Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index

Abstract

We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian Mean Field (HMF) model as a simple example. These systems generically undergo a violent relaxation to a quasi-stationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasi-stationary solution of the Vlasov equation, slowly evolving in time due to finite $N$ effects. For subcritical energies $7/12<U<3/4$, we exhibit cases where the DF is well-fitted by a Tsallis $q$-distribution with an index $q(t)$ slowly decreasing in time from $q\simeq 3$ (semi-ellipse) to $q=1$ (Boltzmann). When the index $q(t)$ reaches a critical value $q_{crit}(U)$, the non-magnetized (homogeneous) phase becomes Vlasov unstable and a dynamical phase transition is triggered, leading to a magnetized (inhomogeneous) state. While Tsallis distributions play an important role in our study, we explain this dynamical phase transition by using only conventional statistical mechanics. For supercritical energies, we report for the first time the existence of a magnetized QSS with a very long lifetime.

5 pages, 3 figures, few stylistic changes from previous version

References in corpus (4)

• A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model
• Lynden-Bell and Tsallis distributions for the HMF model
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

Cited by in corpus (1)

• Unveiling the nature of out-of-equilibrium phase transitions in a system with long-range interactions