Time moment analysis of first passage time, time lag and residence time problems via Taylor expansion of transmission matrix

Abstract

Taylor expansion (with respect to the Laplace variable, s) of the transmission matrix, T(s), has been developed for the diffusion transport with position-dependent diffusivity, D(x) and partition coefficient, K(x). First, we find the relation between the expansion coefficients of the matrix elements and the moments of the first passage times by connecting them to J(s), the Laplace transform of the escaping flux, J(t). The moments can be formulated by repeated integrals of K(x) and [D(x)K(x)](-1) from solving the backward diffusion equation subject to appropriate initial and boundary conditions. In this way, Taylor expansion coefficients of T-11(s), T-21(s), and T-22(s) are expressed in terms of the repeated integrals. Further application of the identity det[T(s)] = 1 leads to the Taylor expansion T-12(s). With the knowledge of the Taylor expansion of T(s), the formulation of the time moments for diffusion problems with position dependent D(x) and K(x) subject to various initial and boundary conditions is then just a simple, algebraic manipulation. Application of this new method is given to the membrane permeation transport and mean residence time problem. (C) 2000 American Institute of Physics. [S0021-9606(00)50810-1].