The random walk’s guide to anomalous diffusion: A fractional dynamics approach. (English) Zbl 0984.82032

Metzler, Ralf; Klafter, Joseph

Phys. Rep. 339, No. 1, 1-77 (2000).

Summary: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have become of age as a complementary tool in the description of anomalous transport processes.

MSC:

82C31	Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C70	Transport processes in time-dependent statistical mechanics

Keywords:

fractional kinetic equations; non-exponential relaxation patterns; Fokker-Planck type equations

Software:

Mathematica

References:

[1]	Goldenfeld, N.; Kadanoff, L. P., Science, 284, 87 (1999)
[2]	F. Guerra, L. Peliti, A. Vulpiani, in L. Peliti, A. Vulpiani (Eds.), Measures of Complexity, Springer, Berlin, 1988.; F. Guerra, L. Peliti, A. Vulpiani, in L. Peliti, A. Vulpiani (Eds.), Measures of Complexity, Springer, Berlin, 1988. · Zbl 0707.68003
[3]	Peliti, L.; Vulpiani (Eds.), A., Measures of Complexity (1988), Springer: Springer Berlin · Zbl 0707.68003
[4]	Takayama (Ed.), H., Cooperative Dynamics in Complex Systems (1989), Springer: Springer Berlin
[5]	Maxwell, J. C., Phil. Mag., 35, 129, 185 (1868) · JFM 01.0379.01
[6]	Debye, P., Polar Molecules (1945), Dover: Dover New York · JFM 55.1179.04
[7]	Lenk, R.; Gellert, W., Fachlexikon Physik (1974), Deutsch: Deutsch Zürich
[8]	Dattagupta, S., Relaxation Phenomena in Condensed Matter Physics (1987), Academic Press: Academic Press Orlando
[9]	Reiner, M., Rheologie in Elementarer Darstellung (1969), VEB Fachbuchverlag: VEB Fachbuchverlag Leipzig
[10]	Tschoegl, N. W., The Phenomenological Theory of Linear Viscoelastic Behavior (1989), Springer: Springer Heidelberg · Zbl 0681.73022
[11]	Kohlrausch, R., Ann. Phys. (Leipzig), 12, 393 (1847)
[12]	Williams, G.; Watts, D. C., Trans. Faraday Soc., 66, 80 (1970)
[13]	Klafter, J.; Shlesinger, M. F., Proc. Natl. Acad. Sci. USA, 83, 848 (1986)
[14]	Blumen, A.; Klafter, J.; Zumofen, G.; Zschokke, I., Optical Spectroscopy of Glasses (1986), Reidel: Reidel Dordrecht
[15]	Vlad, M. O.; Metzler, R.; Nonnenmacher, T. F.; Mackey, M. C., J. Math. Phys., 37, 2279 (1996) · Zbl 0872.60097
[16]	Reiner, M.; Ridgen, J. P.; Thrower, E. N., J. Soc. Chem. Ind., 69, 257 (1939)
[17]	Scott Blair, G. W., Kolloid-Z., 47, 46 (1929)
[18]	Scott Blair, G. W., J. Dairy Res., 9, 347 (1938)
[19]	Scott Blair, G. W., Nature, 152, 412 (1943)
[20]	Glöckle, W. G.; Nonnenmacher, T. F., Macromolecules, 24, 6426 (1991)
[21]	Glöckle, W. G.; Nonnenmacher, T. F., Rheol. Acta, 33, 337 (1994)
[22]	Glöckle, W. G.; Nonnenmacher, T. F., Biophys. J., 68, 46 (1995)
[23]	Einstein, A.; Fürth, R., Investigations on the theory of the Brownian movement (1956), Dover: Dover New York · Zbl 0071.41205
[24]	Wax (Ed.), N., Selected Papers on Noise and Stochastic Processes (1954), Dover: Dover New York · Zbl 0059.11903
[25]	Carslaw, H. S.; Jaegher, J. C., Conduction of Heat in Solids (1959), Clarendon Press: Clarendon Press Oxford
[26]	Nelson, E., Dynamical Theories of Brownian Motion (1967), Princeton University Press: Princeton University Press Princeton · Zbl 0165.58502
[27]	Lévy, P., Processus stochastiques et mouvement Brownien (1965), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0137.11602
