DYNAMICS AND THERMODYNAMICS OF SYSTEMS WITH LONG RANGE INTERACTIONS: Theory and Experiments

The study of the equilibrium and of the dynamical properties of long‐range interacting systems
View Description Hide DescriptionAfter reviewing the main results governing the statistical mechanics description of short‐range systems, we show how most of them do not extend their validity to long‐range systems. In particular, we emphasize the problem of the ensemble inequivalence and its physical implications. Then we show the peculiarities of the dynamics of long‐range systems, in particular the phenomenon of very slow relaxation of states that thermodynamically are very far from equilibrium or metastable states. Finally, we give a short description of the papers presented in this section of the volume.

Statistical Mechanics of systems with long range interactions
View Description Hide DescriptionRecent theoretical studies of statistical mechanical properties of systems with long range interactions are briefly reviewed. In these systems the interaction potential decays with a rate slower than at large distances r in d dimensions. As a result, these systems are non‐additive and they display unusual thermodynamic and dynamical properties which are not present in systems with short range interactions. In particular, the various statistical mechanical ensembles are not equivalent and the microcanonical specific heat may be negative. Long range interactions may also result in breaking of ergodicity, making the maximal entropy state inaccessible from some regions of phase space. In addition, in many cases long range interactions result in slow relaxation processes, with time scales which diverge in the thermodynamic limit. Various models which have been found to exhibit these features are discussed.

Dynamics and thermodynamics of systems with long‐range interactions: interpretation of the different functionals
View Description Hide DescriptionWe discuss the dynamics and thermodynamics of systems with weak long‐range interactions. Generically, these systems experience a violent collisionless relaxation in the Vlasov regime leading to a (usually) non‐Boltzmannian quasi stationary state (QSS), followed by a slow collisional relaxation leading to the Boltzmann statistical equilibrium state. These two regimes can be explained by a kinetic theory, using an expansion of the BBGKY hierarchy in powers of where N is the number of particles. We discuss the physical meaning of the different functionals appearing in the analysis: the Boltzmann entropy, the Lynden‐Bell entropy, the “generalized” entropies arising in the reduced space of coarse‐grained distribution functions, the Tsallis entropy, the generalized H‐functions increasing during violent relaxation (not necessarily monotonically) and the convex Casimir functionals used to settle the formal nonlinear dynamical stability of steady states of the Vlasov equation. We show the connection between the different variational problems associated with these functionals. We also introduce a general class of nonlinear mean field Fokker‐Planck (NFP) equations that can be used as numerical algorithms to solve these constrained optimization problems.

Statistical equilibrium dynamics
View Description Hide DescriptionWe study the mean‐field thermodynamic limit for a class of isolated Newtonian N‐body systems whose Hamiltonian admits several additional integrals of motion. Examples are systems which are isomorphic to plasma models consisting of one specie of charged particles moving in a neutralizing uniform background charge. We find that in the limit of infinitely many particles the stationary ensemble measures with prescribed values of the integrals of motion are supported on the set of maximum entropy solutions of a (time‐independent) nonlinear fixed point equation of mean‐field type. Each maximum entropy solution of this fixed point equation can in turn be either a static or a stationary solution for the entropy‐conserving Vlasov evolution, or even belong to a one‐dimensional orbit of maximum entropy solutions which evolve into one another by the Vlasov dynamics. In short, the macrostates of individual members of an equilibrium ensemble are not necessarily themselves in a state of global statistical equilibrium in the strict sense. Yet they are always locally in thermodynamic equilibrium, and always global maximizers of the pertinent maximum entropy principle.

