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 and Continuous Dynamical Systems - S, 2014, 7 (2) : 207-238. doi: 10.3934/dcdss.2014.7.207
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), 7270-7286.

[2]

E. G. Altmann and T. Tel, Poincaré recurrences and transient chaos in systems with leaks, Phys. Rev. E (3), 79 (2009), 016204, 12 pp. doi: 10.1103/PhysRevE.79.016204.

[3]

E. Barkai, R. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138. doi: 10.1103/PhysRevE.61.132.

[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), 532-539. doi: 10.1016/j.advwatres.2009.01.002.

[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), RG2003. doi: 10.1029/2005RG000178.

[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), W10425. doi: 10.1029/2007WR006062.

[7]

M. Bromly and C. Hinz, Non-Fickian transport in homogeneous unsaturated repacked sand, Water Resour. Res., 40 (2004), W07402. doi: 10.1029/2003WR002579.

[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), 161-173.

[9]

S. Carmi, L. Turgeman and E. Barkai, On distributions of functionals of anomalous diffusion paths, J. Stat. Phys., 141 (2010), 1071-1092. doi: 10.1007/s10955-010-0086-6.

[10]

C. Choquet and M. C. Néel, Rigorous derivation of Feynman-Kac equations for arrested dispersion, Technical report, Lab. MIA, Univ. La Rochelle, 2013.

[11]

P. Clément, S.-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Diff. Equ., 196 (2004), 418-447. doi: 10.1016/j.jde.2003.07.014.

[12]

G. A. Coon and D. L. Bernstein, Some properties of the double Laplace transformation, Trans. Amer. Math. Soc., 74 (1953), 135-176. doi: 10.1090/S0002-9947-1953-0052556-4.

[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), 041108. doi: 10.1103/PhysRevE.70.041108.

[14]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Diff. Equ., 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002.

[15]

A. Einstein, Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen, Ann. der Physik, 17 (1905), 549-560. doi: 10.1002/andp.19053220806.

[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), 1043-1128. doi: 10.1023/A:1015157624384.

[17]

W. Feller, "An Introduction to Probability Theory and its Applications. Vol.II," Wiley, New York, 1970.

[18]

R. Friedrich, F. Jenko, A. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles, Phys. Rev. Lett., 96 (2006), 230601. doi: 10.1103/PhysRevLett.96.230601.

[19]

S. Gheorghiu and M. O. Coppens, Heterogeneity explains features of anomalous thermodynamics and statistics, PNAS, 101 (2004), 15852-15856. doi: 10.1073/pnas.0407191101.

[20]

G. Gripenberg, S.-O. Londen and O. Staffans, "Volterra Integrals and Functional Equations," Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[21]

P. Hanggi, P. Talkner and M. Borkovec, Reaction-rate theory, 50 years after Kramers, Rev. Mod. Phys., 62 (1990), 251-350. doi: 10.1103/RevModPhys.62.251.

[22]

S. Havlin and D. Ben Avraham, Diffusion in disordered media, Adv. Phys., 51 (2002), 187-292.

[23]

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

[24]

M. Kac, On distributions of certain Wiener functionals, Trans. Am. Math. Soc., 65 (1949), 1-13. doi: 10.1090/S0002-9947-1949-0027960-X.

[25]

I. Kaj, L. Leskela, I. Norros and V. Schmidt, Scaling limits for random fields with long-range dependence, Ann. of Prob., 35 (2007), 528-550. doi: 10.1214/009117906000000700.

[26]

P. Lévy, "Théorie de l'Addition des Variables Aléatoires," Gauthier-Villars, Paris, 1937.

[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), 169-201. doi: 10.1007/978-3-0348-7924-8_9.

[28]

M. Magdziarz, Langevin picture of subdiffusion with infinitely divisible waiting times, J. Stat. Phys., 135 (2009), 763-772. doi: 10.1007/s10955-009-9751-z.

[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), 210601, 4 pp. doi: 10.1103/PhysRevLett.101.210601.

[30]

A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Phys. Rep., 314 (1999), 237-574. doi: 10.1016/S0370-1573(98)00083-0.

[31]

S. N. Majumdar, Brownian functionals in physics and computer science, Curr. Sci., 89 (2005), 2076-2092.

[32]

B. B. Mandelbrot, "The Fractal Geometry of Nature," Schriftenreihe für den Referenten [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982.

[33]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437. doi: 10.1137/1010093.

[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, John Wiley & Sons, Inc., New York, 2001.

[35]

R. Metzler and J. Klafter, The restaurant at the end of the random walk, J. Phys. A: Math. Gen., 37 (2004), 161-208. doi: 10.1088/0305-4470/37/31/R01.

[36]

M. S. Mommer and D. Lebiedz, Modeling subdiffusion using reaction diffusion systems, SIAM J. Appl. Math., 70 (2009), 112-129. doi: 10.1137/070681648.

[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), 056301.

