Long-range interacting classical systems: universality in mixing weakening

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.

4 pages, 3 figures; Submitted to Physical Review Letters

References in corpus (4)

• Long-range interacting rotators: connection with the mean--field approximation
• The rate of entropy increase at the edge of chaos
• Canonical solution of a system of long-range interacting rotators on a lattice
• Chaos in the thermodynamic limit

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