Physics of Long-Range Interacting Systems

Abstract

This book deals with an important class of many-body systems: those where the interaction potential decays slowly for large inter-particle distance. In particular, systems where the decay is slower than the inverse inter-particle distance raised to the dimension of the embedding space. Gravitational and Coulomb interactions are the most prominent examples. However, it has become clear that long-range interactions are more common than previously thought. This has stimulated a growing interest in the study of long-range interacting systems, which has led to a much better understanding of the many peculiarities in their behaviour. The seed of all particular features of these systems, both at equilibrium and out-of-equilibrium, is the lack of additivity. It is now well understood that this does not prevent a statistical mechanics treatment. However, it does require a more in-depth study of the thermodynamic limit and of all related theoretical concepts. A satisfactory understanding of properties generally considered as oddities only a couple of decades ago has now been reached: ensemble inequivalence, negative specific heat, negative susceptibility, ergodicity breaking, out-of-equilibrium quasi-stationary-states, anomalous diffusion, etc. The first two parts describe the theoretical and computational instruments needed for addressing the study of both equilibrium and dynamical properties of systems subject to long-range forces. The third part of the book is devoted to discussing the applications of such techniques to the most relevant examples of long-range systems. The only prerequisite is a basic course in statistical mechanics.

Generalized Stochastic Fokker-Planck Equations

“…Until now, systems with long-range interactions and generalized thermodynamics have been studied by different communities (see, for example, the references in the books [10] and [19]) with, sometimes, violent polemics between them. The present contribution tries to make the link between these two topics by showing how a class of generalized Fokker-Planck equations can describe complex systems experiencing both small-scale constraints (generalized thermodynamics) and long-range interactions.…”

“…The first one concerns the physics of systems with long-range interactions [10]. We can distinguish two types of systems with long-range interactions depending whether they are isolated or dissipative.…”

Microscopic aspects of magnetic lattice demagnetizing factors

“…This is because demagnetizing fields arise solely from the long-range dipolar interactions. The thermodynamic limit for short-range models is well defined [44][45][46], and inclusion of short-range interactions does not alter the thermodynamic limit results for N ; see Appendix E. Thermal fluctuations also appear irrelevant in this limit. For ellipsoids, N is calculated from averaged macroscopic fields that do not include thermal fluctuations and, similarly, our mean-field-like iterative method captures the essential demagnetizing effects also for cuboids.…”

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

“…Recent exploration of long-range interacting systems, and in particular, of their static and dynamic properties, has focussed on an analytically tractable and representative model called the Hamiltonian mean-field (HMF) model [13,34]. Long-range interacting (LRI) systems are those in which the inter-particle interaction potential decays slower than 1/r d , with d being the dimension of the embedding space [35,36,37,38,39]. Unlike short-range ones, LRI systems are intrinsically nonadditive, namely, they cannot be trivially divided into independent macroscopic subparts.…”

“…It proves convenient to reduce the number of parameters in the dynamics (37). To this end, we note that the effect of σ may be made explicit by replacing ω j in the second equation by σω j .…”