Breakdown of Exponential Sensitivity to Initial Conditions: Role of the Range of Interactions

Abstract

Within a microcanonical scenario we numerically study an $N$-sized linear chain classical inertial $\mathrm{XY}$ model including ferromagnetic couplings which decrease with distance as $r^{-\alpha }$ ( $\alpha \ge 0$). We show that for $N\to \infty$ (thermodynamic limit): (i) The energy per particle $E_N/N$ scales like $N^{*}\equiv (N^{1-\alpha }-1\mathrm{})/(1-\alpha )$; (ii) The properly scaled maximum Lyapunov exponent ${\tilde{\lambda }}_N^{\max }$ tends, for $E_N/\left({\mathrm{NN}}^{*}\right)$ above a threshold, to zero for and only for $\alpha \le 1$. These results are analogous to those observed in low-dimensional and in self-organized critical dissipative systems. This entire picture suggests a connection with the nonextensive thermostatistics recently introduced by one of us.