Application of fractional calculus to the description of anomalous diffusion and relaxation processes in complex media provided one of the most impressive impulses to the development of statistical physics during the last decade. In particular the so-called fractional diffusion equation enabled one to capture the main features of anomalous diffusion. However the price for this achievement is rather high - the fractional diffusion coefficient becomes an involved function of a characteristic of the media (e.g., that of the radius of pores in the case of the porous one). Revealing this dependence from the first principles is one of the main problems in this field of science. Another one still remains that of extracting this dependence from the experiment. The latter problem is tackled in the present paper. Our aim is to provide detailed and pedagogical deriving the relationship of the fractional diffusion coefficient with experimentally observable value from nuclear magnetic resonance (NMR) spin-lattice relaxation data. The result obtained promotes the NMR relaxation method to become a powerful tool in solving the problem of experimental measuring the fractional diffusion coefficient. Also the merits and limitations of NMR relaxation method and pulsed-field gradient (PFG) NMR for the research of anomalous diffusion are compared and discussed.
The Larmor frequency shift is found in porous media consisting of NMR-reporting fluid filling a connected pore within an NMR-invisible matrix for the case of fast diffusion in the fluid. The matrix material has a distinct location-independent anisotropic magnetic susceptibility tensor that induces a heterogeneous microscopic magnetic field when exposed to the strong main field of an NMR device. Aside from the connectivity of the pore, the matrix geometry is arbitrary.
Wave localization is ubiquitous in disordered media -- from amorphous materials, where soft-mode localization is closely related to materials failure, to semi-conductors, where Anderson localization leads to metal-insulator transition. Our main understanding, though, is based on discrete models. Here, we provide a continuum perspective on the wave localization in two-phase disordered elastic media by studying the scalar wave equation with heterogeneous modulus and/or density. At low frequencies, soft modes arise as a result of disordered elastic modulus, which can also be predicted by the loca…
Non-Gaussian diffusion is commonly considered as a result of fluctuating diffusivity, which is correlated in time or in space or both. In this work, we investigate the non-Gaussian diffusion in static disordered media via a quenched trap model, where the diffusivity is spatially correlated. Several unique effects due to quenched disorder are reported. We analytically estimate the diffusion coefficient $D_{\text{dis}}$ and its fluctuation over samples of finite size. We show a mechanism of population splitting in the non-Gaussian diffusion. It results in a sharp peak in the distribution of disp…
