April  2014, 7(2): 207-238. doi: 10.3934/dcdss.2014.7.207

From particles scale to anomalous or classical convection-diffusion models with path integrals

1. 

Univ. La Rochelle, MIA CNRS EA 3165, F-17000 La Rochelle, France

2. 

Univ. d'Avignon et des Pays de Vaucluse, UMR 1114 EMMAH, F-84018 Avignon Cedex, France

Received  April 2013 Revised  July 2013 Published  September 2013

The present paper is devoted to the rigorous upscaling of some particles displacement model with trapping events, to the continuum scale. It focuses especially on the transitions between sub-diffusive and diffusive models. The work gives emphasis to the following points: 1. The distribution of waiting times in passing to the continuum limit. The common idea is that the distributions with slowly decaying long tails produce anomalous diffusion while the classical diffusion model corresponds to distributions with short tails. This is shown to be not always true by introducing a simple model of geometrical heterogeneity leading to trapping events without characteristic time scale. 2. The extension of the Feynman-Kac theory to some non-Brownian setting. 3. Constructing a microscopic random walk model that, thought based on a MIM approach, gives both fMIM and FFPE at the mesoscopic limit.
Citation: Catherine Choquet, Marie-Christine Néel. From particles scale to anomalous or classical convection-diffusion models with path integrals. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 207-238. doi: 10.3934/dcdss.2014.7.207
References:
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[2]

E. G. Altmann and T. Tel, Poincaré recurrences and transient chaos in systems with leaks,, Phys. Rev. E (3), 79 (2009). doi: 10.1103/PhysRevE.79.016204. Google Scholar

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E. Barkai, R. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation,, Phys. Rev. E, 61 (2000), 132. doi: 10.1103/PhysRevE.61.132. Google Scholar

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B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk,, Rev. Geophys., 44 (2006). doi: 10.1029/2005RG000178. Google Scholar

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F. Boano, A. I. Packman, A. Cortis, R. Revelli and L. Ridolfi, A continuous time random walk approach to the stream transport of solutes,, Water Resour. Res., 43 (2007). doi: 10.1029/2007WR006062. Google Scholar

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M. Bromly and C. Hinz, Non-Fickian transport in homogeneous unsaturated repacked sand,, Water Resour. Res., 40 (2004). doi: 10.1029/2003WR002579. Google Scholar

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R. Brown, A brief account of microscopical observations made in the months of June, July, and August 1827, on the particles contained in the pollen of plants, and on the general existence of active molecule in organic and inorganic bodies,, Philosophical Magazine, 4 (1828), 161. Google Scholar

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C. Choquet and M. C. Néel, Rigorous derivation of Feynman-Kac equations for arrested dispersion,, Technical report, (2013). Google Scholar

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I. Kaj, L. Leskela, I. Norros and V. Schmidt, Scaling limits for random fields with long-range dependence,, Ann. of Prob., 35 (2007), 528. doi: 10.1214/009117906000000700. Google Scholar

[26]

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S.-O. Londen, H. Petzeltová and J. Prüss, Global well-posedness and stability of a partial integro-differential equation with applications to viscoelasticity,, J. Evol. Equ., 3 (2002), 169. doi: 10.1007/978-3-0348-7924-8_9. Google Scholar

[28]

M. Magdziarz, Langevin picture of subdiffusion with infinitely divisible waiting times,, J. Stat. Phys., 135 (2009), 763. doi: 10.1007/s10955-009-9751-z. Google Scholar

[29]

M. Magdziarz, A. Weron and J. Klafter, Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: The case of time-dependent force,, Phys. Rev. E, 101 (2008). doi: 10.1103/PhysRevLett.101.210601. Google Scholar

[30]

A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena,, Phys. Rep., 314 (1999), 237. doi: 10.1016/S0370-1573(98)00083-0. Google Scholar

[31]

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[34]

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[35]

R. Metzler and J. Klafter, The restaurant at the end of the random walk,, J. Phys. A: Math. Gen., 37 (2004), 161. doi: 10.1088/0305-4470/37/31/R01. Google Scholar

[36]

M. S. Mommer and D. Lebiedz, Modeling subdiffusion using reaction diffusion systems,, SIAM J. Appl. Math., 70 (2009), 112. doi: 10.1137/070681648. Google Scholar

[37]

M. C. Neel, A. Zoia and M. Joelson, Mass transport subject to time-dependent flow with uniform sorption in porous media,, Phys. Rev. E, 80 (2009). Google Scholar

[38]

J. P. Nolan, "Stable Distributions - Models for Heavy Tailed Data,", In progress, (2012). Google Scholar

[39]

