Long-range interacting classical systems: universality in mixing weakening

Details Long-range interacting classical systems: universality in mixing weakening Long-range interacting classical systems: universality in mixing weakening

2000-07-06

arxiv.org/abs/cond-mat/0007104v1

Phys. Lett. A 286, 251 (2001)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 3 figures; Submitted to Physical Review Letters

Scientific paper

10.1016/S0375-9601(01)00440-6

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 $$ 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.

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