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Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium

  • Catherine Choquet
  • Marie-Christine Néel
Article
  • 5 Downloads

Abstract

We derive rigorously the fractional counterpart of the Feynman–Kac equation for a transport problem with trapping events characterized by fat-tailed time distributions. Our starting point is a random walk model with an inherent time subordination process due to the difference between clock times and operational times. We pass to the hydrodynamic limit using weak convergence arguments. Due to the lack of regularity of physical data, we use the framework of measure theory. We finally derive a Bloch–Torrey type equation for nuclear magnetic resonance data in this subdiffusive context.

Keywords

Continuous time random walk Fat-tailed time distributions Hydrodynamic limit Subdiffusion 

Mathematics Subject Classification (2010)

60J27 26A33 60J75 60J70 

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Notes

Acknowledgments

The authors have been supported by GNR MoMaS, CNRS-2439 (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN), project Non-Fickian effects and fractional models for diffusion in porous media. C.C. was also supported by CNRS (NEEDS project DYMHOM2). M.C.N. was also supported by Agence Nationale de la Recherche (ANR project ANR-09-SYSC-015).

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de La RochelleLa Rochelle CedexFrance
  2. 2.Université d’Avignon et des Pays de VaucluseAvignon CedexFrance

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