We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in N -particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated N → ∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite N , dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via non stationary states: we characterize numerically this dynamical instability in the finite N system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite N system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N 1.7 . Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the q-exponential distributions derived from Tsallis statistics.

Systems with long-range interactions display a short-time relaxation towards quasistationary states whose lifetime increases with system size. With reference to the Hamiltonian mean field model, we here show that a maximum entropy principle, based on Lynden-Bell's pioneering idea of "violent relaxation," predicts the presence of out-of-equilibrium phase transitions separating the relaxation towards homogeneous (zero magnetization) or inhomogeneous (nonzero magnetization) quasistationary states. When varying the initial condition within a family of "water bags" with different initial magnetization and energy, first- and second-order phase transition lines are found that merge at an out-of-equilibrium tricritical point. Metastability is theoretically predicted and numerically checked around the first-order phase transition line.

The dynamical behavior and the relaxation to equilibrium of long range interacting systems of particles still offer several open problems. It is possible to sketch the general theory as follows: the short time behavior, on the order of the dynamical time scale, is described by the Vlasov, or a Vlasov-like equation. The system then settles in a quasi-stationary state (hereafter called QSS), which is a stationary solution of the Vlasov equation. On time scales diverging with the number of particles, the system evolves towards the thermodynamic equilibrium, following the analog of a Lenard-Balescu equation (for a recent review, see [1]). The open questions include for instance the selection of the QSS among the stationary states of the Vlasov equation; the possible selection of periodic solutions of the Vlasov equation instead of a QSS; the complete understanding of the timescales for relaxation to equilibrium, especially around inhomogeneous QSS and close to dynamical transitions. We may mention also the rigorous derivation of the Lenard-Balescu equation, which, in contrast to the Vlasov equation, does not have a mathematical foundation. The stationary solutions of the Vlasov equation may be homogeneous in space, as is always the case in plasma physics, or inhomogeneous, as is the rule in self gravitating systems. In general, there is less understanding of the inhomogeneous cases, even at the level of linear perturbations of the Vlasov stationary state, because of the much greater technical difficulties. This linear understanding is an essential ingredient to derive a Lenard-Balescu like equation, and also to characterize possible undamped periodic solutions [2]. The linearization around a homogeneous stationary state corresponds to the usual theory of Landau damping in plasma physics [3]. Technically, the equation are solved by a Fourier transform in space and a Laplace transform in time. The equations for different Fourier modes decouple, and the computation results in the complex dielectric function (k, ω) for the kth Fourier mode. Because (k, ω) is defined in the upper half ω-plane due to the convergence condition of the Laplace transform, one needs to analytically continue (k, ω) in the lower half ω-plane to obtain the Landau frequency and damping rate. Linearization around a non homogeneous stationary state brings additional technical difficulties. First, one needs to use angle-action variables, to simplify the particle dynamics; but now the equations for different Fourier modes in the angle variables do not decouple any more; the "dielectric function" now becomes the determinant of an infinite matrix. Linearization around inhomogeneous stationary states has been studied in the context of self-gravitating systems, and, despite the technical difficulties, the above procedure has been carried out entirely to study the instability of some galactic models. However, studying the analog of Landau damping requires the analytic continuation part, and this is, in the words of Weinberg [4], a "daunting task". This dif...

The relation between relaxation and diffusion is investigated in a Hamiltonian system of globally coupled rotators. Diffusion is anomalous if and only if the system is going towards equilibrium. The anomaly in diffusion is not anomalous diffusion taking a power-type function, but is a transient anomaly due to nonstationarity. For a certain type of initial condition, in quasistationary states, diffusion can be explained by a stretched exponential correlation function, whose stretching exponent is almost constant and correlation time is linear as functions of degrees of freedom. The full time evolution is characterized by varying stretching exponent and correlation time.