Low dimensional features of the Hamiltonian Mean Field model
View Description Hide DescriptionThe order parameter of the Hamiltonian Mean Field (HMF) model, which describes the motion of N globally coupled rotors, is a two‐component vector Its dynamics is found to be “cyclic”: the vector approaches a fixed norm and rotates around the origin. We here show that, using the equations of motion of the HMF model and making a crucial Ansatz, one can derive an infinite set of second order differential equations involving the “moments” of the particle distribution The first of such equations describes the motion of the order parameter. It is also shown that, at equilibrium, the amplitude of the moments rapidly decreases with k and, hence, only a few moments (expecially at high enough energy) should describe the dynamics of the system. This could explain the prominent low‐dimensional features found in the dynamics of the HMF model. Numerical experiments partly confirm this picture, but show the presence of strong instabilities in the set of differential equations. Finally, we find a specific solution of the infinite set of equations, which has some correspondence with numerical observations.

Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows
View Description Hide DescriptionIn self‐gravitating stars, two dimensional or geophysical flows and in plasmas, long range interactions imply a lack of additivity for the energy; as a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the equilibrium statistical mechanics of such systems is a well understood subject. In this proceeding, we explain briefly the classical approach to equilibrium and non equilibrium statistical mechanics for these systems, starting from first principles. We emphasize recent and new results, mainly a classification of equilibrium phase transitions, new unobserved equilibrium phase transition, and out of equilibrium phase transitions. We briefly discuss what we consider as challenges in this field.

Thermodynamics of Small Systems
View Description Hide DescriptionAfter an informal review of the basic ideas in thermodynamics and statistical mechanics, I collect sparse observations on the statistical physics of small systems (small N, few degrees of freedom, nanometer scale in size). Inspired by the contributions of Philippe Chomaz, Francesca Gulminelli and Michel Farizon, presented in this section, I would like to envisage connections with other contributions in the volume, and with other themes of research, close and not so close; the hamiltonian mean field model, the biophysical chemistry of dilute protein solutions and the non equilibrium properties of small systems. Further investigations on the analogies between effectively and real long‐range systems could open new perspectives for a thermodynamic understanding of matter at the nanoscale, but also on larger scales…it seems a matter of relaxation times.

Fragment size distributions and caloric curve in collision induced cluster fragmentation
View Description Hide DescriptionWe report on a cluster fragmentation study involving collisions of high‐energy (60 keV/amu) hydrogen cluster ions with atomic helium. The experimental characterisation of the cluster fragmentation not only by the average fragment size distribution but also by a statistical analysis of the fragmentation events has become possible owing to a developed multi‐coincidence technique in which all the fragments of all collisions occurring in the experiment are mass analysed on an event‐by‐event basis. Measurements and analysis are carried out on a large number of cluster ions prepared at the same total energy, as opposed to observing the evolution of a single system over time (time averaged ergodic hypothesis). By selecting specific decay reactions we can start after the energising collision with a micro‐canonical cluster ion ensemble of fixed excitation energy. From the respective fragment distributions for these selected decay reactions we derive corresponding temperatures of the decaying cluster ions. The relation between this temperature and the excitation energy (caloric curve) exhibits the typical prerequisites of a first order phase transition in a finite system, in the present case signalling the transition from a bound cluster type situation to the free gas phase

Phase Transitions in Finite Systems using Information Theory
View Description Hide DescriptionIn this paper, we present the issues we consider as essential as far as the statistical mechanics of finite systems is concerned. In particular, we emphasize our present understanding of phase transitions in the framework of information theory. Information theory provides a thermodynamically‐consistent treatment of finite, open, transient and expanding systems which are difficult problems in approaches using standard statistical ensembles. As an example, we analyze the problem of boundary conditions, which in the framework of information theory must also be treated statistically. We recall that out of the thermodynamical limit the different ensembles are not equivalent and in particular they may lead to dramatically different equations of state, in the region of a first order phase transition. We recall the recent progresses achieved in the understanding of first‐order phase transition in finite systems: the equivalence between the Yang‐Lee theorem and the occurrence of bimodalities in the intensive ensemble and the presence of inverted curvatures of the thermodynamic potential of the associated extensive ensemble. We come back to the concept of order parameters and to the role of constraints on order parameters in order to predict the expected signature of first‐order phase transition: in absence of any constraint (intensive ensemble) bimodality of the event distribution is expected while an inverted curvature of the thermodynamic potential is expected at a fixed value of the order parameter (extensive ensemble) in between the phases (co‐existence zone). We stress that this discussion is not restricted to the possible occurrence of negative specific heat, but can also include negative compressibilities and negative susceptibilities, and in fact any curvature anomaly of the thermodynamic potential.

