  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.
