Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium
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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.
KeywordsContinuous time random walk Fat-tailed time distributions Hydrodynamic limit Subdiffusion
Mathematics Subject Classification (2010)60J27 26A33 60J75 60J70
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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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