Continuous-time random walk description of anomalous spin transport in dilute dipolar networks

Abstract

Nuclear spin diffusion is often summarized by a single diffusion coefficient, but this coarse-grained description can fail in dilute solids where positional disorder and long-range dipolar couplings generate a broad distribution of hopping rates. We develop a continuous-time random-walk (CTRW) description of 13C polarization transport in natural-abundance diamond (1.1%), constructing the rate matrix from dipolar-mediated flip-flop couplings and sampling exact continuous-time trajectories. Although site-to-site hopping is Markovian, the disorder-averaged dynamics give rise to emergent, anomalous transport. The empirical waiting-time distribution exhibits a heavy tail with exponent α = 0.64 and exponential cutoff tcutoff = 19 s; the mean jump length becomes correlated with the waiting time τ for τ ≳ 0.1 s; and the mean-squared displacement grows sublinearly in both step number and physical time, with exponents γ = 0.56 and δ = 0.87, respectively. We trace the microscopic origin of these signatures to geometric trapping: polarization can rapidly exchange within strongly coupled clusters, including dimers, while weak inter-cluster links control long-range exploration. A kinetic percolation construction links global transport to inter-cluster crossing times and identifies a corresponding crossing time of ∼20 s, consistent with tcutoff. Finally, mapping paramagnetic impurities onto hard-sphere traps connects the CTRW framework to classic studies of trapping in reaction-diffusion theory and reproduces the qualitative timescale of experimentally measured relaxation, whereas a continuum diffusion equation description does not. These results show that dilute dipolar spin networks require a microscopic, network-resolved transport description beyond the Fickian diffusion equation.