Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter $\lambda$, and on the other hand our numerics show that it follows a power law for a broad range of values of $\lambda$. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.

Figure 1

Mathematical constraints on the decorrelation time are shown as straight lines on this plot. The true curve has to lie above all these lines (like the three displayed black curves).

Figure 2

The decorrelation $\eta \left(t\right)$ as a function of rescaled time ${\lambda }^{4}t$. With this rescaling, we observe collapse of decorrelation profiles for small anharmonicities $\lambda$.

Figure 3

The decorrelation time extracted from Fig. 2, with ${\eta }_0=0.2$ and ${\eta }_0=0.5$. We indicated by straight lines the regime for which $\tau \propto {\lambda }^{-4}$, and a short regime for which $\tau \propto {\lambda }^{-2}$.

Figure 4

Correlation $\text{corr}(p_i,p_{i+1})$. The blue curve is a numerical estimate for the ergodic average of $\text{corr}(p_i,p_{i+1})$ after evolving the system of length $L={10}^3$ from the same initial conditions as considered in Ref. [4] for $t={10}^8$. The orange curve shows $\text{corr}(p_i,p_{i+1})$ in the prethermal ensemble constructed from data in a window of size 10 around each site. The black line is the value of $\text{corr}(p_i,p_{i+1})$ in a thermal state. See Appendix pp4 for further explanations.

Figure 5

The decorrelation $\eta$ as a function of rescaled time ${\lambda }^{4}t$, for a system of length $L=160$. We used here $D=100$ realizations of disorder and measured decorrelation on a numerical grid $t_j=j\times {10}^s, j=0,1,\cdots ,{10}^5$ with $s=4$ for $\lambda =4\times {10}^{-4}$ and $s=3$ otherwise. All other parameters were taken the same as for the system of length $L=80$.

Figure 6

The decorrelation $\eta$ as a function of rescaled time ${\lambda }^{4}t$, for systems of length $L=80$ and $L=160$. We plotted the same curves as in Figs. 2 and 5.

Figure 7

Plots of (a) $H\left(t\right)$ and (b) ${\tilde{H}}_F\left(t\right)$ with $\lambda =2\times {10}^{-4}$ evolving from thermal initial conditions as described in Appendix pp3.