Chaotic and ballistic dynamics in time-driven quasiperiodic lattices

We investigate the nonequilibrium dynamics of classical particles in a driven quasiperiodic lattice based on the Fibonacci sequence. An intricate transient dynamics of extraordinarily long ballistic flights at distinct velocities is found. We argue how these transients are caused and can be understood by a hierarchy of block decompositions of the quasiperiodic lattice. A comparison to the cases of periodic and fully randomized lattices is performed.

Figure 1

(a) Sketch of a laterally oscillating lattice build of two barrier types $\mathcal{A}$ and $\mathcal{B}$ distinguished by their potential heights $V_{\mathcal{A}}$ and $V_{\mathcal{B}}$. Between consecutive barriers are the Poincaré surfaces (orange). (b) Extract of the Poincaré surface of section of a periodic lattice consisting only of $\mathcal{A}$-type barriers with $V_{\mathcal{A}}=1.5$. The horizontal dashed line indicates the maximal velocity $v_{\text{max}}^{\text{C}}$ that particles can acquire in the chaotic sea. Also shown is the period three fixed point of the Poincaré map ${\mathcal{M}}_{\mathcal{A}}$ centering the three corresponding regular islands. Each arrow indicates one application of ${\mathcal{M}}_{\mathcal{A}}$ on a trajectory in the fixed point. Remaining parameters are $\omega =A=m=l=1.0$ and $L=5.0$.

Figure 2

Exemplary trajectories $v\left(t\right)$ for the periodic (a), randomized (b), and quasiperiodic lattices (c). In all three cases, the horizontal line denotes $\pm v_{\text{max}}^{\text{C}}$ [see Fig. 1]. The inset in (a) shows a zoom into a typical stickiness event. Parameters are $V_{\mathcal{A}}=1.5$ [same as in Fig. 1] and $V_{\mathcal{B}}=1.0$. Remaining parameters are the same as in Fig. 1.

Figure 3

(a) Flight length distribution $\mathrm{\Gamma }\left(\mathrm{\Delta }x\right)$ for large $\mathrm{\Delta }x$ in double logarithmic representation for the randomized lattice. (b) $\mathrm{\Gamma }\left(\mathrm{\Delta }x\right)$ for periodic and quasiperiodic lattice. Shaded intervals correspond to extraordinarily long flights. (c) Ratio of $\mathrm{\Gamma }\left(\mathrm{\Delta }x\right)$ of the periodic or quasiperiodic lattices and $\mathrm{\Gamma }\left(\mathrm{\Delta }x\right)$ of the randomized lattice. Shaded intervals are identical as in (b). Also shown are velocity-resolved flight length distributions $\mathrm{\Gamma }(\mathrm{\Delta }x,\overline{v})$ for the random (d), periodic (e), and quasiperiodic lattices (f). In the latter case, the dashed rectangle highlights the horizontal branch of long flights.

Figure 4

Flight length $\mathrm{\Delta }x(x_0,{\varphi }_0,v_0)$ starting at $x_0=0$ as a function of the initial phase and initial velocity for the periodic (a), randomized (b), and quasiperiodic (c) lattice. (The shorter cutoff in the PL is for illustrative purpose only). In (d) we show $\mathrm{\Delta }x(x_0=100\times L,{\varphi }_0,v_0)$ for the quasiperiodic case. Dashed rectangles highlight plateaus of extraordinarily long flights. Parameters are the same as in Fig. 2.

Figure 5

Block decomposition of the Fibonacci lattice in symbolic notation according to the decomposition rule given in Eqs. (6) and (7). The first row depicts the first few elements of the Fibonacci sequence [cf. Eq. (3), where a 1 (0) corresponds to a barrier of type $\mathcal{A}$ ($\mathcal{B}$)]. At the same time, this first row is the “zeroth generation” of the decomposition hierarchy. All further rows show block decompositions of increasing generations.

Figure 6

Poincaré surfaces of section of lattices consisting of periodic repetitions of blocks: ${\mathcal{B}}_1$ (a), ${\mathcal{A}}_1$ (b), ${\mathcal{B}}_4$ (c), ${\mathcal{A}}_4$ (d), ${\mathcal{B}}_7$ (e), and ${\mathcal{A}}_7$ (f). Dashed rectangles are at the same locations as in Fig. 4 denoting a domain of long flights in the quasiperiodic lattice. Remaining parameters are as in Fig. 2.