Energy transport in the presence of long-range interactions

We study energy transport in the paradigmatic Hamiltonian mean-field (HMF) model and other related long-range interacting models using molecular dynamics simulations. We show that energy diffusion in the HMF model is subdiffusive in nature, which confirms a recently obtained intriguing result that, despite being globally interacting, this model is a thermal insulator in the thermodynamic limit. Surprisingly, when additional nearest-neighbor interactions are introduced to the HMF model, an energy superdiffusion is observed. We show that these results can be consistently explained by studying energy localization due to thermally generated intrinsic localized excitation modes (discrete breathers) in nonlinear discrete systems. Our analysis for the HMF model can also be readily extended to more generic long-range interacting models where the interaction strength decays algebraically with the (shortest) distance between two lattice sites. This reconciles many of the apparently counterintuitive results presented recently [C. Olivares and C. Anteneodo, Phys. Rev. E 94, 042117 (2016); D. Bagchi, Phys. Rev. E 95, 032102 (2017)] concerning energy transport in two such long-range interacting models.

Figure 1

Subdiffusion of energy in the HMF model for $N=200$ and $u=1.0$: (a) the correlation function ${\rho }_{{}_E}(r,t)$ at different times $t$ (only a small window around $r=0$ is shown here), (b) the MSD grows as ${\sigma }_{{}_E}^2\left(t\right)\sim t^{0.43}$, and (c) the height decreases with time as ${\rho }_{{}_E}(0,t)\sim t^{-0.21}$ at large times, indicating energy subdiffusion.

Figure 2

Superdiffusion of energy in the GHMF model for $N=200$ and $u=1.0$: (a) the correlation function ${\rho }_{{}_E}(r,t)$ at different times $t$, (b) the MSD grows as ${\sigma }_{{}_E}^2\left(t\right)\sim t^{1.5}$, and (c) the height decreases with time as ${\rho }_{{}_E}(0,t)\sim t^{-0.75}$ at large times, indicating energy superdiffusion.

Figure 3

The MSD ${\sigma }_{{}_E}^2\left(t\right)$ as a function of time $t$ with $N=200$, 400, and 800 for (a) the HMF model and (b) the GHMF model. The energy density is fixed at $u=1.0$.

Figure 4

The temperature profile $T_i$ of the HMF model for different system sizes $N=100$, 150, and 200. The $x$ axis represents the particle indices $i$ and has been rescaled by the system size $N$. The dashed horizontal line indicates the initial temperature $T_b=3.0$. Inset: The number of localized excitations $n$ having $T_i>T_b=3.0$ increases as the system size $N$ is increased.

Figure 5

Variation of the particles' temperature with time $t$ for (a) the HMF model, (b) the GHMF model, and (c) the inertial XY (coupled rotor) model, all for $N=50$. The $y$ axis represents the particle indices on the lattice denoted by $i$, and the $x$ axis represents time $t$. The colors indicate the average temperature $T_i$ of each particle ranging from $T_i=0$ to $T_i=10$, as shown in the scale beside each figure.

Figure 6

Temperature profile for the LR-XY model for different values of $\delta$ with $N=500$. Inset: The number $n$ is estimated as a function of $\delta$ with the temperature threshold set at $T_{\text{th}}=1.0$, represented by the horizontal dashed line in the main figure.