  This paper proposes an effective diffusion equation method to analyze nuclear
magnetic resonance (NMR) relaxation. NMR relaxation is a spin system recovery
process, where the evolution of the spin system is affected by the random field
due to Hamiltonians, such as dipolar couplings. The evolution of magnetization
can be treated as a random walk in phase space described either by a normal or
fractional phase diffusion equation. Based on these phase diffusion equations,
the NMR relaxation rates and equations can be obtained, exemplified in the
analysis of relaxations affected by an arbitrary random field, and by dipolar
coupling for both like and unlike spins. The obtained theoretical results are
consistent with the reported results in the literature. Additionally, the
anomalous relaxation expression obtained from the Mittag-Leffler function based
time correlation function can successfully fit the previously reported 13C T1
NMR experimental data of polyisobutylene (PIB) in the blend of PIB and
head-to-head poly(propylene) (hhPP). Furthermore, the proposed phase diffusion
approach provides an intuitive way to interpret NMR relaxation, particularly
for the fractional NMR relaxation, which is still a challenge to explain by the
available theoretical methods. The paper provides additional insights into NMR
and magnetic resonance imaging (MRI) relaxation experiments.
