Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System

In flat space, changing a system’s velocity requires the presence of an external force. However, an isolated nonrigid system can freely change its orientation due to the nonholonomic nature of the angular momentum conservation law. Such nonrigid isolated systems may thus manifest their internal dynamics as rotations. In this work, we show that for such systems chaotic internal dynamics may lead to macroscopic rotational random walk resembling thermally induced motion. We do so by studying the classical harmonic three-mass system in the strongly nonlinear regime, the simplest physical model capable of zero angular momentum rotation as well as chaotic dynamics. At low energies, the dynamics are regular and the system rotates at a constant rate with zero angular momentum. For sufficiently high energies a rotational random walk is observed. For intermediate energies the system performs ballistic bouts of constant rotation rates interrupted by unpredictable orientation reversal events, and the system constitutes a simple physical model for Lévy walks. The orientation reversal statistics in this regime lead to a fractional rotational diffusion that interpolates smoothly between the ballistic and regular diffusive regimes.

Figure 1

The symmetric harmonic three-mass system. (a) Sketch of the system with equal masses $m$, spring constants $k$, and rest lengths $L$. $\theta$ is the angle between $r_{12}$ and the $x$ axis, used to determine the orientation of the three-mass triangle. Panels (b)–(d) present the normal modes and corresponding frequencies, commonly known as the symmetric stretch, isometric bend, and asymmetric bend, respectively.

Figure 2

Four typical trajectories of the orientation of the three-mass triangle as a function of time (a)–(d), their power spectra (e)–(h), and a long-exposure image of their dynamics with each mass colored in a different color (i)–(l). The initial conditions differ in energies: (a),(e),(i) $E=0.02$, (b),(f),(j) $E=0.28$, (c),(g),(k) $E=0.62$, (d),(h),(l) $E=1.29$. The energy units fit the simulation parameters $m=1$, $L=2$, $k=1$ for which the typical energy scale reads $E_s=3k{L}^2/2=6$. (a),(e),(i) At low energies $E\ll 1$, the system behaves quasiperiodically, with frequencies fitting the linear modes of the system, peaking at the degenerate linear frequency $\sqrt{3/2}$. (b),(f),(j) At slightly higher energies $0.05\lesssim E\lesssim 0.3$, the degenerate frequencies split, leading to beating, and the system displays nonlinear yet regular oscillations. (c),(g),(k) At higher energies $0.3\lesssim E\lesssim 0.7$, the system becomes observably chaotic, exhibiting quasiperiodic motion on short timescales, and unpredictable transitions between constant angular velocity bouts on long timescales that show high sensitivity to initial conditions. The power spectrum fills up, adopting a constant ${\omega }^{-2}$ slope yet continues to display a significant peak around the linear frequency $\sqrt{3/2}$. (d),(h),(l) Around the energy $E\gtrsim 0.7$, the system loses its short-time periodicity and the orientation resembles a random walk. The power spectrum shows that all frequencies are excited with approximately the same power, and the linear frequencies lose significance.

Figure 3

Comparison between the angular MSD and the bout-length PDF in the regime of motion $0.3\lesssim E\lesssim 0.7$, where the motion is composed from bouts of constant averaged angular velocity of varying times. Panels (a) and (b) show the process of calculating the bout-length PDF, $\mathrm{\Psi }(T)$, for a trajectory with $E=0.31$ by identifying the turning points between segments of constant averaged angular velocity. Panel (a) shows the given trajectory with red asterisks marking the turning points identified by the algorithm. Panel (b) shows the resulting $\mathrm{\Psi }(T)$ in a log-log plot along with the linear fit. Panels (c) and (d) show the process of calculating the MSD for the same trajectory, by averaging ${[\theta (T)-\theta (0)]}^2$ over $10^4$ runs (see SM [8]). When $T$ is big enough, the MSD is an excellent fit to a power law $⟨\mathrm{\Delta }\theta {(T)}^2⟩\propto T^{\alpha }$ with an anomalous diffusion exponent $\alpha$, as depicted in (d). (e) The PDF power $\nu$ as a function of the anomalous diffusion exponent $\alpha$ for several initial conditions in the relevant energy range $0.3\lesssim E\lesssim 0.7$. The black solid line and red dotted line show the Lévy-walk model prediction, $\alpha =3-\nu$. For the lower energies $E\approx 0.3$ (the enlarged region) the two exponents follow the Lévy-walk prediction well. As the energy grows, the average bout length shortens and $\alpha$ decreases, changing from almost 2 to 1 at high enough energies, $E\gtrsim 0.7$. The PDF fit to a single power law gradually deteriorates with increasing energy and we observe significant deviations from the power-law-based Lévy-walk prediction.

Figure 4

The diffusion exponent $\alpha$ (a) and diffusion coefficient $D$ (b) satisfying $⟨{\theta }^2⟩=2D{t}^{\alpha }$ plotted as a function of the energy for various types of initial conditions (IC). Error bars are smaller than the symbols. The (blue) circle, (red) asterisk, and (green) square symbol correspond to initial conditions set by different phase differences between the degenerate modes. The (gray) pentagram plot corresponds to irrational initial phase difference. The (magenta) diamond plot corresponds to random initial conditions. For most initial conditions the energy suffices for predicting $\alpha$. Near $E=0.35$ trajectories begin to change from ballistic angular motion where $\alpha =2$ (dark gray background) to anomalous diffusion, where $1<\alpha <2$ (light gray background), and at around $E=0.7$ trajectories begin to change to regular diffusion (clear background), where $\alpha =1$. In this regime, the diffusion coefficient appears to be linear in the energy with a slope 0.32 and an offset at $E\approx 0.5$. The MSD is calculated by averaging over a large ensemble of similar initial conditions; see SM for details [8].