Range Interactions: Theory and experiments (AIP Conf. Proc. 965) 122 [3] Dauxois T, Ruffo S and Cugliandolo L (eds.), 2009 Long-Range Interacting Systems (Les Houches Summer School 2008) (Oxford: Oxford University Press) [4] Campa A, Dauxois T and Ruffo S, 2009 Phys. Rep [5] Bouchet F, Gupta S and Mukamel D, 2010 Physica A