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 that decrease with distance as $r^{-\alpha } \left(\alpha \ge 0\right)$, 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 $\alpha >{\alpha }_c\left(T\right)$. For $\alpha <{\alpha }_c\left(T\right)$, a kind of insulator behavior emerges at any $T$.

Figure 1

Temperature profiles for different values of $\alpha$ indicated on the figure. For each $\alpha$, averages over 50 realizations are computed. $T\equiv (T_L+T_R)/2=0.8,\mathrm{\Delta }T\equiv T_L-T_R=0.2$, and $N=100$.

Figure 2

Scaled flux $N\mathcal{J}$ vs. $N$, for different values of $\alpha$ indicated on the figure. Dashed lines are guides to the eye. The relative standard error associated to each symbol is about 100%, 10%, and 1% for $\alpha \in (0,1),(1,2)$, and $(2,\infty )$, respectively. The solid line, with slope $-1$, was drawn for comparison. $T\equiv (T_L+T_R)/2=0.8$ and $\mathrm{\Delta }T\equiv T_L-T_R=0.2$.

Figure 3

Conductivity $\kappa$ versus temperature $T$, for different system sizes and fixed $\alpha$ indicated on each panel. In case $\alpha =3$, we show a power-law fit to the plots of $\kappa \left(T\right)$ in the high-temperature range. In the inset, we show the dependency of $\kappa$ with $N$ at a low temperature $\left(T=0.08\right)$, for different values of $\alpha$. In all cases, $\mathrm{\Delta }T/T=0.25$. Dotted lines are guides to the eye.

Figure 4

Diagram in the plane $T\text{−}\alpha$ of the different thermal regimes of the conductivity $\kappa$: conductivity attains a finite value (F) or not, either decreasing (D) or increasing (I) with $N$, for the investigated range of $N$. From plots of $\kappa \sim N\mathcal{J}$ vs. $N$, like in Fig. 2, we obtained the classification indicated by characters for each pair $(T,\alpha )$, with $\mathrm{\Delta }T/T=0.25$. The arrow highlights the outcomes for $T=0.8$, in agreement with the results shown in Fig. 2, although $\mathrm{\Delta }T$ is slightly different. The black solid line represents the critical value ${\alpha }_c\left(T\right)$.