LYAPUNOV EXPONENTS AND LOCALIZATION IN RANDOMLY LAYERED MEDIA

A variety of problems involving disordered systems can be formulated m athematically in terms of products of random transfer matrices, includ ing Ising spin systems, optical and continuum mechanical wave propagat ion, and lattice dynamical systems. The growth or decay of solutions t o these problems is governed by the Lyapunov spectrum of the product o f these matrices. For continuum mechanical or optical wave propagation , the transfer matrices arise from the application of boundary conditi ons at the discontinuities of the medium. Similar matrices arise in la ttice-based systems when the equations of motion are solved recursivel y. For the disordered lattice mechanical system, on which we focus in this paper, the scattering effects of the heterogeneities on a propaga ting pulse can be characterized by the frequency-dependent localizatio n length-effectively the ''skin depth'' for multiple-scattering attenu ation. Thus there is a close connection in these transfer matrix-based systems between localization and the Lyapunov spectrum. For the one-d imensional lattice, the matrices are 2 x 2 and, assuming certain model s of disorder, both Lyapunov exponents are nonzero and sum to zero. Th us all propagating solutions are either exponentially growing or decay ing. Fbr higher dimensions the situation is more complicated since the re is then a spectrum of exponents, making the calculations more diffi cult, and it is less clear just how to relate the Lyapunov exponents t o a single localization length. Further, unlike for the Schrodinger eq uation, the transfer matrices associated with the lattice mechanical s ystem are not symplectic. We describe a robust numerical procedure for estimating the Lyapunov spectrum of products of random matrices and s how application of the method to the propagation of waves on a lattice . In addition, we show how to estimate the uncertainties of these expo nents. (C) 1997 Academic Press.