[38]

J. P. Nolan, "Stable Distributions - Models for Heavy Tailed Data," In progress, Birkhäuser, Boston, 2012. Chapter 1 available from: http://www.academic2.american.edu/~jpnolan.

[39]

H. Owhadi, Anomalous slow diffusion from perpetual homogenization, Ann. Prob., 31 (2003), 1935-1969. doi: 10.1214/aop/1068646372.

[40]

G. Papanicolaou and H. Kesten, A limit theorem for stochastic acceleration, Comm. Math. Phys., 78 (1980), 19-63. doi: 10.1007/BF01941968.

[41]

J. B. Perrin, Mouvement Brownien et réalité moléculaire, Annales de chimie et de physique, 8 (1909), 5-114.

[42]

A. S. Pikovsky, Escape exponent for transient chaos scattering in non-hyperbolic Hamiltonian systems, J. Phys. A, 25 (1992), L477-L481. doi: 10.1088/0305-4470/25/8/016.

[43]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74 (1980), 189-197. doi: 10.1007/BF01197757.

[44]

S. Ross, "Introduction in Probability Models," (Chapter 10) Fifth edition, Academic Press, Inc., Boston, MA, 1993.

[45]

B. Rubin, "Fractional Integrals and Potentials," Pitman Monographs and Surveys in Pure and Applied Mathematics, 82, Longman, Harlow, 1996.

[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), 426-447. doi: 10.1016/j.jmaa.2011.04.058.

[47]

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

[48]

H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12 (1975), 2455-2477. doi: 10.1103/PhysRevB.12.2455.

[49]

R. Schumer, D. A. Benson, M. M. Meerschaert and B. Bauemer, Fractal mobile/immobile solute transport, Water Resour. Res., 39 (2003), 1296. doi: 10.1029/2003WR002141.

[50]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37. doi: 10.1038/363031a0.

[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), 857-867. doi: 10.1016/j.cnsns.2006.08.002.

[52]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. der Physik, 21 (1906), 756-780.

[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), 291-310. doi: 10.1016/0167-2789(96)00082-6.

[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), 1615-1628. doi: 10.1098/rspa.2011.0697.

[55]

W. R. Young, Arrested shear dispersion and other models of anomalous diffusion, J. Fluid Mech., 193 (1988), 129-149. doi: 10.1017/S0022112088002083.

[56]

G. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phys. D, 76 (1994), 110-122. doi: 10.1016/0167-2789(94)90254-2.

[57]

Y. Zhang, D. A. Benson and B. Bauemer, Moment analysis for spatiotemporal fractional dispersion, Water Resour., 44 (2008), W04424. doi: 10.1029/2007WR006291.

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), 7270-7286.

[2]

E. G. Altmann and T. Tel, Poincaré recurrences and transient chaos in systems with leaks, Phys. Rev. E (3), 79 (2009), 016204, 12 pp. doi: 10.1103/PhysRevE.79.016204.

[3]

E. Barkai, R. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138. doi: 10.1103/PhysRevE.61.132.

[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), 532-539. doi: 10.1016/j.advwatres.2009.01.002.

[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), RG2003. doi: 10.1029/2005RG000178.

[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), W10425. doi: 10.1029/2007WR006062.

[7]

M. Bromly and C. Hinz, Non-Fickian transport in homogeneous unsaturated repacked sand, Water Resour. Res., 40 (2004), W07402. doi: 10.1029/2003WR002579.

[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), 161-173.

[9]

S. Carmi, L. Turgeman and E. Barkai, On distributions of functionals of anomalous diffusion paths, J. Stat. Phys., 141 (2010), 1071-1092. doi: 10.1007/s10955-010-0086-6.

[10]

C. Choquet and M. C. Néel, Rigorous derivation of Feynman-Kac equations for arrested dispersion, Technical report, Lab. MIA, Univ. La Rochelle, 2013.

[11]

P. Clément, S.-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Diff. Equ., 196 (2004), 418-447. doi: 10.1016/j.jde.2003.07.014.

[12]

G. A. Coon and D. L. Bernstein, Some properties of the double Laplace transformation, Trans. Amer. Math. Soc., 74 (1953), 135-176. doi: 10.1090/S0002-9947-1953-0052556-4.

[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), 041108. doi: 10.1103/PhysRevE.70.041108.

[14]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Diff. Equ., 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002.

[15]

A. Einstein, Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen, Ann. der Physik, 17 (1905), 549-560. doi: 10.1002/andp.19053220806.

[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), 1043-1128. doi: 10.1023/A:1015157624384.

[17]

W. Feller, "An Introduction to Probability Theory and its Applications. Vol.II," Wiley, New York, 1970.

[18]

R. Friedrich, F. Jenko, A. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles, Phys. Rev. Lett., 96 (2006), 230601. doi: 10.1103/PhysRevLett.96.230601.

[19]

S. Gheorghiu and M. O. Coppens, Heterogeneity explains features of anomalous thermodynamics and statistics, PNAS, 101 (2004), 15852-15856. doi: 10.1073/pnas.0407191101.

