Temperature dependence of heat conduction in the Fermi-Pasta-Ulam-$\beta$ lattice with next-nearest-neighbor coupling

We show numerically that introducing the next-nearest-neighbor interactions (of appropriate strength) into the one-dimensional (1D) Fermi-Pasta-Ulam-$\beta$ (FPU-$\beta$) lattice can result in an unusual, nonmonotonic temperature dependent divergence behavior in a wide temperature range, which is in clear contrast to the universal divergence manner independent of temperature as suggested previously in the conventional 1D FPU-$\beta$ models with nearest-neighbor (NN) coupling only. We also discuss the underlying mechanism of this finding by analyzing the temperature variations of the properties of discrete breathers, especially that with frequencies having the intraband components. The results may provide useful information for establishing the connection between the macroscopic heat transport properties and the underlying dynamics in general 1D systems with interactions beyond NN couplings.

Figure 1

(a) and (b): The temperature profile for two typical average system temperatures $T=0.25$ and $T=30$, respectively. (c) and (d): The corresponding dependent properties of the heat conductivity $\kappa$ on system size $L$; The best fitting (the dashed lines) suggest $\kappa \sim L^{\alpha }$ with $\alpha =0.38\pm 0.01$ and $\alpha =0.28\pm 0.01$, respectively.

Figure 2

The dependence of $\alpha$ on the average system temperature $T$ for FPU-$\beta$ model (the solid circle) and the purely quartic FPU model (the hollow circle). Both models include the NNN couplings with a fixed ratio $\gamma =0.25$. The error bars give the standard error for evaluating $\alpha$ by linearly fitting $ln\kappa$ versus $lnL$. The vertical dashed line indicates the turning point $T_{\mathrm{tr}}=1.5$; the horizontal line indicates $\alpha =1/3$ for the FPU-$\beta$ model without including the NNN couplings.

Figure 3

(a) and (b): Snapshots of two typical examples of DBs that randomly emerge in the lattice after a long enough time for absorption. The initial system temperature for both cases is fixed at $T=1.5$. (c) and (d): The corresponding power spectrum $P\left(\omega \right)$ (take arbitrary units, the same hereinafter) of the DBs. The boxed region in (d) implies that kind of DBs having the intraband components.

Figure 4

Comparison between the frequencies of the intraband components indicated in Fig. 3 (the dashed lines) and the phonons frequencies (the solid lines). For a better visualization, in all plots, the values of $P\left(\omega \right)$ have been rescaled between 0 and 1.

Figure 5

The power spectrum $P\left(\omega \right)$ of the residual thermal fluctuations for three typical initial system temperatures: (a) $T=0.25$; (b) $T=1.5$; (c) $T=30$, respectively. The inset in (b) is a zoom for the boxed intraband components ($\omega \le 2$). (d) The energy portion $\varepsilon$ of the residual thermal fluctuations within the phonon band. The vertical dashed line indicates the turning point $T_{\mathrm{tr}}=1.5$.