First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model

Abstract: The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size $L$ is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures, $T_{h}$ and $T_{l}$ ($T_{h}>T_{l}$), respectively. These particles at extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators ($i=2, \cdots , L-1$) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier's law for the heat flux is verified numerically with the thermal conductivity becoming independent of the lattice size in the limit $L \to \infty$, scaling with the temperature as $\kappa(T) \sim T^{-2.25}$, where $T=(T_{h}+T_{l})/2$. Moreover, the thermal conductance, $\sigma(L,T)=\kappa(T)/L$, is well-fitted by a function, typical of nonextensive statistical mechanics, according to $\sigma(L,T)=A \exp_{q}(-B x^{\eta})$, where $A$ and $B$ are constants, $x=L^{0.475}T$, $q=2.28 \pm 0.04$, and $\eta=2.88 \pm 0.04$.