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Materials under confinement can possess properties that deviate considerablyfrom their bulk counterparts. Indeed, confinement makes all physical propertiesposition-dependent and possibly anisotropic, and characterizing such spatialvariations and directionality is an intense area of focus in experimental andcomputational studies of confined matter. While this task is fairlystraightforward for simple mechanical observables, it is far more daunting fortransport properties such as diffusivity that can only be estimated fromautocorrelations of mechanical observables. For instance, there are wellestablished methods for estimating diffusivity from experimentally observed orcomputationally generated trajectories in the bulk. No rigorous generalizationsof such methods, however, exist for confined systems. In this work, we presenttwo filtered covariance estimators for computing anisotropic andposition-dependent diffusivity tensors and validate them by applying them tostochastic trajectories generated according to known diffusivity profiles.These estimators can accurately capture spatial variations spanning overseveral orders of magnitude and assuming different functional forms. Ourapproach is also very robust to implementation details such as the localizationfunction and time discretization and performs significantly better thanestimators that are solely based on local covariance. Moreover, the kernelfunction does not have to be localized and can instead belong to a dictionaryof orthogonal functions. Therefore, the proposed estimator can be readily usedto obtain functional estimates of diffusivity rather than a tabulatedcollection of pointwise estimates. Nonetheless, the susceptibility of theproposed estimators to time discretization is higher close to hard boundaries.We demonstrate this heightened susceptibility to be common among allcovariance-based estimators.

Robust estimation of position-dependent anisotropic diffusivity tensors from stochastic trajectories