"Superdiffusive Transport and Energy Localization in Disordered Granular Crystals"

A new paper of mine just came out in final form today. Here are the details. (Also see a paper by others published as a consecutive article with ours. Scientifically, it's really good that these articles have appeared as back-to-back papers.)

Title: Superdiffusive Transport and Energy Localization in Disordered Granular Crystals

Authors: Alejandro J. Martínez, P. G. Kevrekidis, and Mason A. Porter

Abstract: We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder—an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)—and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment ˜m2 of the energy 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-like disorder. By contrast, for velocity excitations, we find that a standard scaling m_2 ∼ t^γ (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m_2 and the inverse participation ratio P^{−1} satisfy scaling relations m_2 ∼ t^γ and P^{−1} ∼ t^{−η}, and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η < 0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like 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 (roughly γ ≈ 1.7 and η ≈ 1) for all three types of disorder and for both types of initial conditions.