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 $\eta =2.88\pm 0.04$.

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