H. Owhadi, Anomalous slow diffusion from perpetual homogenization,, Ann. Prob., 31 (2003), 1935. doi: 10.1214/aop/1068646372. Google Scholar

[40]

G. Papanicolaou and H. Kesten, A limit theorem for stochastic acceleration,, Comm. Math. Phys., 78 (1980), 19. doi: 10.1007/BF01941968. Google Scholar

[41]

J. B. Perrin, Mouvement Brownien et réalité moléculaire,, Annales de chimie et de physique, 8 (1909), 5. Google Scholar

[42]

A. S. Pikovsky, Escape exponent for transient chaos scattering in non-hyperbolic Hamiltonian systems,, J. Phys. A, 25 (1992). doi: 10.1088/0305-4470/25/8/016. Google Scholar

[43]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, Commun. Math. Phys., 74 (1980), 189. doi: 10.1007/BF01197757. Google Scholar

[44]

S. Ross, "Introduction in Probability Models,", (Chapter 10) Fifth edition, (1993). Google Scholar

[45]

B. Rubin, "Fractional Integrals and Potentials,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 82 (1996). Google Scholar

[46]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion wave equations and applications to some inverse problems,, J. Math. Anal. Appl., 382 (2011), 426. doi: 10.1016/j.jmaa.2011.04.058. Google Scholar

[47]

S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives: Theory and Applications,", Gordon and Breach Science Publishers, (1993). Google Scholar

[48]

H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids,, Phys. Rev. B, 12 (1975), 2455. doi: 10.1103/PhysRevB.12.2455. Google Scholar

[49]

R. Schumer, D. A. Benson, M. M. Meerschaert and B. Bauemer, Fractal mobile/immobile solute transport,, Water Resour. Res., 39 (2003). doi: 10.1029/2003WR002141. Google Scholar

[50]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics,, Nature, 363 (1993), 31. doi: 10.1038/363031a0. Google Scholar

[51]

A. Shojiguchi, C. B. Li, T. Tomatsuzaki and M. Toda, Dynamical foundation and limitations of statistical reaction theory,, Comm. Nonlinear Sci. Numer. Simul., 13 (2008), 857. doi: 10.1016/j.cnsns.2006.08.002. Google Scholar

[52]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,, Ann. der Physik, 21 (1906), 756. Google Scholar

[53]

E. R. Weeks, J. S. Urbach and H. L. Swinney, Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example,, Phys. D, 97 (1996), 291. doi: 10.1016/0167-2789(96)00082-6. Google Scholar

[54]

K. Weron, A. Stanislavsky, A. Jurlewicz, M. M. Meerschaert and H. P. Scheffler, Clustered continuous time random walks: Diffusion and relaxation consequences,, Proc. Royal Soc. A, 468 (2012), 1615. doi: 10.1098/rspa.2011.0697. Google Scholar

[55]

W. R. Young, Arrested shear dispersion and other models of anomalous diffusion,, J. Fluid Mech., 193 (1988), 129. doi: 10.1017/S0022112088002083. Google Scholar

[56]

G. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos,, Phys. D, 76 (1994), 110. doi: 10.1016/0167-2789(94)90254-2. Google Scholar

[57]

Y. Zhang, D. A. Benson and B. Bauemer, Moment analysis for spatiotemporal fractional dispersion,, Water Resour., 44 (2008). doi: 10.1029/2007WR006291. Google Scholar

show all references

References:
[1]

N. Agman and S. Rabinovich, Diffusive dynamics on potential energy surfaces: Non equilibrium CO binding to heme proteins,, J. Chem. Phys., 92 (1992), 7270. Google Scholar

[2]

E. G. Altmann and T. Tel, Poincaré recurrences and transient chaos in systems with leaks,, Phys. Rev. E (3), 79 (2009). doi: 10.1103/PhysRevE.79.016204. Google Scholar

[3]

E. Barkai, R. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation,, Phys. Rev. E, 61 (2000), 132. doi: 10.1103/PhysRevE.61.132. Google Scholar

[4]

D. A. Benson and M. M. Meerschaert, A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations,, Adv. Water Resources, 32 (2009), 532. doi: 10.1016/j.advwatres.2009.01.002. Google Scholar

[5]

B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk,, Rev. Geophys., 44 (2006). doi: 10.1029/2005RG000178. Google Scholar

[6]

F. Boano, A. I. Packman, A. Cortis, R. Revelli and L. Ridolfi, A continuous time random walk approach to the stream transport of solutes,, Water Resour. Res., 43 (2007). doi: 10.1029/2007WR006062. Google Scholar

[7]

