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

TL;DR

This paper investigates dynamical phase transitions in long-range interacting systems, demonstrating how Tsallis distributions with a time-dependent index describe the evolution of the system's state and identifying conditions for phase transitions.

Contribution

It introduces a novel analysis of dynamical phase transitions using Tsallis distributions with a time-varying index in the Hamiltonian Mean Field model.

Findings

Tsallis q-distributions fit the out-of-equilibrium states.

Critical q-value triggers phase transition to magnetized state.

Long-lived magnetized quasi-stationary states at supercritical energies.

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.