Abstract
Fractional Brownian motion (FBM), a non-Markovian self-similar Gaussian stochastic process with long-ranged correlations, represents a widely applied, paradigmatic mathematical model of anomalous diffusion. We report the results of large-scale computer simulations of FBM in one, two, and three dimensions in the presence of reflecting boundaries that confine the motion to finite regions in space. Generalizing earlier results for finite and semi-infinite one-dimensional intervals, we observe that the interplay between the long-time correlations of FBM and the reflecting boundaries leads to striking deviations of the stationary probability density from the uniform density found for normal diffusion. Particles accumulate at the boundaries for superdiffusive FBM while their density is depleted at the boundaries for subdiffusion. Specifically, the probability density P develops a power-law singularity, P ∼r κ, as a function of the distance r from the wall. We determine the exponent κ as a function of the dimensionality, the confining geometry, and the anomalous diffusion exponent α of the FBM. We also discuss implications of our results, including an application to modeling serotonergic fiber density patterns in vertebrate brains.
Recommended Citation
T. Vojta et al., "Reflected Fractional Brownian Motion in One and Higher Dimensions," Physical Review E, vol. 102, no. 3, American Physical Society (APS), Sep 2020.
The definitive version is available at https://doi.org/10.1103/PhysRevE.102.032108
Department(s)
Physics
Research Center/Lab(s)
Center for High Performance Computing Research
International Standard Serial Number (ISSN)
2470-0045; 2470-0053
Document Type
Article - Journal
Document Version
Final Version
File Type
text
Language(s)
English
Rights
© 2020 American Physical Society (APS), All rights reserved.
Publication Date
08 Sep 2020
PubMed ID
33075869
Comments
This work was supported in part by a Cottrell SEED award from Research Corporation and by the National Science Foundation under Grants No. DMR-1828489 and No. OAC-1919789 (T.V.).