[28]	R. Becker, Theorie der Wärme, Heidelberger Taschenbücher, Vol. 10, Springer, Berlin, 1966; Theory of Heat, Springer, Berlin, 1967.; R. Becker, Theorie der Wärme, Heidelberger Taschenbücher, Vol. 10, Springer, Berlin, 1966; Theory of Heat, Springer, Berlin, 1967.
[29]	de Groot, S. R.; Mazur, P., Non-equilibrium Thermodynamics (1969), North-Holland: North-Holland Amsterdam
[30]	Doob, J. L., Stochastic Processes (1953), Wiley: Wiley New York · Zbl 0053.26802
[31]	Crank, J., The Mathematics of Diffusion (1970), Clarendon Press: Clarendon Press Oxford · Zbl 0212.18102
[32]	Cox, D. R.; Miller, H. D., The Theory of Stochastic Processes (1965), Methuen: Methuen London · Zbl 0149.12902
[33]	Aris, R., The Mathematical Theory of Diffusion and Reaction in Permeable Catalysis, Vols. I and II (1975), Clarendon Press: Clarendon Press Oxford · Zbl 0315.76051
[34]	L.D. Landau, E.M. Lifschitz, Statistische Physik, Akademie, Leipzig, 1989; Statistical Physics, Pergamon, Oxford, 1980.; L.D. Landau, E.M. Lifschitz, Statistische Physik, Akademie, Leipzig, 1989; Statistical Physics, Pergamon, Oxford, 1980.
[35]	van Kampen, N. G., Stochastic Processes in Physics and Chemistry (1981), North-Holland: North-Holland Amsterdam · Zbl 0511.60038
[36]	Risken, H., The Fokker-Planck Equation (1989), Springer: Springer Berlin · Zbl 0665.60084
[37]	Coffey, W. T.; Kalmykov, Yu. P.; Waldron, J. T., The Langevin Equation (1996), World Scientific: World Scientific Singapore · Zbl 0952.82510
[38]	Hughes, B. D., Random Walks and Random Environments, Vol. 1: Random Walks (1995), Oxford University Press: Oxford University Press Oxford · Zbl 0820.60053
[39]	Weiss, G. H.; Rubin, R. J., Adv. Chem. Phys., 52, 363 (1983)
[40]	Alexander, S.; Bernasconi, J.; Schneider, W. R.; Orbach, R., Rev. Mod. Phys., 53, 175 (1981)
[41]	Havlin, S.; Ben-Avraham, D., Adv. Phys., 36, 695 (1987)
[42]	Bouchaud, J.-P.; Georges, A., Phys. Rep., 195, 12 (1990)
[43]	Isichenko, M. B., Rev. Mod. Phys., 64, 961 (1992)
[44]	Haus, J. W.; Kehr, K. W., Phys. Rep., 150, 263 (1987)
[45]	Hänggi, P.; Thomas, H., Phys. Rep., 88, 207 (1982)
[46]	Grifoni, M.; Hänggi, P., Phys. Rep., 304, 229 (1998)
[47]	West, B. J.; Deering, W., Phys. Rep., 246, 1 (1994)
[48]	Balescu, R., Statistical Dynamics, Matter out of Equilibrium (1997), Imperial College Press: Imperial College Press London · Zbl 0997.82505
[49]	Shlesinger, M. F.; Zaslavsky, G. M.; Klafter, J., Nature, 363, 31 (1993)
[50]	Weiss, G. H., Aspects and Applications of the Random Walk (1994), North-Holland: North-Holland Amsterdam · Zbl 0925.60079
[51]	Shlesinger, M. F.; Zaslavsky, G. M.; Frisch (Eds.), U., Lévy Flights and Related Topics (1995), Springer: Springer Berlin
[52]	Klafter, J.; Shlesinger, M. F.; Zumofen, G., Phys. Today, 49, 2, 33 (1996)
[53]	Kutner, R.; Pȩkalski, A.; Sznajd-Weron (Eds.), K., Anomalous Diffusion, from Basics to Applications (1999), Springer: Springer Berlin
[54]	Plonka, A., Ann. Rep. Prog. Chem., Sec. C, 94, 89 (1997)
[55]	Zaslavsky, G. M.; Benkadda, S., Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas (1998), Springer: Springer Berlin · Zbl 0896.00027
[56]	Lévy, P., Calcul des Probabilités (1925), Gauthier-Villars: Gauthier-Villars Paris · JFM 51.0380.02
