Lyapunov Exponents and Localization in Randomly Layered Media

Abstract

A variety of problems involving disordered systems can be formulated mathematically in terms of products of random transfer matrices, including Ising spin systems, optical and continuum mechanical wave propagation, and lattice dynamical systems. The growth or decay of solutions to these problems is governed by the Lyapunov spectrum of the product of these matrices. For continuum mechanical or optical wave propagation, the transfer matrices arise from the application of boundary conditions at the discontinuities of the medium. Similar matrices arise in lattice-based systems when the equations of motion are solved recursively. For the disordered lattice mechanical system, on which we focus in this paper, the scattering effects of the heterogeneities on a propagating pulse can be characterized by the frequency-dependent localization length—effectively the “skin depth” for multiple-scattering attenuation. Thus there is a close connection in these transfer matrix-based systems between localization and the Lyapunov spectrum. For the one-dimensional lattice, the matrices are 2 × 2 and, assuming certain models of disorder, both Lyapunov exponents are nonzero and sum to zero. Thus all propagating solutions are either exponentially growing or decaying. For higher dimensions the situation is more complicated since there is then a spectrum of exponents, making the calculations more difficult, and it is less clear just how to relate the Lyapunov exponents to a single localization length. Further, unlike for the Schrödinger equation, the transfer matrices associated with the lattice mechanical system are not symplectic. We describe a robust numerical procedure for estimating the Lyapunov spectrum of products of random matrices and show application of the method to the propagation of waves on a lattice. In addition, we show how to estimate the uncertainties of these exponents.