Wave propagation in a strongly disordered 1D phononic lattice supporting rotational waves

We first study the dynamics of the lattice under two different initial momentum excitations

i.e., initially exciting either the transverse or the rotational momentum of the central site. Note that this choice of initial conditions corresponds to a total energy of $H=0.5$ for both cases. Some typical time evolutions of the energy densities are shown Figs. 3(a) and (b) for an initial (a) rotational and (b) transverse momentum excitation. We observe that for both cases, a large amount of energy remains localized in the region of the initial excitation at the lattice’s center. This is expected due to the fact that most of the modes are localized and thus the implemented initial condition strongly excites localized modes around the central site. This is also quantified by the time evolution of the averaged participation number $\langle P\rangle$ shown in Figs. 3(c) and (d) for respectively the rotational and transverse initial momentum excitations. In fact we observe that in both cases $\langle P\rangle$ saturates to a small number compared to the total lattice size. However, comparing the two final values we observe a significant difference between the two cases as the transverse initial excitation [Fig. 3(d)] leads to a larger $\langle P\rangle$.

This behavior can be understood by studying the projection of the initial conditions onto the normal modes of a finite but large disordered lattice. For a given initial momentum excitation we define the vector $\vec{V}(0)=[\dot{u}_{1}(0),\ldots,\dot{u}_{N}(0),\dot{\phi}_{1}(0),\ldots,\dot{\phi}_{N}(0)]^{T}$ whose projection on the system’s normal modes is given by $\vec{\dot{R}}=\mathbf{A}^{-1}\vec{V}(0)$ with matrix $\mathbf{A}$ having as columns the lattice eigenvectors. Using this projection, we can calculate the energy given to each normal mode as $E_{k}=\dot{R}_{k}^{2}/2$ where $\dot{R}_{k}$ are the elements of projection vector $\vec{\dot{R}}$. Obviously the system’s total energy is $H=\sum E_{k}$. Figs. 3(e) and (f), although they appear to have a similar form, exhibit important differences regarding low index ($k$) modes (see also Appendix B). Since we sorted the modes with increasing frequency, low indices correspond to low frequency extended modes. In fact, for the initial rotational momentum [Fig. 3(e)], the low frequency extended modes (low index $k$) are the stronger excited ones with an energy up to the order of $10^{-6}$. On the other hand, by initially exciting the transverse momentum, the low frequency modes (low index $k$) as shown in Fig. 3(f), are strongly excited acquiring energies up to $10^{-3}$. These orders of magnitudes difference in energy of low frequency extended modes explains the differences in $\langle P\rangle$ shown in Figs. 3(c) and (d).

Here we also observe major differences between the micropolar lattice and the well studied $1$D harmonic lattice with disorder  disorder_15 ; Kundu ; Lepri . With an initial momentum excitation, instead of exciting all modes with the same energy as in the $1$D harmonic lattice, here we observe a strong excitation of the low frequency modes and another set of modes around the cut-off frequency of the upper branch of the periodic case. As we will see below this has consequences to the energy transport.

To quantify the energy spreading we compute the averaged second moment $\langle m_{2}\rangle$ Kundu ; flach2009 ; SGF13 ; Bob2018 ; malcolmDNA ; arn_gran of the energy distributions, which for an initial excitation in the middle of the lattice is given by $m_{2}=\sum_{n}(n-N/2)^{2}h_{n}/H$. Assuming a polynomial dependence of the spreading, for sufficiently long times, we may write $\langle m_{2}\rangle\propto t^{\beta}$ and the parameter $\beta$ is used to quantify the asymptotic behavior. The exponent $\beta$ is calculated by first smoothing the $m_{2}(t)$ values of each disorder realization through a locally weighted difference algorithm cleveland1 ; cleveland2 . The estimate of the rate of change

is thus obtained numerically through a central finite difference scheme as the values of $m_{2}(t)$ are analyzed in log-log scale.

