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.
Highlights
Nuclear magnetic resonance is often used to study random motion of spins in different systems
The phenomenological model of fractional Brownian motion is one of the most popular and best justified models in the soft condensed matter physics. It generalizes other models of the stochastic dynamics described by the generalized Langevin equation
Various Nuclear Magnetic Resonance (NMR) methods serve as probes for accurate characterization of the random motion of particles in different systems, including the biological ones due to the non-invasive character of these measurements
Summary
We obtain the solution of the generalized Langevin equation (GLE) for this model in an exceedingly simple way. The results for the mean square displacement (MSD) of the particles are applied to calculate the attenuation function S (t) for an ensemble of spins in a magnetic-field gradient. Our work was mainly aimed to contribute to adequate interpretations of the NMR experiments on the BM in complex fluids, in the limit of free particles coupled to a fractal heat bath our results correct the description and give favorable comparison with experiments acquired in human neuronal tissues [12]
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