M. Bromly and C. Hinz, Non-Fickian transport in homogeneous unsaturated repacked sand,, Water Resour. Res., 40 (2004). doi: 10.1029/2003WR002579. Google Scholar

[8]

R. Brown, A brief account of microscopical observations made in the months of June, July, and August 1827, on the particles contained in the pollen of plants, and on the general existence of active molecule in organic and inorganic bodies,, Philosophical Magazine, 4 (1828), 161. Google Scholar

[9]

S. Carmi, L. Turgeman and E. Barkai, On distributions of functionals of anomalous diffusion paths,, J. Stat. Phys., 141 (2010), 1071. doi: 10.1007/s10955-010-0086-6. Google Scholar

[10]

C. Choquet and M. C. Néel, Rigorous derivation of Feynman-Kac equations for arrested dispersion,, Technical report, (2013). Google Scholar

[11]

P. Clément, S.-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces,, J. Diff. Equ., 196 (2004), 418. doi: 10.1016/j.jde.2003.07.014. Google Scholar

[12]

G. A. Coon and D. L. Bernstein, Some properties of the double Laplace transformation,, Trans. Amer. Math. Soc., 74 (1953), 135. doi: 10.1090/S0002-9947-1953-0052556-4. Google Scholar

[13]

A. Cortis, Y. Chen, H. Scher and B. Berkowitz, Quantitative characterization of pore-scale disorder effects on transport in homogeneous granular media,, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.041108. Google Scholar

[14]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations,, J. Diff. Equ., 199 (2004), 211. doi: 10.1016/j.jde.2003.12.002. Google Scholar

[15]

A. Einstein, Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen,, Ann. der Physik, 17 (1905), 549. doi: 10.1002/andp.19053220806. Google Scholar

[16]

L. Erdos, Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field,, J. Stat. Phys., 107 (2002), 1043. doi: 10.1023/A:1015157624384. Google Scholar

[17]

W. Feller, "An Introduction to Probability Theory and its Applications. Vol.II,", Wiley, (1970). Google Scholar

[18]

R. Friedrich, F. Jenko, A. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles,, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.230601. Google Scholar

[19]

S. Gheorghiu and M. O. Coppens, Heterogeneity explains features of anomalous thermodynamics and statistics,, PNAS, 101 (2004), 15852. doi: 10.1073/pnas.0407191101. Google Scholar

[20]

G. Gripenberg, S.-O. Londen and O. Staffans, "Volterra Integrals and Functional Equations,", Encyclopedia of Mathematics and its Applications, 34 (1990). doi: 10.1017/CBO9780511662805. Google Scholar

[21]

P. Hanggi, P. Talkner and M. Borkovec, Reaction-rate theory, 50 years after Kramers,, Rev. Mod. Phys., 62 (1990), 251. doi: 10.1103/RevModPhys.62.251. Google Scholar

[22]

S. Havlin and D. Ben Avraham, Diffusion in disordered media,, Adv. Phys., 51 (2002), 187. Google Scholar

[23]

A. Hunt and R. Ewing, "Percolation Theory for Flow in Porous Media,", Lecture Notes in Physics, 771 (2009). doi: 10.1007/978-3-540-89790-3. Google Scholar

[24]

M. Kac, On distributions of certain Wiener functionals,, Trans. Am. Math. Soc., 65 (1949), 1. doi: 10.1090/S0002-9947-1949-0027960-X. Google Scholar

[25]

I. Kaj, L. Leskela, I. Norros and V. Schmidt, Scaling limits for random fields with long-range dependence,, Ann. of Prob., 35 (2007), 528. doi: 10.1214/009117906000000700. Google Scholar

[26]

P. Lévy, "Théorie de l'Addition des Variables Aléatoires,", Gauthier-Villars, (1937). Google Scholar

[27]

S.-O. Londen, H. Petzeltová and J. Prüss, Global well-posedness and stability of a partial integro-differential equation with applications to viscoelasticity,, J. Evol. Equ., 3 (2002), 169. doi: 10.1007/978-3-0348-7924-8_9. Google Scholar

[28]

M. Magdziarz, Langevin picture of subdiffusion with infinitely divisible waiting times,, J. Stat. Phys., 135 (2009), 763. doi: 10.1007/s10955-009-9751-z. Google Scholar

[29]

M. Magdziarz, A. Weron and J. Klafter, Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: The case of time-dependent force,, Phys. Rev. E, 101 (2008). doi: 10.1103/PhysRevLett.101.210601. Google Scholar

[30]

A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena,, Phys. Rep., 314 (1999), 237. doi: 10.1016/S0370-1573(98)00083-0. Google Scholar

[31]

S. N. Majumdar, Brownian functionals in physics and computer science,, Curr. Sci., 89 (2005), 2076. Google Scholar

[32]