Gravitational clustering: an overview
View Description Hide DescriptionWe discuss the differences and analogies of gravitational clustering in finite and infinite systems. The process of collective, or violent, relaxation leading to the formation of quasi‐stationary states is one of the distinguished features in the dynamics of self‐gravitating systems. This occurs, in different conditions, both in a finite than in an infinite system, the latter embedded in a static or in an expanding background. We then discuss, by considering some simple and paradigmatic examples, the problems related to the definition of a mean‐field approach to gravitational clustering, focusing on role of discrete fluctuations. The effect of these fluctuations is a basic issue to be clarified to establish the range of scales and times in which a collision‐less approximation may describe the evolution of a self‐gravitating system and for the theoretical modeling of the non‐linear phase.

Statistical Mechanics of Infinite Gravitating Systems
View Description Hide DescriptionThe cosmological many‐body problem was stated over 300 years ago, but its solution is quite recent and still incomplete. Imagine an infinite expanding universe essentially containing a very large number of objects moving in response to their mutual gravitational forces. What will be the spatial and velocity distributions of these objects and how will they evolve? This question fascinates on many levels. Though inherently non‐linear, it turns out to be one of the few analytically solvable problems of statistical mechanics with long range forces. The partition function can be calculated. From this all the thermodynamic properties of the system can be obtained for the grand canonical ensemble. They confirm results derived independently directly from the first and second laws of thermodynamics. The behavior of infinite gravitating systems is quite different from their finite relations such as star clusters. Infinite gravitating systems have regimes of negative specific heat, an unusual type of phase transition, and a very close relation to the observed large‐scale structure of our universe. This last feature provides an additional astronomical motivation, especially since the statistical mechanics may be generalized to include effects of dark matter haloes around galaxies. Previously the cosmological many‐body problem has mostly been studied using the BBGKY hierarchy (not so suitable in the non‐linear regime) and by direct computer integrations of the objects' orbits. The statistical mechanics agrees with and substantially extends these earlier results. Most astrophysicists had previously thought that a statistical thermodynamic approach would not be applicable because: a) many‐body gravitational systems have no rigorous equilibrium state, b) the unshielded nature of the long‐range force would cause the partition function to diverge on large scales, and c) point masses would produce divergences on small scales. However, deeper considerations show that these are not significant difficulties. There remain many important questions for which we have only a very preliminary understanding. An example is to find the basin of attraction for initial states which are able to evolve into systems described by this gravitational partition function. It may well be that ideas and techniques developed for other types of statistical mechanical systems will help us answer these questions.

Infinite self‐gravitating systems and cosmological structure formation
View Description Hide DescriptionThe usual thermodynamic limit for systems of classical self‐gravitating point particles becomes well defined, as a dynamical problem, using a simple physical prescription for the calculation of the force, equivalent to the so‐called “Jeans' swindle”. The relation of the resulting intrinsically out of equilibrium problem, of particles evolving from prescribed uniform initial conditions in an infinite space, to the one studied in current cosmological models (in an expanding universe) is explained. We then describe results of a numerical study of the dynamical evolution of such a system, starting from a simple class of infinite “shuffled lattice” initial conditions. The clustering, which develops in time starting from scales around the grid scale, is qualitatively very similar to that seen in cosmological simulations, which begin from lattices with applied correlated displacements and incorporate an expanding spatial background. From very soon after the formation of the first non‐linear structures, a spatio‐temporal scaling relation describes well the evolution of the two‐point correlations. At larger times the dynamics of these correlations converges to what is termed “self‐similar” evolution in cosmology, in which the time dependence in the scaling relation is specified entirely by that of the linearized fluid theory. We show how this statistical mechanical “toy model” can be useful in addressing various questions about these systems which are relevant in cosmology. Some of these questions are closely analagous to those currently studied in the literature on long range interactions, notably the relation of the evolution of the particle system to that in the Vlasov limit and the nature of approximately quasi‐stationary states.

