Heterogeneous Materials - Diffusion, Laplace spectrum and NMR

Abstract: Relations to the effective diffusion constant and the relaxation rate of the time-dependent diffusion coefficient for porous systems are derived from the Laplace operator spectrum. The Padé approximation is then explained in terms of the Laplace operator spectrum. The calculations are made in a finite difference scheme with Neumann conditions defining the boundaries and validated by comparison with Brownian motion simulations. The relation between the surface-to-pore volume and the Laplace spectrum is also discussed. Furthermore, a new perturbation method to solve the diffusion equation is presented. The method is formulated on the boundaries and the computational complexity is estimated to be O(s^2) for s number of boundary points. The method is applied on diffusion NMR.