[20]

G. Gripenberg, S.-O. Londen and O. Staffans, "Volterra Integrals and Functional Equations," Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[21]

P. Hanggi, P. Talkner and M. Borkovec, Reaction-rate theory, 50 years after Kramers, Rev. Mod. Phys., 62 (1990), 251-350. doi: 10.1103/RevModPhys.62.251.

[22]

S. Havlin and D. Ben Avraham, Diffusion in disordered media, Adv. Phys., 51 (2002), 187-292.

[23]

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

[24]

M. Kac, On distributions of certain Wiener functionals, Trans. Am. Math. Soc., 65 (1949), 1-13. doi: 10.1090/S0002-9947-1949-0027960-X.

[25]

I. Kaj, L. Leskela, I. Norros and V. Schmidt, Scaling limits for random fields with long-range dependence, Ann. of Prob., 35 (2007), 528-550. doi: 10.1214/009117906000000700.

[26]

P. Lévy, "Théorie de l'Addition des Variables Aléatoires," Gauthier-Villars, Paris, 1937.

[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), 169-201. doi: 10.1007/978-3-0348-7924-8_9.

[28]

M. Magdziarz, Langevin picture of subdiffusion with infinitely divisible waiting times, J. Stat. Phys., 135 (2009), 763-772. doi: 10.1007/s10955-009-9751-z.

[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), 210601, 4 pp. doi: 10.1103/PhysRevLett.101.210601.

[30]

A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Phys. Rep., 314 (1999), 237-574. doi: 10.1016/S0370-1573(98)00083-0.

[31]

S. N. Majumdar, Brownian functionals in physics and computer science, Curr. Sci., 89 (2005), 2076-2092.

[32]

B. B. Mandelbrot, "The Fractal Geometry of Nature," Schriftenreihe für den Referenten [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982.

[33]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437. doi: 10.1137/1010093.

[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, John Wiley & Sons, Inc., New York, 2001.

[35]

R. Metzler and J. Klafter, The restaurant at the end of the random walk, J. Phys. A: Math. Gen., 37 (2004), 161-208. doi: 10.1088/0305-4470/37/31/R01.

[36]

M. S. Mommer and D. Lebiedz, Modeling subdiffusion using reaction diffusion systems, SIAM J. Appl. Math., 70 (2009), 112-129. doi: 10.1137/070681648.

[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), 056301.

[38]

J. P. Nolan, "Stable Distributions - Models for Heavy Tailed Data," In progress, Birkhäuser, Boston, 2012. Chapter 1 available from: http://www.academic2.american.edu/~jpnolan.

[39]

H. Owhadi, Anomalous slow diffusion from perpetual homogenization, Ann. Prob., 31 (2003), 1935-1969. doi: 10.1214/aop/1068646372.

[40]

G. Papanicolaou and H. Kesten, A limit theorem for stochastic acceleration, Comm. Math. Phys., 78 (1980), 19-63. doi: 10.1007/BF01941968.

[41]

J. B. Perrin, Mouvement Brownien et réalité moléculaire, Annales de chimie et de physique, 8 (1909), 5-114.

[42]

A. S. Pikovsky, Escape exponent for transient chaos scattering in non-hyperbolic Hamiltonian systems, J. Phys. A, 25 (1992), L477-L481. doi: 10.1088/0305-4470/25/8/016.

[43]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74 (1980), 189-197. doi: 10.1007/BF01197757.

[44]

S. Ross, "Introduction in Probability Models," (Chapter 10) Fifth edition, Academic Press, Inc., Boston, MA, 1993.

[45]

B. Rubin, "Fractional Integrals and Potentials," Pitman Monographs and Surveys in Pure and Applied Mathematics, 82, Longman, Harlow, 1996.

[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), 426-447. doi: 10.1016/j.jmaa.2011.04.058.

[47]

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

[48]

H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12 (1975), 2455-2477. doi: 10.1103/PhysRevB.12.2455.

[49]

R. Schumer, D. A. Benson, M. M. Meerschaert and B. Bauemer, Fractal mobile/immobile solute transport, Water Resour. Res., 39 (2003), 1296. doi: 10.1029/2003WR002141.

[50]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37. doi: 10.1038/363031a0.

[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), 857-867. doi: 10.1016/j.cnsns.2006.08.002.

[52]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. der Physik, 21 (1906), 756-780.

[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), 291-310. doi: 10.1016/0167-2789(96)00082-6.

[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), 1615-1628. doi: 10.1098/rspa.2011.0697.

[55]

W. R. Young, Arrested shear dispersion and other models of anomalous diffusion, J. Fluid Mech., 193 (1988), 129-149. doi: 10.1017/S0022112088002083.

[56]

G. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phys. D, 76 (1994), 110-122. doi: 10.1016/0167-2789(94)90254-2.

[57]

Y. Zhang, D. A. Benson and B. Bauemer, Moment analysis for spatiotemporal fractional dispersion, Water Resour., 44 (2008), W04424. doi: 10.1029/2007WR006291.

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