Journal of Statistical Physics

, Volume 79, Issue 5–6, pp 895–922 | Cite as

First passage time in a two-layer system

  • Jysoo Lee
  • Joel Koplik


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 lengthL crosses over fromL−2 behavior in the diffusive limit toL−1 behavior in the convective regime, where the crossover lengthL* 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.

Key Words

First passage problem convection-diffusion equation layered system asymptotic behavior 


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Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Jysoo Lee
    • 1
  • Joel Koplik
    • 1
  1. 1.Benjamin Levich Institute and Department of PhysicsCity College of the City University of New YorkNew York

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