[57]	Lévy, P., Théorie de l’addition des variables Aléatoires (1954), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0056.35903
[58]	Gnedenko, B. V.; Kolmogorov, A. N., Limit Distributions for Sums of Random Variables (1954), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0056.36001
[59]	Zwanzig, R., J. Chem. Phys., 15, 983 (1961)
[60]	Nordholm, S.; Zwanzig, R., J. Stat. Phys., 13, 347 (1975)
[61]	R. Zwanzig, in: Molecular Fluids, Les Houches 1973, R. Balian, G. Weill (Eds.), Gordon and Breach, London, 1976.; R. Zwanzig, in: Molecular Fluids, Les Houches 1973, R. Balian, G. Weill (Eds.), Gordon and Breach, London, 1976.
[62]	Kubo, R.; Toda, M.; Hashitsume, N., Statistical Physics II, Solid State Sciences, Vol. 31 (1985), Springer: Springer Berlin
[63]	Fick, E.; Sauermann, G., Quantenstatistik dynamischer Prozesse, Band 1, Generelle Aspekte (1983), Harri Deutsch: Harri Deutsch Thun
[64]	Langevin, P., Comptes Rendus, 146, 530 (1908)
[65]	Pearson, K., Nature, 72, 294, 342 (1905) · JFM 36.0303.02
[66]	Barkai, E.; Silbey, R., J. Phys. Chem. B, 104, 3866 (2000)
[67]	R. Brown, Phil. Mag. 4 (1828) 161; Ann. Phys. Chem. 14 (1828) 294.; R. Brown, Phil. Mag. 4 (1828) 161; Ann. Phys. Chem. 14 (1828) 294.
[68]	Fick, A., Ann. Phys. (Leipzig), 170, 50 (1855)
[69]	Bachelier, L., Ann. scientif. de l’École norm., 17, 21 (1900) · JFM 31.0241.02
[70]	Cootner (Ed.), P. H., The Random Character of Stock Market Prices (1964), MIT Press: MIT Press Cambridge, Massachusetts
[71]	Perrin, J., Comptes Rendus (Paris), 146, 967 (1908)
[72]	Perrin, J., Ann. chim. et d. phys. VIII, 18, 5 (1909)
[73]	A. Westgren, Z. Phys. Chem. 83 (1913) 151; Z. Phys. Chem. 89 (1914) 63; Arch. f. Mat. Astr. och Fysik 11 (8 & 14) (1916); Z. anorg. Chem. 93 (1915) 231; Z. anorg. Chem. 95 (1916) 39.; A. Westgren, Z. Phys. Chem. 83 (1913) 151; Z. Phys. Chem. 89 (1914) 63; Arch. f. Mat. Astr. och Fysik 11 (8 & 14) (1916); Z. anorg. Chem. 93 (1915) 231; Z. anorg. Chem. 95 (1916) 39.
[74]	Kappler, E., Ann. d. Phys. (Leipzig), 11, 233 (1931)
[75]	M.v. Smoluchowski, Vorträge über kinetische Theorie der Materie, Göttingen, 1914.; M.v. Smoluchowski, Vorträge über kinetische Theorie der Materie, Göttingen, 1914. · JFM 45.0955.02
[76]	Gerlach, W.; Lehrer, E., Naturwiss., 15, 15 (1927)
[77]	Eggers, W., Ann. d. Phys. (Leipzig), 7, 833 (1930)
[78]	Fokker, A. D., Ann. Phys. (Leipzig), 43, 810 (1914)
[79]	M. Planck, Sitzber. Preuß. Akad. Wiss. (1917) 324.; M. Planck, Sitzber. Preuß. Akad. Wiss. (1917) 324.
[80]	Smoluchowski, M. V., Ann. Phys., 48, 1103 (1915)
[81]	Klein, O., Arkiv Mat. Astr. Fys., 16, 5 (1922) · JFM 48.0918.02
[82]	Kramers, H. A., Physica, 7, 284 (1940) · Zbl 0061.46405
[83]	Montroll, E. W., J. SIAM, 4, 241 (1956)
[84]	Montroll, E. W.; Weiss, G. H., J. Math. Phys., 10, 753 (1969)
[85]	Montroll, E. W.; Scher, H., J. Stat. Phys., 9, 101 (1973)
[86]	M.F. Shlesinger (Ed.), The wonderful world of stochastics: a tribute to E.W. Montroll. Studies in statistical mechanics 12, North-Holland, Amsterdam, 1985.; M.F. Shlesinger (Ed.), The wonderful world of stochastics: a tribute to E.W. Montroll. Studies in statistical mechanics 12, North-Holland, Amsterdam, 1985. · Zbl 0591.60001
