NMR signals within the generalized Langevin model for fractional Brownian motion

TL;DR

This paper develops a generalized model for NMR signal attenuation in systems exhibiting fractional Brownian motion, capturing both long- and short-time dynamics, and compares theoretical results with experimental data from neuronal tissues.

Contribution

It introduces a fractional Brownian motion framework for NMR signals that accounts for memory effects and short-time dynamics, extending beyond standard Langevin models.

Findings

The model accurately describes NMR signals in biological tissues.

Solutions for trapped particles are derived simply.

Results align well with experimental data.

Abstract

The methods of Nuclear Magnetic Resonance belong to the best developed and often used tools for studying random motion of particles in different systems, including soft biological tissues. 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 memoryless Langevin description of the Brownian motion (BM). In the present work, the attenuation function S(t) for an ensemble of spin-bearing particles 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 an exceedingly 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. The effect of the trap is demonstrated by introducing a simple model for the generalized diffusion coefficient of the particle.