In Figs. 3(g) and (h) we observe that for both cases $\beta$ reaches an asymptotic value. In fact $\beta\approx 0.75~(\beta\approx 1.25)$ for initial rotational (transverse) momentum excitations corresponding to subdiffusive (superdiffusive) transport. These values are quite different than the ones observed for the $1$D harmonic lattice where momentum excitation is always found to be superdiffusive with $\beta\approx 1.5$ Kundu ; Lepri ; Wagner . To qualitatively explain this difference we first note that the exponent $\beta$ has been shown to depend mainly on two factors: (i) the characteristics of extended modes (group velocity, localization length as function of frequency, total number) and (ii) the projection of the initial condition on the modes Kundu ; Lepri ; Wagner . Regarding point (i), for both models, there is a set of extended modes at $\Omega\ll 1$. Major differences are thus expected since the dispersion relation of Eq. (5) for the micropolar lattice at low frequencies is quadratic with respect to the wavenumber i.e.,  $\Omega\approx 3\sqrt{K^{(2)}}Q^{2}$ in contrast to the $1$D harmonic lattice where $\Omega\approx Q$. Furthermore, for the micropolar lattice, the quasi-extended modes at higher frequencies may influence the energy spreading, as it was shown for example in Kundu ; Luck where additional extended modes were found either due to symmetries or resonances. As far as point (ii) is concerned, the results of Figs. 3(e) and (f) are substantially different from those of the $1$D harmonic lattice indicating that differences between the two models are anticipated.

To further compare the behavior of the micropolar model to that of the $1$D harmonic lattice Ishii ; Kundu , we now study the dynamics induced by the following initial conditions

which correspond to initial rotation or transverse displacement of the central block. In this study, the values of $\phi_{\frac{N}{2}}$ and $u_{\frac{N}{2}}$ are chosen such that the total energy for each realization is again $H=0.5$.

Similarly to Section III.1, the evolution of the energy distribution [Figs. 4(a) and (b)] is characterized by a localized wave-packet at the region of the initial excitation in center of the lattice, and by a portion which is propagating. However, compared to the initial momenta excitations, here the energy carried away from the central site is substantially smaller. For both types of initial conditions, $\langle P\rangle$ attains an asymptotic value of less than $10$ sites as shown in Figs. 4(c) and (d). This behavior can be also understood using the projection of the initial conditions to the normal modes of a large but finite lattice. This is now done by projecting the vector $\vec{U}(0)=[u_{1}(0),\ldots,u_{N}(0),\phi_{1}(0),\ldots,\phi_{N}(0)]^{T}$ onto the normal modes to yield $\vec{R}=\mathbf{A}^{-1}\vec{U}(0)$. In this case, the energy of the $k$th normal mode is $E_{k}=\Omega_{k}^{2}R_{k}^{2}/2$ with $\Omega_{k}$ being the $k$th eigenfrequency. Again the system’s total energy is $H=\sum E_{k}$. The outcome of this projection is shown in Figs. 4(e) and (f) for the initial rotation and transverse displacement respectively. The results are similar to those of Figs. 3(e) and (f) with suppressed contributions of the low frequency modes leading to a small value of $\langle P\rangle$ during the evolution.

Furthermore, we also calculated the exponent $\beta$ for the energy propagation resulting from these two different initial excitations and the results are shown in Figs. 4(g) and (h). In a similar manner as in the $1$D disordered harmonic lattice case, single site displacement excitation lead to subdiffusive behavior. However, our findings show that although the initial rotation excitations leads to the same value ($\beta\approx 0.5$) as in the $1$D harmonic lattice, the initial transverse displacement features extremely slow energy transport with $\beta\approx 0.25$. The discrepancy between the two models is anticipated as it was discussed at the end of Section III.1. However our results for the displacement excitations of the micropolar lattice strongly suggest that the energy transport is indeed mediated by both the low frequency and the quasi-extended modes. To be more precise we compare the results of Fig. 3(e) with Fig. 4(e) and notice that in the latter lower frequency modes are less excited leading to a smaller value of $\beta$ ($\beta\approx 0.75$ for the former and $\beta\approx 0.5$ for the latter). In the same spirit, by comparing Fig. 3(e) with Fig. 4(f) the main difference lies in the quasi-extended modes which are suppressed in the later case leading to a $\beta\approx 0.25$ instead of $0.75$. Thus reducing the amount of energy allocated to either the low frequency extended modes or the quasi-extended modes results in a reduced $\beta$ suggesting that both contribute to the energy spreading.

