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Fractional Modeling in Action: a Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

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Abstract

Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary application areas — subsurface transport, turbulence, and anomalous materials — in which fractional-order differential equations provide accurate and predictive models. For each area, we report on the evidence of anomalous behavior that justifies the use of fractional-order models, and survey both foundational models as well as more expressive state-of-the-art models. We also propose avenues for future research, including more advanced and physically sound models, as well as tools for calibration and discovery of fractional-order models.

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Data Availability

All data generated or analyzed during this study are included in this published article.

Notes

  1. While the word “anomalous” means “not normal”, Klafter and Sokolov [2] and Sancho et al. [268] point out the widespread nature of anomalous diffusion with the statement “anomalous is normal!”.

  2. Another frequently used discrete random walk that leads to Brownian motion simply involves steps of fixed length to the left or right with probability 1/2 each; see Lawler [269]. In the long-time limit, all such discrete walks that draw increments from a finite-variance distribution lead to Brownian motion, due to the central limit theorem [44]

  3. Several parametrizations of the \(\alpha\)-stable characteristic function exist. The parametrization Eq. (7) is due to Samorodnitsky and Taqqu [270]. See Nolan [45, 271] for discussions of alternate forms.

  4. Special cases are \(\alpha = 2\) corresponding to the normal distribution, \(\alpha = 1\) and \(\gamma = 0\) corresponding to the Cauchy distribution, and \(\alpha = 1/2\) and \(\gamma = 1\) corresponding to the Lévy distribution.

  5. The definition of \(\tau\) in terms of D is an example of a right-continuous inverse of an increasing function. Paths of D, thought of as functions of t, are nondecreasing, so that each path of \(\tau\) constructed in this way is a continuous-from-the-right inverse of the parent path of D used to construct it.

  6. This equation was also derived by Scher and Lax [272, 273] and is referred to as the CTRW equation of Scher and Lax by Klafter and Silbey [126]. Other authors, such as Torrejon and Emelianenko [59] refer to this as the master equation of a CTRW.

  7. A Lévy walk may be compared to a non-standard CTRW in which waiting times prior to jumps are correlated to the jump length, e.g., proportional to the jump length, so that long excursions are penalized by long waiting times. See Zaburdaev et al. [44]

  8. For more recent works and novel definitions of variable-order fractional Laplacians we refer the reader to [63, 65].

  9. The units of D are given by \(\mathrm{L}^\alpha /\mathrm{T}\) where \(\alpha\) is the fractional order, L indicates space and T indicates time.

  10. We refer to [148] for more details regarding the FD-ADE and FFD-ADE models.

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Acknowledgements

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration contract number DE-NA0003525. This paper, SAND2021-11291 R, describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

Funding

Marta D’Elia and Mamikon Gulian were partially supported by the U.S. Department of Energy, Office of Advanced Scientific Computing Research under the Collaboratory on Mathematics and Physics-Informed Learning Machines for Multiscale and Multiphysics Problems (PhILMs) project. Marta D’Elia was also supported by the Sandia National Laboratories Laboratory-directed Research and Development (LDRD) program, project 218318. Mamikon Gulian was also supported by John von Neumann fellowship at Sandia National Laboratories. Mohsen Zayernouri and Jorge L. Suzuki were supported by the ARO Young Investigator Program (YIP) award (W911NF-19-1-0444), the National Science Foundation award (DMS-1923201), and the MURI/ARO grant (W911NF-15-1-0562).

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  1. Department of Mechanical Engineering & Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA

    Jorge L. Suzuki

  2. Quantitative Modeling and Analysis, Sandia National Laboratories, Livermore, CA, USA

    Mamikon Gulian & Marta D’Elia

  3. Department of Mechanical Engineering & Statistics and Probability, Michigan State University, East Lansing, MI, USA

    Mohsen Zayernouri

  4. Microvast Power Solutions, Lake Mary, FL, USA

    Jorge L. Suzuki

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Suzuki, J.L., Gulian, M., Zayernouri, M. et al. Fractional Modeling in Action: a Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials. J Peridyn Nonlocal Model 5, 392–459 (2023). https://doi.org/10.1007/s42102-022-00085-2

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