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 ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ ( $\ensuremath{\alpha}\ensuremath{\ge}0$). We show that for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ (thermodynamic limit): (i) The energy per particle ${E}_{N}/N$ scales like ${N}^{*}\ensuremath{\equiv}({N}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1)/(1\ensuremath{-}\ensuremath{\alpha})$; (ii) The properly scaled maximum Lyapunov exponent ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\lambda}}}_{N}^{\mathrm{max}}$ tends, for ${E}_{N}/({\mathrm{NN}}^{*})$ above a threshold, to zero for and only for $\ensuremath{\alpha}\ensuremath{\le}1$. 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.