Metastability, negative specific heat and weak mixing in classical long-range many-rotator system

TL;DR

This study investigates metastability, negative specific heat, and weak mixing in a classical long-range rotator system, revealing how these phenomena relate to phase transitions and nonextensive statistical mechanics.

Contribution

It uncovers the existence of long-lived metastable states and their properties, including Lyapunov exponents, in a long-range interacting system, linking anomalies across the phase transition.

Findings

Metastable states have longer durations with increasing system size.

Lyapunov exponents scale with system size, indicating weak mixing.

The scaling exponent for metastable states is one-third of that for stable states.

Abstract

We perform a molecular dynamical study of the isolated $d=1$ classical Hamiltonian ${\cal H} = {1/2} \sum_{i=1}^N L_i^2 + \sum_{i \ne j} \frac{1-cos(\theta_i-\theta_j)}{r_{ij}^\alpha} ;(\alpha \ge 0)$, known to exhibit a second order phase transition, being disordered for $u \equiv U/N{\tilde N} \ge u_c(\alpha,d)$ and ordered otherwise ($U\equiv$ total energy and ${\tilde N} \equiv \frac{N^{1-\alpha/d}-\alpha/d}{1-\alpha/d}$). We focus on the nonextensive case $\alpha/d \le 1$ and observe that, for $u<u_c$, a basin of attraction exists for the initial conditions for which the system quickly relaxes onto a longstanding metastable state (whose duration presumably diverges with $N$ like ${\tilde N}$) which eventually crosses over to the microcanonical Boltzmann-Gibbs stable state. The temperature associated with the (scaled) average kinetic energy per particle is lower in the metastable state than in the stable one. It is exhibited for the first time that the appropriately scaled maximal Lyapunov exponent $\lambda_{u<u_c}^{max}(metastable) \propto N^{-\kappa_{metastable}} ;(N \to \infty)$, where, for all values of $\alpha/d$, $\kappa_{metastable}$ numerically coincides with {\it one third} of its value for $u>u_c$, hence decreases from 1/9 to zero when $\alpha/d$ increases from zero to unity, remaining zero thereafter. This new and simple {\it connection between anomalies above and below the critical point} reinforces the nonextensive universality scenario.