Laplacian eigenfunctions in NMR. II. Theoretical advances

In this article, theoretical advances in the study of restricted diffusion in NMR that have been achieved by using Laplacian eigenfunctions are described. The macroscopic signal is represented as the characteristic function of a random phase shift that a nucleus acquires during its motion in an inhomogeneous magnetic field. The moments of this random variable are written in a matrix form that is based on Laplacian eigenfunctions. This article focuses on the zeroth, first, and second moments which provide the major contribution to the macroscopic signal. The asymptotic behavior of the macroscopic signal in both short-time and long-time regimes is considered when the diffusion length is either much smaller or much larger than the size of a diffusion-confining domain, respectively. Apparent diffusion coefficient, localization regime, inverse spectral problem, and many other issues are discussed.