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

  • Catherine ChoquetEmail author
  • Marie-Christine Néel


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.


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

Mathematics Subject Classification (2010)

60J27 26A33 60J75 60J70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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).


  1. Baule A, Friedrich R (2007) Two-point correlation function of the fractional Ornstein-Uhlenbeck process. EuroPhys Lett 79:6004MathSciNetCrossRefGoogle Scholar
  2. Benson DA, Meerschaert MM (2009) A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Adv Water Resour 32:532–539CrossRefGoogle Scholar
  3. Bloch F (1946) Nuclear induction. Phys Rev 70:460–474CrossRefGoogle Scholar
  4. Boccardo L, Gallouët T (1989) Nonlinear elliptic and parabolic equations involving measure data. J Funct Anal 87:149–169MathSciNetCrossRefGoogle Scholar
  5. Callaghan PT (1991) Principles of nuclear magnetic resonance microscopy. Clarendon Press, OxfordGoogle Scholar
  6. Carmi S, Turgeman L, Barkai E (2010) On distributions of functionals of anomalous diffusion paths. J Stat Phys 141:1071–1092MathSciNetCrossRefGoogle Scholar
  7. Choquet C, Néel M-C (2014) From particles scale to anomalous or classical convection–diffusion models with path integrals. DCDS Ser S 7:207–238MathSciNetCrossRefGoogle Scholar
  8. Coats KH, Smith BD (1964) Dead-end pore volume and dispersion in porous media. Soc Rec Eng J 4:73Google Scholar
  9. Cooke JM, Kalmykov YP, Coffey WT, Kerskens CM (2009) Langevin equation approach to diffusion magnetic resonance imaging. Phys Rev E 80:061102CrossRefGoogle Scholar
  10. Fremlin DH (2003) Measure theory, vol. 4, Chapter 2, 347Yo. Torres Fremlin, ColchesterGoogle Scholar
  11. Friedrich R, Jenko F, Baule A, Eule S (2006) Anomalous diffusion of inertial, weakly damped particles. Phys Rev Lett 96:230601CrossRefGoogle Scholar
  12. Grebenkov D (2007) NMR Survey of reflected Brownian motion. Rev Mod Phys 79(3):1077–1133CrossRefGoogle Scholar
  13. Haggerty R, Mac Kenna SA, Meigs L (2000) On the late-time behavior of tracer test breakthrough curves, vol 36CrossRefGoogle Scholar
  14. Hahn EL (1950) Spin echoes. Phys Rev 80:580–594CrossRefGoogle Scholar
  15. Kac M (1949) On distributions of certain Wiener functionals. Trans Am Math Soc 65:1–13MathSciNetCrossRefGoogle Scholar
  16. Kaj I, Leskela L, Norros I, Schmidt V (2007) Scaling limits for random fields with long-range dependence. Ann Prob 35(2):528–550MathSciNetCrossRefGoogle Scholar
  17. Kern P, Meerschaert MM, Scheffler HP (2004) Limits theorems for coupled continuous time random walks. Ann Prob 32:730–756MathSciNetzbMATHGoogle Scholar
  18. Kimmic R (1997) NMR Tomography, Diffusometry, relaxometry. Springer, HeidelbergGoogle Scholar
  19. Londen S-O, Petzeltová H, Prüss J (2002) Global well-posedness and stability of a partial integro-differential equation with applications to viscoelasticity. J Evol Equ 3:169–201MathSciNetCrossRefGoogle Scholar
  20. Magin RL, Abdullah 0, Baleanu D, Zhou XJ (2008) Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation. J Magn Resonance 190:255–270CrossRefGoogle Scholar
  21. Majumdar SN (2005) Brownian functionals in physics and computer science. Curr Sci 89:2076MathSciNetGoogle Scholar
  22. Maryshev B, Joelson M, Lyubimov D, Lyubimova T, Néel MC (2009) Non Fickian flux for advection-dispersion with immobile periods. J Phys A: Math Theor 42:115001MathSciNetCrossRefGoogle Scholar
  23. Meerschaert MM, Straka P (2014) Semi–Markov approach to continuous time random walk limit processes. Ann Prob 42:1699–1723MathSciNetCrossRefGoogle Scholar
  24. Néel MC, Zoia A, Joelson M (2009) Mass transport subject to time-dependent flow with non-uniform sorption in porous media. Phys Rev E 80:056301CrossRefGoogle Scholar
  25. Schumer R, Benson DA, Meerschaert MM, Bauemer B (2003) Fractal mobile/immobile solute transport. Water Resour Res 39(10):1296CrossRefGoogle Scholar
  26. Slichter CP (1990) Arrested shear dispersion and other models of anomalous diffusion. In: Principles of Magnetic Resonance Springer Seried in Solid-State Sciences 1 SpringerGoogle Scholar
  27. Torrey HC (1956) Bloch equations with diffusion terms. Phys Rev 104:563–565CrossRefGoogle Scholar
  28. Turgeman L, Carmi S, Barkai E (2009) Fractional Feyman-Kac equations for Non-Brownian functionals. Phys Rev Lett 103:190201MathSciNetCrossRefGoogle Scholar
  29. Van Genuchten MT, Wierenga PJ (1976) Mass transfer studies in sorbing porous media, I Analytical solutions. Soil Sci Soc Am J 40(4):473CrossRefGoogle Scholar
  30. Young WR (1988) Arrested shear dispersion and other models of anomalous diffusion. J Fluid Mech 193:129–149CrossRefGoogle Scholar
  31. Zhang Y, Benson DA, Bauemer B (2008) Moment analysis for spatiotemporal fractional dispersion. Water Resour. 44:W05404Google Scholar
  32. Zhang H, Li GH, Luo MK (2013) Fractional Feyman-Kac equation with space-dependent exponent. J Stat Phys 152:1194–1206MathSciNetCrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations