Heterogeneous Materials - Diffusion, Laplace Spectrum, and Nuclear Magnetic Resonance

Abstract: In this thesis diffusion in heterogeneous materials is studied using spectral analysis of the Laplace operator. 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’s 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. Furthermore, a new method to approximately solve the diffusion equation is presented. The method uses a mixture of free diffusion eigenfunctions and surface functions describing the influence of the boundaries. The method is completely formulated on the boundaries and the number of operations scale as O(s^2) for s number of boundary points. The result is an approximation to the first low frequency eigenfunctions and eigenvalues to the Laplace operator in bounded domains. The method is applied on diffusion NMR in the SGP-limit and also extended beyond the SGP-limit to cover general gradient forms and pulse sequences. Finally, an iterative scheme to find exact eigenvalues and eigenfunctions to the Laplace operator in bounded domains by surface integrals is presented with a theoretical O(s log(s)) scaling.