Mechanical energy fluctuations in granular chains: The possibility of rogue fluctuations or waves

The existence of rogue or freak waves in the ocean has been known for some time. They have been reported in the context of optical lattices and the financial market. We ask whether such waves are generic to late time behavior in nonlinear systems. In that vein, we examine the dynamics of an alignment of spherical elastic beads held within fixed, rigid walls at zero precompression when they are subjected to sufficiently rich initial conditions. Here we define such waves generically as unusually large energy fluctuations that sustain for short periods of time. Our simulations suggest that such unusually large fluctuations (“hot spots”) and occasional series of such fluctuations through space and time (“rogue fluctuations”) are likely to exist in the late time dynamics of the granular chain system at zero dissipation. We show that while hot spots are common in late time evolution, rogue fluctuations are seen in purely nonlinear systems (i.e., no precompression) at late enough times. We next show that the number of such fluctuations grows exponentially with increasing nonlinearity whereas rogue fluctuations decrease superexponentially with increasing precompression. Dissipation-free granular alignment systems may be possible to realize as integrated circuits and hence our observations may potentially be testable in the laboratory.

Figure 1

The number of hot spots $n_H$ for $n=2.01,2.1,2.2,2,3,2.4,2.5$ are shown. The number of hot spots has been calculated for periods across the total run time of 2 ms in our nondissipative ultrahard systems. The number of hot spots is strongly dependent upon the definition used to find them.

Figure 2

Plot shows how the number of rogue waves $n_R$ grows with increasing $n$, for various perturbation velocity amplitudes $v$ (0.01 mm/$\mu \mathrm{s}$, 0.03 mm/$\mu \mathrm{s}$, and 0.05 mm/$\mu \mathrm{s}$). The exponential growth in $n_R$ is robust (see text for details).

Figure 3

The number of rogue waves $n_R$ decays superexponentially with increasing precompression $\frac{\Delta }{R}$ (see text for details).