Chaos and Anderson localization in disordered classical chains: Hertzian versus Fermi-Pasta-Ulam-Tsingou models

We numerically investigate the dynamics of strongly disordered 1D lattices under single-particle displacements, using both the Hertzian model, describing a granular chain, and the $\alpha +\beta$ Fermi-Pasta-Ulam-Tsingou model (FPUT). The most profound difference between the two systems is the discontinuous nonlinearity of the granular chain appearing whenever neighboring particles are detached. We therefore sought to unravel the role of these discontinuities in the destruction of Anderson localization and their influence on the system's chaotic dynamics. Our results show that the dynamics of both models can be characterized by: (i) localization with no chaos; (ii) localization and chaos; (iii) spreading of energy, chaos, and equipartition. The discontinuous nonlinearity of the Hertzian model is found to trigger energy spreading at lower energies. More importantly, a transition from Anderson localization to energy equipartition is found for the Hertzian chain and is associated with the “propagation” of the discontinuous nonlinearity in the chain. On the contrary, the FPUT chain exhibits an alternate behavior between localized and delocalized chaotic behavior which is strongly dependent on the initial energy excitation.

Figure 1

(a) Mean (over 1000 disorder realizations) participation number $\langle P\rangle$ of the eigenmodes for varying disorder strengths $\alpha$, sorted in descending order $k$ for each realization. The standard deviation at each point is shown by the error bars. (b) The eigenfrequencies of a particular disordered chain of 40 sites for $\alpha =5$ sorted by increasing frequency. The insert shows the profile of the $34\text{th}$ mode.

Figure 2

(a) and (b) The spatiotemporal evolution of the energy distribution for the Hertzian and FPUT chains respectively for $H=0.25$. The black curves indicate the running mean position of the energy distributions. The color bars on the right sides of (a) and (b) are in logarithmic scale. (c) The locally weighted smoothed values of $P$ as a function of time for the Hertzian chain (red curve) and the FPUT chain (blue curve). (d) The time evolution of $\mathrm{\Lambda }\left(t\right)$ for the Hertzian chain (red curve) and the FPUT chain (blue curve). Both lines practically overlap and the dashed line indicates the law $\mathrm{\Lambda }\left(t\right)\propto t^{-1}$.

Figure 3

(a) [(b)] The spatiotemporal evolution of the deviation vector density (DVD) for the Hertzian [FPUT] disordered chain. The color bars on the right sides of (a) and (b) are in logarithmic scale. (c) [(d)] Deviation vector profiles for three time instances of $t\approx {10}^1$ indicated by the blue (b) curve, $t\approx {10}^3$ indicated by the red (r) curve and $t\approx {10}^5$ indicated by the black (bl) curve. These times correspond, respectively, to the blue, red, and black horizontal lines in panel (a) [b]. (e) [(f)] The time evolution of the participation number $P_D$ of the DVD for the Hertzian [ FPUT] model. All results are obtained for $H=0.25$.

Figure 4

Panels (a) and (b) show the spatiotemporal evolution of the energy distribution for the Hertzian model with $H=0.5$ and $H=1.8,$ respectively. Black curves indicate the running mean position of the energy distributions. The color [41, 42, 43] bars on the right sides of (a, b) are in logarithmic scale. Panels (c) and (d) are the same as Figs. 2 and 2; for the Hertzian model $H=0.5$ and $H=1.8$ and for the FPUT model with $H=1.8$.

Figure 5

Panel (a) shows the spatiotemporal evolution of the DVD for the Hertzian model at $H=0.5$ whilst panel (b) shows the profiles of the DVDs at $t\approx 1.7\times {10}^1$ red (r) curve, $t\approx 1.7\times {10}^4$ magenta (m) curve and $t\approx 8.2\times {10}^4$ blue (b) curve. (c) Same as (a) but for $H=1.8$. (d) Same as (b) but for $H=1.8$ at $t\approx 1.7\times {10}^1$ red (r) curve, $t\approx 4.9\times {10}^3$ magenta (m) curve, $t\approx 3.5\times {10}^4$ blue (b) curve and $t\approx 4.8\times {10}^4$ black (bl) curve. The color bars in (a) and (c) are in logarithmic scale.

Figure 6

Panels (a–d) depict the energy density, $P$, DVD, $P_{\mathrm{D}}$ and $\mathrm{\Lambda }\left(t\right),$ respectively, for the FPUT with $H=2.9$. The second, and third rows correspond to energies $H=4$ and $H=8.7381$ respectively. The color bars on the right sides of panels (a, c, e, g, i, and k) are in logarithmic scale.

Figure 7

The spatiotemporal evolution of the gaps in the Hertzian model for energies $H=0.5$ (a) and $H=1.8$ (b). The yellow (lighter) color corresponds to the lattice points where $[u_n\left(t\right)-u_{n-1}\left(t\right)]>{\delta }_n$. The instantaneous total number of gaps for the Hertzian model for energies $H=0.5$ (c) and $H=1.8$ (d).

Figure 8

Top row: The time evolution of the normalized spectral entropy $\eta \left(t\right)$ for the Hertzian model. The dashed horizontal line in panels (c) and (d) show the mean value $\langle \eta \rangle$ given by Eq. (12). Bottom row: The evolution of the weighted harmonic energy of eigenmodes as a function of time. The modes are sorted by increasing frequency [c.f. Fig. 1]. The values of the energy are $H=0.25$ (a–e), $H=0.5$ (b–f), $H=1.8$ (c–g), and $H=3$ (d–h). The color bars on the right sides of panels (e–h) are in logarithmic scale.

Figure 9

(a) Temporal evolution of $\eta \left(t\right)$ for the 9 most localized modes of the distribution corresponding to Fig. 1. (b) Temporal evolution of $\eta \left(t\right)$ obtained by exciting site $n=20$ of 9 different disordered realizations with $\alpha =5$. In both panels, the results correspond to the Hertzian model with an energy $H=3$. The dashed horizontal lines show the average (on the different initial conditions), mean entropy at equipartition $\langle \eta \rangle$.

Figure 10

Same as in Fig. 8 but for the FPUT model. The dashed horizontal line in panels (b) and (d) show the mean value $\langle \eta \rangle$ given by Eq. (12). The values of the energy in this case are $H=0.25$ (a–e), $H=2.9$ (b–f), $H=4$ (c–g), and $H=8.7381$ (d–h).