Dynamics of Collision‐less Self‐gravitating Systems and Dark Turbulence
View Description Hide DescriptionMany scaling relations have been reported so far for the self‐gravitating systems in the sky, such as the density‐linear scale relation, mass‐angular momentum relation, velocity dispersion‐mass relations. For the consistent understanding of all such relations, we explore the basic principle of (a) local virial relation and (b) the turbulent property, which are expected to hold in self‐gravitating systems, especially in the late non‐linear stage. A special focus is put on the collisions‐less dark matter in the universe, in which the flow lines of particles can cross with each other and form chaotic dynamics which naturally leads to the turbulence, thus we call it dark turbulence.

Long‐range interactions in cold atomic systems: A foreword
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Strongly Correlated Ion Coulomb Systems
View Description Hide DescriptionAs a specific example of strongly correlated Coulomb systems, three‐dimensional long‐range ordered structures in smaller and near‐spherically symmetric Coulomb crystals of ions confined in a linear rf Paul trap are discussed. Though the observed structures are not expected from ground state molecular dynamics (MD) simulations, similar structures are found as metastable ion configurations in MD simulations at low temperatures.

Long Range Interactions in Magneto‐Optical Traps
View Description Hide DescriptionWe present in this paper a review of our work on the impact of long‐range interactions on the static and dynamic behaviors of magneto‐optical traps (MOT) for neutral atoms. In such systems, all forces (confinement and interaction) are mediated by light. The most obvious signature of long‐range interactions is the dependence of the MOT size with number N of trapped atoms. We study in detail the scaling laws for the MOT size versus N, in a range of large We then show that the competition between confinement and repulsive interaction can lead, for N large, to a new dynamical instability characterized by spontaneous, self‐sustained oscillations of the MOT. This instability is investigated experimentally and theoretically, using both a simple analytical model and a more refined numerical one.

Collective Atomic Recoil Lasing and Superradiant Rayleigh Scattering in a high‐Q ring cavity
View Description Hide DescriptionCold atoms in optical high‐Q cavities are an ideal model system for long‐range interacting particles. The position of two arbitrary atoms is, independent on their distance, coupled by the back‐scattering of photons within the cavity. This mutual coupling can lead to collective instability and self‐organization of a cloud of cold atoms interacting with the cavity fields. This phenomenon (CARL, i.e. Collective Atomic Recoil Lasing) has been discussed theoretically for years, but was observed only recently in our lab. The CARL‐effect is closely linked to superradiant Rayleigh scattering, which has been intensely studied with Bose‐Einstein condensates in free space. By adding a resonator the coherence time of the system, in which the instability occurs, can be strongly enhanced. This enables us to observe cavity‐enhanced superradiance with both Bose‐Einstein condensates and thermal clouds and allows us to close the discussion about the role of quantum statistics in superradiant scattering.

Dipolar interaction in ultra‐cold atomic gases
View Description Hide DescriptionUltra‐cold atomic systems provide a new setting where to investigate the role of long‐range interactions. In this paper we will review the basics features of those physical systems, in particular focusing on the case of Chromium atoms. On the experimental side, we report on the observation of dipolar effects in the expansion dynamics of a Chromium Bose‐Einstein condensate. By using a Feshbach resonance, the scattering length characterising the contact interaction can be strongly reduced, thus increasing the relative effect of the dipole‐dipole interaction. Such experiments make Chromium atoms the strongest candidates at present for the achievement of the strong dipolar regime. On the theoretical side, we investigate the behaviour of ultra‐cold dipolar systems in the presence of a periodic potential. We discuss how to realise this situation experimentally and we characterise the system in terms of its quantum phases and metastable states, discussing in detail the differences with respect to the case of zero‐range interactions.

Back Matter
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