Note that the result of initial transverse momentum, corresponding to Fig. 3(f) is not compared with the other three since in that case the low frequency modes are highly excited. We thus conclude that the complete picture is comprehended by casting one eye on the low frequency extended and the other onto the quasi-extended ones.

The participation number measures the localization of the total energy and the exponent $\beta$ is a measure of how fast the energy is spreading, however, none of them carries any information of what amount of this energy is attributed to the rotational or the transverse DOFs. We can have an indication of how much energy is attributed to each of the two DOFs by decomposing the total energy of the system into two parts i.e., $H=H_{R}+H_{T}$ as follows

Before discussing the dynamical behavior of the system in these two distinct regions, it is relevant to show how the two different energy contributions are shared between the modes of a finite disordered lattice. The result is shown in Fig. 5(a) where the red (blue) curve depicts the transverse (rotational) energy contribution. As a general observation, we mention that the modes with lower $k$ values are dominated by transverse motion, while the high frequency ones are dominated by rotational motion. As it is shown in Fig. 5(b), for both initial conditions concerning the rotational DOFs [$P_{n}^{(\phi)}(0)=\sqrt{I}\delta_{n,N/2}$ and $\phi_{n}(0)=\phi_{N/2}\delta_{n,N/2}$], the central, localized part of the energy distribution is dominated by the rotational motion with almost the same ratios. This is so as the majority of the localized modes are dominated by rotation [see Fig. 5(a) and Fig. 2(b)]. By initially exciting the transverse DOF we end up with the two energy contributions in the central part shown in Fig. 5(c). We find that the central part of the energy distribution for the initial displacement excitation [$u_{n}(0)=u_{N/2}\delta_{n,N/2}$] is still dominated by rotation as indicated by the solid curve in Fig. 5(c). Interestingly, for the case of initial transverse momentum excitation [$P_{n}^{(u)}(0)=\delta_{n,N/2}$], the energy contribution of each motion in the central part is inverted with respect to all other cases. This is due to the fact that this initial condition excites more strongly the low frequency modes [Fig. 3(f)] which according to Fig. 5(a) are dominated by transverse motion.

Let us now turn our attention to the energy distribution far away from the central site following the propagating tails that are responsible for the energy transfer. The corresponding results for the initial transverse momentum excitation is shown in Fig. 5(d). After the arrival of the propagating front at the chosen sites $n=N/2\pm 5000$, it is readily seen that the energy at the edges is completely carried by the transverse motion. We have confirmed the same quantitative result for all types of initial conditions. This behavior can be comprehended since we have shown that energy is carried away mostly from the low frequency extended modes hence as shown in Fig. 5(a) these modes (corresponding to small $k$) are almost completely constituted by transverse motion and so is the energy at the edges.

Now, we focus on a special case of the system i.e., in the limit of vanishing bending stiffness $K^{(2)}$. Note that such a case is very relevant to situations where the bending stiffness is so small that it may be neglected, see for example Ref. Flornew .Then, the corresponding dispersion relation of the periodic system [dashed lines in Fig. 2(a)] is substantially altered. In particular instead of two propagating bands, it consists of a zero frequency non-propagative band and a dispersive band emerging after a cut-off frequency $\Omega=2$. The zero frequency branch is made possible due to a counterbalance between the shear and bending forces pichard .

To study the energy transfer for this special case of $K^{(2)}=0$, we have performed simulations for the different single site initial conditions considered before and a characteristic example is shown in Fig. 6. There it is readily seen that the energy remains localized around the center, and in addition there is no energy transfer to the rest of the lattice. This is also confirmed by the time dependence of the exponent $\beta$ in $\langle m_{2}(t)\rangle\propto t^{\beta}$, which becomes zero (see inset of Fig. 6) thus signaling no energy spreading.

Trying to explain the absence of energy transport we found (by solving the corresponding eigenvalue problem numerically) that the lower branch of the system’s frequency spectrum, in the limit case of $K^{(2)}=0$, still remains at zero frequency even in the presence of strong disorder due to a counterbalance of the transverse and rotational motions. As such, in this limit, the micropolar lattice is similar to a $1$D KG model i.e., featuring a single propagating band emerging after a lower cut-off frequency and thus the system is expected to exhibit AL.