Differential Equations Seminar

Abstract: We study the homogenization limit of a biased random walk on a periodic graph embedded in the multidimensional Euclidean space. Our model is inspired by the problem of solute diffusion in an aqueous polymer environment. Prediction of effective solute diffusivity in aqueous polymer environments enables the design of polymer-based drug and protein delivery devices as well as cell scaffolds for tissue engineering, among other applications.

Homogenization theory provides a large scale homogeneous approximation of a medium that is heterogeneous in the fine-scale. We are interested in computing the diffusivity of the approximating homogeneous medium, which we shall call the effective diffusivity.

Our rationale for considering a random walk on a graph is to account for the fact that the mean path length of a diffusing solute may not be much smaller than the finescale length of the structures of interest. When the finescale length of interest is only a few nanometers, as in the case of polymer gel networks, the case for our starting point is quite strong.

First, the effective diffusivity is formally derived via an intuitive approach that assumes the probability mass function of the random walk admits an asymptotic expansion. This expansion yields a set unit-cell problems that must be solved in order to compute the effective diffusivity. Next, the effective diffusivity is rigorously derived using the random time change representation, functional law of large numbers, and the martingale central limit theorem. The main results are 1) agreement between the formal and rigorous derivations and 2) a mathematical formula for the effective diffusivity in terms of the embedded graph and the solutions to the so-called unit-cell problems.