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.
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BibTeX
@misc{selco2026continuoustime,
title = {{Continuous-Time Random Walk Description of Anomalous Spin Transport in Dilute Dipolar Networks}},
author = {Cooper M. Selco and Christian Bengs and Ashok Ajoy},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.22626},
eprint = {2607.22626},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.22626},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Continuous-Time Random Walk Description of Anomalous Spin Transport in Dilute Dipolar Networks
AU - Cooper M. Selco
AU - Christian Bengs
AU - Ashok Ajoy
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2607.22626
UR - https://arxiv.org/abs/2607.22626
AB - 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 t_cutoff=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 t_cutoff. 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.
ER -