First-principle validation of Fourier's law in \ensuremath{d=1,2,3} classical systems

We numerically study the thermal transport in the classical inertial nearest-neighbor XY ferromagnet in $d=1,2,3$, the total number of sites being given by $N=L^d$, where $L$ is the linear size of the system. For the thermal conductance $\sigma$, we obtain $\sigma(T,L)\, L^{\delta(d)} = A(d)\, e_{q(d)}^{- B(d)\,[L^{\gamma(d)}T]^{\eta(d)}}$ (with $e_q^z \equiv [1+(1-q)z]^{1/(1-q)};\,e_1^z=e^z;\,A(d)>0;\,B(d)>0;\,q(d)>1;\,\eta(d)>2;\,\delta \ge 0; \,\gamma(d)>0)$, for all values of $L^{\gamma(d)}T$ for $d=1,2,3$. In the $L\to\infty$ limit, we have $\sigma \propto 1/L^{\rho_\sigma(d)}$ with $\rho_\sigma(d)= \delta(d)+ \gamma(d) \eta(d)/[q(d)-1]$. The material conductivity is given by $\kappa=\sigma L^d \propto 1/L^{\rho_\kappa(d)}$ ($L\to\infty$) with $\rho_\kappa(d)=\rho_\sigma(d)-d$. Our numerical results are consistent with 'conspiratory' $d$-dependences of $(q,\eta,\delta,\gamma)$, which comply with normal thermal conductivity (Fourier law) for all dimensions.