Ergodicity and slow relaxation in the one-dimensional self-gravitating system.

We show that homogeneous and nonhomogeneous states of the one-dimensional self-gravitating sheets models have different ergodic properties. The former are nonergodic and the one-particle distribution function has a zero collision term if a proper limit is taken for the periodic boundary conditions. As a consequence, homogeneous states of the sheets model are nonergodic and do not relax to the equilibrium state, while nonhomogeneous states are ergodic in a time window of the order of the relaxation time to equilibrium, as similarly observed in other systems with a long range interaction.

Classical Goldstone modes in long-range interacting systems.

For a classical system with long-range interactions, a soft mode exists whenever a stationary state spontaneously breaks a continuous symmetry of the Hamiltonian. Besides that, if the corresponding coordinate associated to the symmetry breaking is periodic, then the same energy of the different stationary states and finite N thermal fluctuations result in a superdiffusive motion of the center of mass for total zero momentum, that tends to a normal diffusion for very long times. As examples of this, we provide a two-dimensional self-gravitating system, a free electron laser, and the Hamiltonian mean-field (HMF) model. For the latter, a detailed theory for the motion of the center of mass is given. We also discuss how the coupling of the soft mode to the mean-field motion of individual particles may lead to strong chaotic behavior for a finite particle number, as illustrated by the HMF model.

Relaxation processes in long-range lattices.

The relaxation to equilibrium of lattice systems with long-range interactions is investigated. The timescales involved depend polynomially on the system size, potentially leading to diverging equilibration times. A kinetic equation for long-range lattices is proposed, which explain these timescales as well as a threshold in the interaction range reported in [Phys. Rev. Lett. 110, 170603 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.170603]. Non-Markovian effects are shown to play an important role in the relaxation of systems of up to thousands of particles.

Truncated Lévy flights and weak ergodicity breaking in the Hamiltonian mean-field model.

The dynamics of the Hamiltonian mean-field model is studied in the context of continuous-time random walks. We show that the sojourn times in cells in the momentum space are well described by a one-sided truncated Lévy distribution. Consequently, the system is nonergodic for long observation times that diverge with the number of particles. Ergodicity is attained only after very long times both at thermodynamic equilibrium and at quasistationary out-of-equilibrium states.

Entropy of classical systems with long-range interactions.

We discuss the form of the entropy for classical Hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a N particle in the limit N-->infinity. The stationary states of the Hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence, the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.