Diffusion in one-dimensional inhomogeneous media

Abstract

Formal solution of the diffusion equation (\ensuremath{\partial}/\ensuremath{\partial}t)V=(\ensuremath{\partial}/\ensuremath{\partial}x)[D(x)(\ensuremath{\partial}/\ensuremath{\partial}x)V] is obtained as a functional of D(x). The solution is expressed by means of certain operators, which are somewhat similar to the time-ordering operator of quantum mechanics. This formalism provides us with a good perspective of the problem and serves as a basis for approximation methods. In particular, the equation is exactly solvable when D(x) is a discrete (stepwise) function. Coarse graining of the diffusion coefficient is also discussed. It is shown that the reciprocal of the coarse-grained diffusion coefficient is equal to the arithmetical average of 1/D(x). Finally, a stochastic equation obtained by adding a decay term and noise terms to the diffusion equation is considered. Spatial correlation function is calculated exactly for the case where D(x) is a step function.