Role of the range of the interactions in thermal conduction

We investigate thermal transport along a one-dimensional lattice of classical inertial rotators, with attractive couplings which decrease with distance as $r^{-\alpha}$ ($\alpha \ge 0$), subject at its ends to Brownian heat reservoirs at different temperatures with average value $T$. By means of numerical integration of the equations of motion, we show the effects of the range of the interactions in the temperature profile and energy transport, and determine the domain of validity of Fourier's law in this context. We find that Fourier's law, as signaled by a finite $\kappa$ in the thermodynamic limit, holds only for sufficiently short range interactions, with $\alpha >\alpha_c(T)$. For $\alpha<\alpha_c(T)$, a kind of insulator behavior emerges at any $T$.