Nonequilibrium stationary states of 3D self-gravitating systems.

Three-dimensional self-gravitating systems do not evolve to thermodynamic equilibrium but become trapped in nonequilibrium quasistationary states. In this Letter, we present a theory which allows us to a priori predict the particle distribution in a final quasistationary state to which a self-gravitating system will evolve from an initial condition which is isotropic in particle velocities and satisfies a virial constraint 2K=-U, where K is the total kinetic energy, and U is the potential energy of the system.

Mean-field model for the density of states of jammed soft spheres.

We propose a class of mean-field models for the isostatic transition of systems of soft spheres, in which the contact network is modeled as a random graph and each contact is associated to d degrees of freedom. We study such models in the hypostatic, isostatic, and hyperstatic regimes. The density of states is evaluated by both the cavity method and exact diagonalization of the dynamical matrix. We show that the model correctly reproduces the main features of the density of states of real packings and, moreover, it predicts the presence of localized modes near the lower band edge. Finally, the behavior of the density of states D(ω)∼ω^{α} for ω→0 in the hyperstatic regime is studied. We find that the model predicts a nontrivial dependence of α on the details of the coordination distribution.

Chaos and relaxation to equilibrium in systems with long-range interactions.

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become trapped in nonequilibrium stationary states. For a finite number of particles a nonequilibrium state has a finite lifetime, so that eventually a system will relax to thermodynamic equilibrium. The time that a system remains trapped in a quasistationary state (QSS) scales with the number of particles as N(δ), with δ>0, and diverges in the thermodynamic limit. In this paper we will explore the role of chaotic dynamics on the time that a system remains trapped in a QSS. We discover that chaos, measured by the Lyapunov exponents, favors faster relaxation to equilibrium. Surprisingly, weak chaos favors faster relaxation than strong chaos.

Ergodicity breaking and quasistationary states in systems with long-range interactions.

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become trapped in quasistationary states (qSS), the life time of which diverges with the number of particles. In this paper we will explore the relaxation of the Hamiltonian Mean-Field model to qSS for a class of initial conditions of the multilevel water-bag form. We will show that if the initial distribution satisfies the virial condition, thereby reducing mean field changes, the final distribution in the qSS can be predicted very accurately using a reduced exactly integrable model. The calculated distribution functions obtained using this approach are found to be more accurate than the ones predicted by the Lynden-Bell theory.

Ensemble inequivalence in a mean-field XY model with ferromagnetic and nematic couplings.

We explore ensemble inequivalence in long-range interacting systems by studying an XY model of classical spins with ferromagnetic and nematic coupling. We demonstrate the inequivalence by mapping the microcanonical phase diagram onto the canonical one, and also by doing the inverse mapping. We show that the equilibrium phase diagrams within the two ensembles strongly disagree within the regions of first-order transitions, exhibiting interesting features like temperature jumps. In particular, we discuss the coexistence and forbidden regions of different macroscopic states in both the phase diagrams.