First passage time in a two-layer system

As a first step in the first passage problem for passive tracer in stratified porous media, we consider the case of a two-dimensional system consisting of two layers with different convection velocities. Using a lattice generating function formalism and a variety of analytic and numerical techniques, we calculate the asymptotic behavior of the first passage time probability distribution. We show analytically that the asymptotic distribution is a simple exponential in time for any choice of the velocities. The decay constant is given in terms of the largest eigenvalue of an operator related to a half-space Green's function. For the anti-symmetric case of opposite velocities in the layers, we show that the decay constant for system length L crosses over from L-2 behavior in the diffusive limit to L-1 behavior in the convective regime, where the crossover length L* is given in terms of the velocities. We also have formulated a general self-consistency relation, from which we have developed a recursive approach which is useful for studying the short-time behavior