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.

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...

The weakly nonlinear localization of obliquely modulated high-frequency electromagnetic waves in an electron-positron-ion plasma is considered. It is shown that the amplitude of the wave turns out to be a strongly dependent function of the angle between the slow modulations and the fast spatial variations and that the possibility appears of spontaneous generation of low-frequency magnetic fields. These magnetic fields are also functions of this angle and of the high-frequency wave polarization. The analysis of colinear modulation in electron-positron plasmas shows that some restriction must be made regarding the validity of previous calculations.

A calculation is presented which quantitatively accounts for the terminal velocity of a cylindrical magnet falling through a long copper or aluminum pipe. The experiment and the theory are a dramatic illustration of the Faraday's and Lenz's laws and are bound to capture student's attention in any electricity and magnetism course.

We investigate the concept of a standard map for the interaction of relativistic particles and electrostatic waves of arbitrary amplitudes, under the action of external magnetic fields. The map is adequate for physical settings where waves and particles interact impulsively, and allows for a series of analytical result to be exactly obtained. Unlike the traditional form of the standard map, the present map is nonlinear in the wave amplitude and displays a series of peculiar properties. Among these properties we discuss the relation involving fixed points of the maps and accelerator regimes.

We analyze the behavior of a relativistic particle moving under the influence of a uniform magnetic field and a stationary electrostatic wave. We work with a set of pulsed waves that allows us to obtain an exact map for the system. We also use a method of control for near-integrable Hamiltonians that consists of the addition of a small and simple control term to the system. This control term creates invariant tori in phase space that prevent chaos from spreading to large regions, making the controlled dynamics more regular. We show numerically that the control term just slightly modifies the system but is able to drastically reduce chaos with a low additional cost of energy. Moreover, we discuss how the control of chaos and the consequent recovery of regular trajectories in phase space are useful to improve regular particle acceleration.

This work analyzes the dynamics of inhomogeneous, magnetically focused high-intensity beams of charged particles. While for homogeneous beams the whole system oscillates with a single frequency, any inhomogeneity leads to propagating transverse density waves which eventually result in a singular density build up, causing wave breaking and jet formation. The theory presented in this paper allows to analytically calculate the time at which the wave breaking takes place. It also gives a good estimate of the time necessary for the beam to relax into the final stationary state consisting of a cold core surrounded by a halo of highly energetic particles.Comment: Accepted in Physics of Plasma Letter