Numerical Analysis of the Harmonic and Spread Harmonic Measures

Diffusive transport across irregular interfaces is ubiquitous in physics, biology, chemistry and material sciences [1,2]. Diffusing particles first arrive onto the boundary and then may explore some neighboring area before reacting or being transferred across this interface. The overall functioning of the interface is then governed by a “competition ” between its accessi-bility (how easy to reach the boundary) and permeability (how easy to cross the boundary). This competition is controlled by two transport parameters: diffusion coefficient D and surface permeability or reactivity W. Their ratio D/W, which is homogeneous to a length, parameter-izes a transition from a purely reactive boundary (W = ∞) and a purely reflecting boundary (W = 0). In mathematical terms, the arrival points for diffusion are known to be characterized by the harmonic measure. By analogy, one can introduce a family of the spread harmonic mea-sures (parameterized by the ratio D/W) in order to characterize the distribution of transfer or reaction points [3-5]. These measures can either be generated by partially reflected Brownian motion, or obtained as solutions of the related boundary value problems for Laplace operator [4-5]