Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary state (qSS), the lifetime of which diverges with the number of particles. Therefore, in thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle-wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.

A theoretical framework is presented which allows us to quantitatively predict the final stationary state achieved by a non-neutral plasma during a process of collisionless relaxation. As a specific application, the theory is used to study relaxation of charged-particle beams. It is shown that a fully matched beam relaxes to the Lynden-Bell distribution. However, when a mismatch is present and the beam oscillates, parametric resonances lead to a core-halo phase separation. The approach developed accounts for both the density and the velocity distributions in the final stationary state.

We study, using both theory and molecular dynamics simulations, the relaxation dynamics of a microcanonical two dimensional self-gravitating system. After a sufficiently large time, a gravitational cluster of N particles relaxes to the Maxwell-Boltzmann distribution. The time to reach the thermodynamic equilibrium, however, scales with the number of particles. In the thermodynamic limit, N → ∞ at fixed total mass, equilibrium state is never reached and the system becomes trapped in a non-ergodic stationary state. An analytical theory is presented which allows us to quantitatively described this final stationary state, without any adjustable parameters. I. INTRODUCTIONSystems interacting through long-range forces behave very differently from those in which particles interact through short-range potentials. For systems with short-range forces, for arbitrary initial condition, the final stationary state corresponds to the thermodynamic equilibrium and can be described equivalently by either microcanonical, canonical, or grand-canonical ensembles. On the other hand, for systems with unscreened long-range interactions, equivalence between ensembles breaks down [1,2]. Often these systems are characterized by a negative specific heat [3][4][5] in the microcanonical ensemble and a broken ergodicity [6, 7]. In the infinite particle limit, N → ∞, these systems never reach the thermodynamic equilibrium and become trapped in a stationary out of equilibrium state (SS) [8,9]. Unlike normal thermodynamic equilibrium, the SS does not have Maxwell-Boltzmann velocity distribution. For finite N , relaxation to equilibrium proceeds in two steps. First, the system relaxes to a quasi-stationary state (qSS), in which it stays for time τ × (N ), after which it crosses over to the normal thermodynamic equilibrium with the Maxwell-Boltzmann (MB) velocity distribution [10]. In the limit N → ∞, the life time of qSS diverges, τ × → ∞, and the thermodynamic equilibrium is never reached.Unlike the equilibrium state, which only depends on the global invariants such as the total energy and momentum and is independent of the specifics of the initial particle distribution, the SS explicitly depends on the initial condition. This is the case for self-gravitating systems [11], confined one component plasmas [12,13], geophysical systems [14], vortex dynamics [15][16][17], etc [18], for which the SS state often has a peculiar core-halo structure [12]. In the thermodynamic limit, none of these systems can be described by the usual equilibrium statistical mechanics, and new methods must be developed.In this paper we will restrict our attention to self-gravitating systems. Unfortunately, it is very hard to study these systems in 3d [19,20]. The reason for this is that the 3d Newton potential is not confining. Some particles can gain enough energy to completely escape from the gravitational cluster, going all the way to infinity. In the thermodynamic limit, one must then consider three distinct populations: particles which will relax to form the central c...

Temperature inversions occur in nature, e.g., in the solar corona and in interstellar molecular clouds: somewhat counterintuitively, denser parts of the system are colder than dilute ones. We propose a simple and appealing way to spontaneously generate temperature inversions in systems with long-range interactions, by preparing them in inhomogeneous thermal equilibrium states and then applying an impulsive perturbation. In similar situations, short-range systems would typically relax to another thermal equilibrium, with uniform temperature profile. By contrast, in long-range systems, the interplay between wave-particle interaction and spatial inhomogeneity drives the system to nonequilibrium stationary states that generically exhibit temperature inversion. We demonstrate this mechanism in a simple mean-field model and in a two-dimensional self-gravitating system. Our work underlines the crucial rôle the range of interparticle interaction plays in determining the nature of steady states out of thermal equilibrium. Stationary states far from thermal equilibrium occur in nature. In some cases, e.g., in the solar corona and in interstellar molecular clouds, such states exhibit temperature inversion: denser parts of the system are colder than dilute ones. This work is motivated by an attempt to explain how such a counterintuitive effect may spontaneously arise in nonequilibrium states, unveiling its minimal ingredients and the underlying physical mechanism. To this end, we start with asking a simple yet physically relevant question: What happens if an isolated macroscopic system in thermal equilibrium is momentarily disturbed, e.g., by an impulsive force or a "kick"? If the interactions among the system constituents are shortranged, collisions redistribute the kick-injected energy among the particles, yielding a fast relaxation to a new equilibrium, with a Maxwellian velocity distribution and a uniform temperature across the system.Is the scenario the same if instead the interactions are long-ranged [1]? For long-range systems, collisional effects act over a characteristic time τ coll that, unlike shortrange systems, scales with the system size N , diverging as N → ∞ [2]. As a result, a macroscopic system with longrange interactions starting from generic initial conditions will attain thermal equilibrium only after extremely long times, often exceeding typical observation times. Examples of long-range systems are self-gravitating systems, for which, e.g., τ coll ≃ 10 10 years for globular clusters and orders of magnitude larger than the age of the universe * tarcisio.nteles@gmail.com; S. Gupta and T. N. Teles contributed equally to the work. † shamikg1@gmail.com ‡ pierfrancesco.dicintio@unifi.it § lapo.casetti@unifi.it for galaxies [3,4]. The collisionless evolution of longrange interacting systems for times shorter than τ coll is governed by the Vlasov (or collisionless Boltzmann) equation [2]. When kicked out of thermal equilibrium, a longrange interacting system relaxes to a Vlasov-stationary state, and thermal eq...