Fractional Langevin Equation Model for Characterization of Anomalous Brownian Motion from NMR Signals

¹ Laboratory of Radiation Biology, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
² Department of Physics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Park Komenského 2, 042 00 Košice, Slovakia

^* e-mail: lisy@jinr.ru
^** e-mail: jana.tothova@tuke.sk

Published online: 14 February 2018

Abstract

Nuclear magnetic resonance is often used to study random motion of spins in different systems. In the long-time limit the current mathematical description of the experiments allows proper interpretation of measurements of normal and anomalous diffusion. The shorter-time dynamics is however correctly considered only in a few works that do not go beyond the standard Langevin theory of the Brownian motion (BM). In the present work, the attenuation function S (t) for an ensemble of spins in a magnetic-field gradient, expressed in a form applicable for any kind of stationary stochastic dynamics of spins with or without a memory, is calculated in the frame of the model of fractional BM. The solution of the model for particles trapped in a harmonic potential is obtained in a simple way and used for the calculation of S (t). In the limit of free particles coupled to a fractal heat bath, the results compare favorably with experiments acquired in human neuronal tissues.