B. B. Mandelbrot, "The Fractal Geometry of Nature,", Schriftenreihe für den Referenten [Series for the Referee], (1982). Google Scholar

[33]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications,, SIAM Rev., 10 (1968), 422. doi: 10.1137/1010093. Google Scholar

[34]

M. M. Meerschaert and H.-P. Scheffler, "Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice,", Wiley Series in Probability and Statistics: Probability and Statistics, (2001). Google Scholar

[35]

R. Metzler and J. Klafter, The restaurant at the end of the random walk,, J. Phys. A: Math. Gen., 37 (2004), 161. doi: 10.1088/0305-4470/37/31/R01. Google Scholar

[36]

M. S. Mommer and D. Lebiedz, Modeling subdiffusion using reaction diffusion systems,, SIAM J. Appl. Math., 70 (2009), 112. doi: 10.1137/070681648. Google Scholar

[37]

M. C. Neel, A. Zoia and M. Joelson, Mass transport subject to time-dependent flow with uniform sorption in porous media,, Phys. Rev. E, 80 (2009). Google Scholar

[38]

J. P. Nolan, "Stable Distributions - Models for Heavy Tailed Data,", In progress, (2012). Google Scholar

[39]

H. Owhadi, Anomalous slow diffusion from perpetual homogenization,, Ann. Prob., 31 (2003), 1935. doi: 10.1214/aop/1068646372. Google Scholar

[40]

G. Papanicolaou and H. Kesten, A limit theorem for stochastic acceleration,, Comm. Math. Phys., 78 (1980), 19. doi: 10.1007/BF01941968. Google Scholar

[41]

J. B. Perrin, Mouvement Brownien et réalité moléculaire,, Annales de chimie et de physique, 8 (1909), 5. Google Scholar

[42]

A. S. Pikovsky, Escape exponent for transient chaos scattering in non-hyperbolic Hamiltonian systems,, J. Phys. A, 25 (1992). doi: 10.1088/0305-4470/25/8/016. Google Scholar

[43]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, Commun. Math. Phys., 74 (1980), 189. doi: 10.1007/BF01197757. Google Scholar

[44]

S. Ross, "Introduction in Probability Models,", (Chapter 10) Fifth edition, (1993). Google Scholar

[45]

B. Rubin, "Fractional Integrals and Potentials,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 82 (1996). Google Scholar

[46]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion wave equations and applications to some inverse problems,, J. Math. Anal. Appl., 382 (2011), 426. doi: 10.1016/j.jmaa.2011.04.058. Google Scholar

[47]

S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives: Theory and Applications,", Gordon and Breach Science Publishers, (1993). Google Scholar

[48]

H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids,, Phys. Rev. B, 12 (1975), 2455. doi: 10.1103/PhysRevB.12.2455. Google Scholar

[49]

R. Schumer, D. A. Benson, M. M. Meerschaert and B. Bauemer, Fractal mobile/immobile solute transport,, Water Resour. Res., 39 (2003). doi: 10.1029/2003WR002141. Google Scholar

[50]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics,, Nature, 363 (1993), 31. doi: 10.1038/363031a0. Google Scholar

[51]

A. Shojiguchi, C. B. Li, T. Tomatsuzaki and M. Toda, Dynamical foundation and limitations of statistical reaction theory,, Comm. Nonlinear Sci. Numer. Simul., 13 (2008), 857. doi: 10.1016/j.cnsns.2006.08.002. Google Scholar

[52]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,, Ann. der Physik, 21 (1906), 756. Google Scholar

[53]

E. R. Weeks, J. S. Urbach and H. L. Swinney, Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example,, Phys. D, 97 (1996), 291. doi: 10.1016/0167-2789(96)00082-6. Google Scholar

[54]

K. Weron, A. Stanislavsky, A. Jurlewicz, M. M. Meerschaert and H. P. Scheffler, Clustered continuous time random walks: Diffusion and relaxation consequences,, Proc. Royal Soc. A, 468 (2012), 1615. doi: 10.1098/rspa.2011.0697. Google Scholar

[55]

W. R. Young, Arrested shear dispersion and other models of anomalous diffusion,, J. Fluid Mech., 193 (1988), 129. doi: 10.1017/S0022112088002083. Google Scholar

[56]

G. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos,, Phys. D, 76 (1994), 110. doi: 10.1016/0167-2789(94)90254-2. Google Scholar

[57]

Y. Zhang, D. A. Benson and B. Bauemer, Moment analysis for spatiotemporal fractional dispersion,, Water Resour., 44 (2008). doi: 10.1029/2007WR006291. Google Scholar

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