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 a paradigmatic system with long-range interactions: the Hamiltonian mean-field (HMF) model. It is shown that in the thermodynamic limit this model does not relax to the usual equilibrium Maxwell-Boltzmann distribution. Instead, the final stationary state has a peculiar core-halo structure. In the thermodynamic limit, HMF is neither ergodic nor mixing. Nevertheless, we find that using dynamical properties of Hamiltonian systems it is possible to quantitatively predict both the spin distribution and the velocity distribution functions in the final stationary state, without any adjustable parameters. We also show that HMF undergoes a nonequilibrium first-order phase transition between paramagnetic and ferromagnetic states. DOI: 10.1103/PhysRevLett.106.200603 PACS numbers: 05.20.Ày, 05.45.Àa, 05.70.Ln Since the early work of Clausius, Boltzmann, and Gibbs it has been known that for particles interacting through short-range potentials, the final stationary state reached by a system corresponds to the thermodynamic equilibrium [1]. Although no exact proof exists, in practice it is found that nonintegrable systems with a fixed energy and number of particles (microcanonical ensemble) always relax to a unique stationary state which only depends on the global conserved quantities: energy, momentum, and angular momentum. The equilibrium state does not depend on the specifics of the initial particle distribution. The situation is very different for systems in which particles interact through long-ranged unscreened potentials. This is the case for gravitational systems and confined one component plasmas [2,3]. For these systems, in the thermodynamic limit, the collision duration time diverges, and the thermodynamic equilibrium is never reached [4]. Instead, as time t ! 1, these systems become trapped in a stationary state characterized by a broken ergodicity [5][6][7]. Unlike the thermodynamic equilibrium, the stationary state depends explicitly on the initial particle distribution. Over the last 50 years, there has been a great effort to predict the final stationary state without having to explicitly solve the N-body dynamics or the collisionless Boltzmann (Vlasov) equation. Qualitatively, it has been observed that for many different systems the nonequilibrium stationary state has a peculiar core-halo shape. Recently, an ansatz solution to the Vlasov equation has been proposed which allowed us to explicitly calculate the core-halo distribution function for confined plasmas and self-gravitating systems [2,3]. In this Letter we will show that an ansatz solution is also possible for the Hamiltonian mean-field (HMF) model. The theory proposed allows us also to locate the nonequilibrium para-to-ferromagnetic phase transition, which earlier theories incorrectly predicted to be of second order [8]. All of the results are compared with the molecular dynamics simulations performed using a symplectic integrator, and are found to be in excellent agreement.The HMF model consists of N, XY interacting spins, whose dynamics i...

Theory and simulations are used to study collisionless relaxation of a gravitational N -body system. It is shown that when the initial one-particle distribution function satisfies the virial condition--potential energy is minus twice the kinetic energy--the system quickly relaxes to a metastable state described quantitatively by the Lynden-Bell distribution with a cutoff. If the initial distribution function does not meet the virial requirement, the system undergoes violent oscillations, resulting in a partial evaporation of mass. The leftover particles phase-separate into a core-halo structure. The theory presented allows us to quantitatively predict the amount and the distribution of mass left in the central core, without any adjustable parameters. On a longer time scale tauG-N , collisionless relaxation leads to a gravothermal collapse.

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...