Local diffusion analysis using square displacement averaged in subspace

There is a well-known linear relation between the mean square displacement (MSD) and time, with the slope corresponding to the global diffusion coefficient of the particle of interest. Similarly, the local MSD computed as a statistical average over a subspace of the system exhibits a nonlinear relationship with time, reflecting the local diffusion coefficient within the subspace. Thus, by analyzing this relationship, we can extract the local diffusion coefficient. To enable this analysis, in this work, we derived an analytical expression for the nonlinear behavior of the local MSD using the Edgeworth expansion up to fourth order. Combining this expression with regression analysis, we estimated the local diffusion coefficient of a hydrogen molecule in homogeneous bulk water as a proof of concept. As a result, at a spatial resolution as high as 0.23 nm, the estimated local diffusion coefficients deviate by less than 20% from the reference value obtained from the standard linear relation between the regular MSD and time. We also demonstrated the utility of our approach in heterogeneous systems by analyzing the diffusion of a methane molecule in aqueous solution near a hydrophobic interface. The estimated local diffusion coefficients of the methane molecule are in good agreement with previous studies, except in a specific region ∼0.6 nm from the interface, where the influence of higher-order terms of the Edgeworth expansion omitted in this work becomes significant. This limitation is shared by previous approaches, and resolving this issue remains a challenge for future work in local diffusion analysis.