Self-diffusion in periodic porous media: A comparison of numerical simulation and eigenvalue methods

Random walk computer simulations are an important tool in understanding magnetic resonance measurements in porous media. In this paper we focus on the description of pulsed field gradient spin echo (PGSE) experiments that measure the probability, P(R, t), that a diffusing water molecule will travel a distance R in a time t. Because PGSE simulations are often limited by statistical considerations, we will see that valuable insight can be gained by working with simple periodic geometries and comparing simulation data to the results of exact eigenvalue expansions. In this connection, our attention will be focused on (1) the wavevector, k, and time dependent magnetization, M,(k, t); and (2) the normalized probability, Ps(ΔR, t), that a diffusing particle will return to within ΔR of the origin after time t.