Super heat diffusion in one-dimensional momentum-conserving nonlinear lattices

Heat diffusion processes in various one-dimensional total-momentum-conserving nonlinear lattices with symmetric interaction and asymmetric interaction are systematically studied. It is revealed that the asymmetry of interaction largely enhances the heat diffusion; while according to our existing studies for heat conduction in the same lattices, it slows the divergence of heat conductivity in a wide regime of system size. These findings violate the proposed relations that connect anomalous heat conduction and super heat diffusion. The generality of those expectations is thus questioned.

Figure 1

Left column: Snapshots of the rescaled excess energy distribution ${\delta }_{\mathrm{E}}(i,t)$ at time $t=100$ for various lattices and three different energy pulse values. Right column: The corresponding ${\sigma }^2\left(t\right)$. Generally the side peaks move a little faster for the case $\mathrm{\Delta }E=10\langle E\rangle$, and correspondingly ${\sigma }^2\left(t\right)$ is also larger. The difference vanishes when $t$ increases. It is reasonable, because the system is closer to equilibrium state as $t$ increases. The results for $\mathrm{\Delta }E=\langle E\rangle$ and $3\langle E\rangle$ basically overlap each other.

Figure 2

(a)–(c) Energy diffusion in various lattices, $\mathrm{\Delta }E=3\langle E\rangle$. Data binning over contiguous time $t$ has been performed to reduce statistical fluctuations. (d) The corresponding ${\sigma }^2\left(t\right)$. Inset: The comparison between the results from different ways of energy injection, the kinetic energy and potential energy, for the FPU-$\alpha 2\beta$ lattice.

Figure 3

(a)–(c) The rescaled local energy correlation function ${\rho }_{\mathrm{E}}(i,t)$ in various lattices. (d) The corresponding $r^2\left(t\right)$. The error bars are not plotted since they are smaller than the symbol size.

Figure 4

(a) ${\kappa }_{\mathrm{NE}}$ and (b) ${\kappa }_{\mathrm{GK}}$ vs lattice length $L$ for various lattices. They basically agree with each other. These data have been published in Refs. [20, 21]. The lines in (b) are the best linear fits of ${\kappa }_{\mathrm{GK}}$ in the middle region of $L$. $\kappa$ diverges much more slowly in the lattices with asymmetric interaction, in a wide region of $L$.

Figure 5

Calorie curves for the four lattices. Symbols correspond to the numerical simulations, and the lines are their linear fit around $T=1$.

Figure 6

(a) Comparison between the left-hand (symbols) and right-hand (curves) sides of Eq. (9) for various lattices. The agreement for those with symmetric interaction (the FPU-$\beta$ and the purely quartic lattices) is very good. However, evident discrepancies are observed for the lattices with asymmetric interaction, i.e., the FPU-$\alpha \beta$ ones. (b) Comparison between the results from different definitions of the local energy $E_i$ and the local current $j_i$ for the FPU-$\beta$ lattice. Symbols and curves correspond to the left- and right-hand sides of Eq. (9), respectively.