Localization and the invariant probability measure for a structural dynamic system

Abstract

In the one-dimensional classical analogs to Anderson localization, whether optical, acoustical or structural dynamic, the periodic system has its periodicity disrupted by having one or more of its parameters randomly disordered. Such randomized systems can be modeled via an infinite product of random transfer matrices. In the case where the transfer matrices are 2x2, the upper (and positive) Lyapunov exponent of the random matrix product is identified as the localization factor (inverse localization length) for the disordered one-dimensional model. It is this localization factor which governs the confinement of energy transmission along the disordered system, and for which the localization phenomenon has been of interest. The theorem of Furstenberg for infinite products of random matrices allows us to calculate this upper Lyapunov exponent. In Furstenberg's master formula we integrate with respect to the probability measure of the random matrices, but also with respect to the invariant probability measure of the direction of the vector propagated by the long chain of random matrices. This invariant measure is difficult to find analytically, and, as a result, either an approximating assumption is frequently made, or, less frequently, the invariant measure is determined numerically. Here we calculate the invariant measure numerically using a Monte Carlo bin counting technique and then numerically integrate Furstenberg's formula to arrive at the localization factor for both continuous and discrete disorder of the mass. This result is cross checked with the (modified) Wolf algorithm.