Heat transport in oscillator chains with long-range interactions coupled to thermal reservoirs

We investigate thermal conduction in arrays of long-range interacting rotors and Fermi-Pasta-Ulam (FPU) oscillators coupled to two reservoirs at different temperatures. The strength of the interaction between two lattice sites decays as a power $\alpha$ of the inverse of their distance. We point out the necessity of distinguishing between energy flows towards or from the reservoirs and those within the system. We show that energy flow between the reservoirs occurs via a direct transfer induced by long-range couplings and a diffusive process through the chain. To this aim, we introduce a decomposition of the steady-state heat current that explicitly accounts for such direct transfer of energy between the reservoir. For $0\le \alpha <1$, the direct transfer term dominates, meaning that the system can be effectively described as a set of oscillators each interacting with the thermal baths. Also, the heat current exchanged with the reservoirs depends on the size of the thermalized regions: In the case in which such size is proportional to the system size $N$, the stationary current is independent on $N$. For $\alpha >1$, heat transport mostly occurs through diffusion along the chain: For the rotors transport is normal, while for FPU the data are compatible with an anomalous diffusion, possibly with an $\alpha$-dependent characteristic exponent.

Figure 1

A schematic representation of the nonequilibrium setup. Regions $A$ and $B$ (rounded boxes) identify the oscillators that interact with heat baths at temperature $T_A$ and $T_B$, respectively, with $l_A=3$ and $l_B=2$. Heat currents from the reservoirs to the system are denoted with $Q_A$ and $Q_B$ and those flowing from two generic sites of the chain by $J_{ij}$, respectively.

Figure 2

Stationary $XY$ temperature profiles for the setup in Fig. 3 and $N=128$. Panel (a) refers to the extensive-coupling case with $l_A=l_B=N/16$, while panel (b) shows the profiles for the finite-coupling case, with $l_A=l_B=1$. Simulations have been performed by setting $t_s={10}^7$. All the other parameters are the same of Fig. 3.

Figure 3

Scaling of the stationary heat current $Q$ in the $XY$ chain as a function of the system size $N$ and for different range parameters $\alpha$. Panel (a) refers to the extensive thermal coupling, with a size $l=N/16$ for both the thermalized regions A and B, while panel (b) shows the results for finite thermal coupling with $l_A=l_B=1$. In both panels, the black dashed line refers to the power-law scaling $N^{-1}$. Simulations have been performed by setting $T_A=0.45,T_B=0.35$, and $t_s={10}^6$.

Figure 4

Stationary current-matrix $\widehat{J}$, represented as a function of the intensive variables $i/N$ and $j/N$ for a rotor chain with $2N=64$ (periodic boundary conditions) and $t_s=5\times {10}^7$. The chain is in contact with two reservoirs at temperature $T_A=0.45$ and $T_B=0.35$ with extensive coupling, i.e., $l_A=l_B=N/16$. The color code points out the main channels through which heat flows. Panels (a), (b), (c), and (d) refer to $\alpha =2,1,0.5,0$, respectively. Panels (e) and (f) are pictorial representations of the heat transfer process for $\alpha =\infty$ and $\alpha =0$, respectively. Thermalized sites are contained in the rectangular boxes while the bulk system is represented in the ellipse. Black lines identify the relevant transport channels.

Figure 5

Spatial profiles of the modulus of $J_{ik}\left(\alpha \right)$ in the rotor chain for different system size $N$ and extensive coupling $l_A=l_B=N/16$: $\alpha =0$ (a), $\alpha =1$ (b), and $\alpha =2$ (c). To obtain a cleaner evidence of the data collapse in the lower and upper inset of panel (b) and for the profiles in panel (c) we have just reported $J_{-1k}\left(\alpha \right)$. The setup for numerical simulations is the same adopted for Fig. 4.

Figure 6

Scaling of the stationary heat current $Q$ of the FPU chain as a function of the system size $N$ for different range parameters $\alpha$. The black dashed line refers to a power law $Q\sim N^{-\frac{3}{5}}$. Simulations refer to an extensive thermal coupling with $l_A=l_B=N/16$ and fixed boundary conditions. Nonequilibrium states were sampled by choosing $t_s={10}^6$. The inset shows the temperature profiles for $\alpha =2$ and different chain lengths $N$. Note that the horizontal axis reports the rescaled spatial variable $x=n/N$.

Figure 7

Stationary current-matrix $\widehat{J}$ as a function of the intensive variables $i/N$ and $j/N$ for a long-range FPU chain with $N=64$ and periodic boundary conditions and $t_s={10}^7$. The chain is in contact with two reservoirs at temperature $T_A=1.1$ and $T_B=0.9$ with coupling lengths $l_A=l_B=N/16$. Panels (a), (b), (c), and (d) refer to $\alpha =2,1,0.5,0$, respectively.

Figure 8

Scaling of the stationary heat current $Q$ of the FPU chain as a function of the system size $N$ for different values of the range parameter $\alpha >1$ and $t_s=5\times {10}^6$. Extensive coupling and periodic boundary conditions have been adopted for these numerical simulations. The black dashed line refers to a power law $Q\sim N^{-\frac{3}{5}}$.