On quasi-stationary states of the conservative dynamics of long-range interacting systems

Resumo: Long-range interacting (LRI) systems are ubiquitous in nature. They range from the astronomical scale (e.g. self-gravitating systems), to the macroscopic (e.g. non-neutral plasmas, wave-plasma interacting systems, two-dimensional geophysical vortex systems), down to the atomic scale (e.g. classical and quantum cold atoms interacting via quasi-resonant lasers). Despite their importance, much of the behaviour of these systems still remains poorly understood. It is known that LRI systems can exhibit ergodicity-breaking, anomalous relaxation and diffusion, quasi-stationary states (qSS), vanishing Lyapunov exponents, ensemble inequivalence, negative specific heat, temperature discontinuities. Thermodynamic anomalies result from the non-additivity of energy, while dynamical ones arise from the complexity of collisionless relaxation driven by the wave-particle interactions.

In the thermodynamic limit, dynamics is governed by the Vlasov equation, which has an infinity of stationary solutions. Approaching the thermodynamic limit, LRI systems do not relax to equilibrium, but become trapped in qSS, whose life-time diverges with the particle number, whence the importance of studying Vlasov stationary states.

In this talk we will explore the relaxation to qSS of two long-range potential models, namely, the Hamiltonian Mean-Field (HMF) model and gravitation in three dimensions, for some classes of initial conditions (ICs) of interest. In the case of the HMF we 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. For gravitation in 3d, we have tested spherically symmetric initial conditions satisfying the virial condition and isotropy in velocity. Again, the integrable approximation fits well qSS radial and velocity marginal distributions obtained through molecular dynamics. The distribution functions obtained using this approach are found to be more accurate than the ones predicted by the ergodic Lynden-Bell theory.

For ICs not obeying the virial condition, ergodicity-breaking has been shown to happen through parametric resonances between certain particles and the strong mean-field oscillations felt by particles. This mechanism is analogous to the nonlinear Landau damping seen in particle-wave interactions in plasmas. For these ICs Lynden-Bell distributions do not coincide with qSS distributions obtained through molecular dynamics. The present work goes a step beyond, showing that the ergodicity hypothesis is neither applicable to ICs obeying the virial condition, since an exactly integrable model provides a better approximation to the aforementioned distributions.

* Ana C. Ribeiro-Teixeira, Fernanda P. C. Benetti, Renato Pakter, Yan Levin, Phys. Rev. E 89 (2014) 022130

* Fernanda P. C. Benetti, Ana C. Ribeiro-Teixeira, Renato Pakter, Yan Levin, Phys. Rev. Lett. 113 (2014) 100602.