Energy spreading in strongly nonlinear disordered lattices

M Mulansky,A Pikovsky

doi:10.1088/1367-2630/15/5/053015

Abstract

We study the scaling properties of energy spreading in disordered strongly nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled local linear or nonlinear oscillators, and demonstrate a rather slow, subdiffusive spreading of initially localized wave packets. We use a fractional nonlinear diffusion equation as a heuristic model of this process, and confirm that the scaling predictions resulting from a self-similar solution of this equation are indeed applicable to all studied cases. We show that the spreading in nonlinearly coupled linear oscillators slows down compared to a pure power law, while for nonlinear local oscillators a power law is valid in the whole studied range of parameters.

Highlights

The general understanding of the relation between chaos in classical systems, and ergodicity and thermalization is still far from complete nowadays
We report on extensive numerical simulations of strongly nonlinear lattices, trying to check the predictions of the nonlinear diffusion equation (NDE) framework
We can find a relation between the order of the fractional derivative γ, the nonlinearity of the fractional nonlinear diffusion equation (FNDE) a and the parameter κ for this homogeneous case

Summary

Introduction

The general understanding of the relation between chaos in classical systems, and ergodicity and thermalization is still far from complete nowadays. One expects from high-dimensional, non-integrable complex systems to demonstrate strong chaos and it seems reasonable to expect thermalization. This is essentially the fundamental assumption of classical thermodynamics [1]. The conditions under which this assumption can be safely made, is still an open question. It is not known what level of “chaoticity” or “complexity” is required to ensure thermalizing behavior. Starting with only a few initially excited modes, one can view thermalization as spreading in the mode-space, i.e. the excitation of new modes, due to the nonlinear chaotic interactions

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Conclusion