Long-range interacting classical systems: universality in mixing weakening

TL;DR

This study investigates long-range interacting classical systems, revealing a universal decay in chaos indicators linked to interaction range, supporting nonextensive statistical mechanics and the transition from exponential to power-law mixing.

Contribution

It demonstrates a universal relationship between the decay of Lyapunov exponents and interaction range in classical rotator models, linking nonextensivity to mixing properties.

Findings

Lyapunov exponent scales as N^{-kappa} with a universal kappa function of alpha/d

kappa decreases from 1/3 to 0 as alpha/d increases from 0 to 1

Mixing transitions from exponential to power-law in the nonextensive regime

Abstract

Through molecular dynamics, we study the $d=2,3$ classical model of $N$ coupled rotators (inertial XY model) assuming a coupling constant which decays with distance as $r_{ij}^{-\alpha}$ ($\alpha \ge 0$). The total energy $<H>$ is asymptotically $\propto N {\tilde N}$ with ${\tilde N} \equiv [N^{1-\alpha/d}-(\alpha/d)]/[1-\alpha/d]$, hence the model is thermodynamically extensive if $\alpha/d>1$ and nonextensive otherwise. We numerically show that, for energies above some threshold, the (appropriately scaled) maximum Lyapunov exponent is $\propto N^{-\kappa}$ where $\kappa$ is an {\it universal} (one and the same for $d=1,2$ and 3, and all energies) function of $\alpha/d$, which monotonically decreases from 1/3 to zero when $\alpha/d$ increases from 0 to 1, and identically vanishes above 1. These features are consistent with the nonextensive statistical mechanics scenario, where thermodynamic extensivity is associated with {\it exponential} mixing in phase space, whereas {\it weaker} (possibly {\it power-law} in the present case) mixing emerges at the $N \to \infty$ limit whenever nonextensivity is observed.