NMR measurements and all-time Brownian movement with memory

In the present work, by using the method of accumulation of phase shifts in the rotating frame, the attenuation function S(t) of the NMR signal from an ensemble of spin-bearing particles in a magnetic-field gradient is expressed through the particle mean square displacement in a form applicable for any kind of stationary stochastic dynamics of spins and for any times. S(t) is evaluated providing that the random motion of particles can be modeled by the generalized Langevin equation (GLE) with a colored random force driving the particles. The memory integral in this equation is the convolution of the particle velocity or its acceleration with a memory kernel related to the random force by the fluctuation-dissipation theorem. We consider three popular models of the BM with memory: the model of viscoelastic (Maxwell) fluids with the memory exponentially decaying in time, the fractional BM model, and the model of the hydrodynamic BM. In all the cases the solutions of the GLEs are obtained in an exceedingly simple way. The corresponding attenuation functions are then found for the free-induction NMR signal and the pulsed and steady-gradient spin-echo experiments. The results for the free-particle fractional BM compare favorably with experiments acquired in human neuronal tissues and with the observed subdiffusion dynamics in proteins.