Superdiffusive Transport and Energy Localization in Disordered Granular Crystals

Abstract

We study the spreading of initially localized excitations in 1D disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to be fundamentally different from localization in linear and weakly nonlinear systems. We compare wave dynamics in chains with 3 different types of disorder: an uncorrelated (Anderson-like) disorder and 2 types of correlated disorders (random dimer arrangements), and for 2 types of initial conditions: displacement excitations and velocity excitations. For strongly precompressed chains, the dynamics depend strongly on the initial condition. For displacement excitations, the long-time behavior of the second moment $\tilde{m}_2$ has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson disorder. For velocity excitations, we find a scaling $\tilde{m}_2\sim t^γ$ (for a constant $γ$) for all 3 types of disorder. For weakly precompressed (strongly nonlinear) chains, $\tilde{m}_2$ and the inverse participation ratio $P^{-1}$ satisfy $\tilde{m}_2\sim t^γ$ and $P^{-1}\sim t^{-η}$, and the dynamics is superdiffusive for all examined cases. When precompression is strong, the IPR decreases slowly for all 3 types of disorder, and we observe a partial localization around the core and the leading edge of the wave. For an Anderson disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar in all cases.