Classical transport in quasiperiodic media

Abstract. Classical transport coefficients such as the effective conductivity or diffusivity of a quasiperiodic medium were observed [1] to depend discontinuously on the frequencies of the quasiperiodicity. For example, for a one-dimensional medium with a potential V(x) = cosx + coskx , the effective diffusion coefficient D*(k) has the same value D for all irrational k , but differs from D and depends on k for k rational, where it is thus discontinuous. Here we review some recent progress [2-4] in understanding this discontinuous behavior. In particular, a class of examples which explicitly exhibit the discontinuity in dimensions d >_ 2 is constructed. In addition, we examine some rather surprising consequences of the discontinuity for the rate of approach to limiting behavior of diffusion or conduction in quasiperiodic media as time or volume becomes infinite. It is found that these rates can be "arbitrarily slow," which contrasts with the power laws observed for random media.