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 ${\tilde{m}}_2\sim t^{\gamma }$ (for a constant $\gamma$) for all 3 types of disorder. For weakly precompressed (strongly nonlinear) chains, ${\tilde{m}}_2$ and the inverse participation ratio $P^{-1}$ satisfy ${\tilde{m}}_2\sim t^{\gamma }$ and $P^{-1}\sim t^{-\eta }$, 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.

23 pages, 11 figures; the abstract in this website field is a shortened version of the paper's full abstract because of spatial limits

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• Quasiperiodic granular chains and Hofstadter butterflies
• Computational efficiency of symplectic integration schemes: Application to multidimensional disordered Klein-Gordon lattices
• Chaos and Anderson Localisation in Disordered Classical Chains: Hertzian vs FPUT models
• Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method
• Polydispersed Granular Chains: From Long-lived Chaotic Anderson-like Localization to Energy Equipartition
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• Wave propagation in a strongly disordered 1D phononic lattice supporting rotational waves
• Numerical study of anomalous diffusion of light in semi-crystalline polymer structures
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• Elastic Wannier-Stark Ladders and Bloch Oscillations in 1D Granular Crystals
• Nanoptera in weakly nonlinear woodpile and diatomic granular chains
• Chaotic behaviour of disordered nonlinear lattices
• Using Random Walks to Establish Wavelike Behavior in an FPUT System with Random Coefficients
• Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems
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