Superdiffusive transport and energy localization in disordered granular crystals

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 m̃(2) 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 m̃(2)∼t(γ) (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m̃(2) and the inverse participation ratio P(-1) satisfy scaling relations m̃(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.

Nonlinear coherent structures in granular crystals

“…In the on-site case, nonlinearity promotes localization (in the form of discrete breathers), rather than enabling mobility (in the form of traveling waves, as in FPU-type lattices). As a result, energy transport is only subdiffusive, whereas (as indicated above) superdiffusivity can occur for disordered granular crystals [6,127]. The study of disorder and nonlinear analogs of Anderson localization are important directions to pursue; they are accessible experimentally, and a systematic theoretical understanding of associated superdiffusive and subdiffusive transport should lead to important insights more generally on the interaction between disorder and nonlinearity.…”

“…In the broad context of condensed-matter physics, disorder (e.g., in the form of randomness) causes strong localization-so-called "Anderson localization"that affects the diffusivity of waves in disordered media [9,22]. Recent studies in the context of granular crystals have demonstrated computationally that such a phenomenon can be altered dramatically and can be even reversed (as superdiffusive transport can occur) by manipulating the nonlinearity strength in mechanical systems [6,127] (see also, e.g., [150,170]) and its potential "competition" with disorder. Such versatile dynamics associated with disorder and nonlinearity have only been studied very recently in granular crystals (and, more generally, they have not been studied much in lattice systems with inter-site interactions).…”

Scattering of waves by impurities in precompressed granular chains

“…In this region, multiple scatterings occur because the presence of impurities has broken discrete translation symmetry, and successive interferences can then lead to complicated dynamics that depend on the distribution of impurities. A particular example of this phenomenon was investigated recently in the context of disordered granular chains [29].…”

Nanoptera in a Period-2 Toda Chain

“…Numerous studies have considered 'uniform' (i.e., monoatomic) one-dimensional (1D) lattices, which are chains in which each particle in the lattice is identical. However, particles need not be identical, and examinations of such 'heterogenous' lattices [51], with either periodic [43,44,65,79] or random [32,54] distributions of different particles, reveal a wealth of fascinating dynamics that do not arise in uniform lattices. Such dynamics include new families of solitary waves that have been observed in diatomic granular chains and which can exist only for discrete values of the ratio between the masses of the two particles in a diatomic unit [43].…”