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

Abstract

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.

Contribution to the Festschrift celebrating the 80th anniversary of Professor Giorgio Benedek. 11 pages including 3 figures (v2). Published in Physica D: Nonlinear Phenomena (2023)

References in corpus (4)

• Heat Transport in low-dimensional systems
• Thermal conductance of graphene and dimerite
• Fourier's Law in a Quantum Spin Chain and the Onset of Quantum Chaos
• Temperature dependence of thermal conductivities of coupled rotator lattice and the momentum diffusion in standard map

Cited by in corpus (2)

• Ising chain: Thermal conductivity and first-principle validation of Fourier law
• First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model