[87]	N. Wiener, J. Math. and Phys. (MIT) 2 (1923) 131; Bull. Soc. math. (France) 52 (1924) 569; Acta Math. 55 (1930) 117; Amer. J. Math. 60 (1938) 897.; N. Wiener, J. Math. and Phys. (MIT) 2 (1923) 131; Bull. Soc. math. (France) 52 (1924) 569; Acta Math. 55 (1930) 117; Amer. J. Math. 60 (1938) 897.
[88]	Feller, W., An Introduction to Probability Theory and its Applications, Vol. 2 (1971), Wiley: Wiley New York · Zbl 0219.60003
[89]	Mandelbrot, B. B., The Fractal Geometry of Nature (1983), Freeman: Freeman New York · Zbl 0504.28001
[90]	Feder, J., Fractals (1988), Plenum: Plenum New York · Zbl 0648.28006
[91]	Takayasu, H., Fractals in the Physical Sciences (1990), Manchester University Press: Manchester University Press Manchester · Zbl 0689.58001
[92]	Gouyet, J.-F., Physique et Structures Fractales (1992), Masson: Masson Paris · Zbl 0773.58015
[93]	Falconer, K., Fractal Geometry: Mathematical Foundations and Applications (1990), Wiley: Wiley Chichester, UK · Zbl 0689.28003
[94]	Richardson, L. F., Proc. Roy. Soc., 110, 709 (1926)
[95]	Scher, H.; Montroll, E. W., Phys. Rev. B, 12, 2455 (1975)
[96]	Pfister, G.; Scher, H., Phys. Rev. B, 15, 2062 (1977)
[97]	Pfister, G.; Scher, H., Adv. Phys., 27, 747 (1978)
[98]	Zumofen, G.; Blumen, A.; Klafter, J., Phys. Rev. A, 41, 4558 (1990)
[99]	Blom, P. W.M.; Vissenberg, M. C.J. M., Phys. Rev. Lett., 80, 3819 (1998)
[100]	Shlesinger, M. F., J. Stat. Phys., 10, 421 (1974)
[101]	Scher, H.; Shlesinger, M. F.; Bendler, J. T., Phys. Today, 44, 26 (1991)
[102]	Berkowitz, B.; Scher, H., Phys. Rev. E, 57, 5858 (1998)
[103]	Gu, Q.; Schiff, E. A.; Grebner, S.; Schwartz, R., Phys. Rev. Lett., 76, 3196 (1996)
[104]	Rinn, B.; Dieterich, W.; Maass, P., Phil. Mag. B, 77, 1283 (1998)
[105]	Klemm, A.; Müller, H.-P.; Kimmich, R., Phys. Rev. E, 55, 4413 (1997)
[106]	Klemm, A.; Müller, H.-P.; Kimmich, R., Physica, 266A, 242 (1999)
[107]	Klammler, F.; Kimmich, R., Croat. Chem. Acta, 65, 455 (1992)
[108]	Kimmich, R., NMR: Tomography, Diffusometry, Relaxometry (1997), Springer: Springer Berlin
[109]	Weber, H. W.; Kimmich, R., Macromol., 26, 2597 (1993)
[110]	Fischer, E.; Kimmich, R.; Fatkullin, N., J. Chem. Phys., 104, 9174 (1996)
[111]	Fischer, E.; Kimmich, R.; Beginn, U.; Moeller, M.; Fatkullin, N., Phys. Rev. E, 59, 4079 (1999)
[112]	Kimmich, R.; Seitter, R.-O.; Beginn, U.; Moeller, M.; Fatkullin, N., Chem. Phys. Lett., 307, 147 (1999)
[113]	Binder, K.; Paul, W., J. Polym. Sci. B Pol. Phys., 35, 1 (1997)
[114]	de Gennes, P.-G., Scaling Concepts in Polymer Physics (1979), Cornell University Press: Cornell University Press Ithaca
[115]	Doi, M.; Edwards, S. F., The Theory of Polymer Dynamics (1986), Clarendon Press: Clarendon Press Oxford
[116]	Havlin, S.; Movshovitz, D.; Trus, B.; Weiss, G. H., J. Phys. A, 18, L719 (1985)
[117]	Porto, M.; Bunde, A.; Havlin, S.; Roman, H. E., Phys. Rev. E, 56, 1667 (1997)
[118]	Young, W.; Pumir, A.; Pomeau, Y., Phys. Fluids A, 1, 462 (1989) · Zbl 0659.76097
[119]	Cardoso, O.; Tabeling, P., Europhys. Lett., 7, 225 (1988)
[120]	Amblard, F.; Maggs, A. C.; Yurke, B.; Pargellis, A. N.; Leibler, S., Phys. Rev. Lett., 77, 4470 (1996)
[121]	Barkai, E.; Klafter, J., Phys. Rev. Lett., 81, 1134 (1998)
[122]	Weeks, E. R.; Swinney, H. L., Phys. Rev. E, 57, 4915 (1998)
[123]	Luedtke, W. D.; Landmann, U., Phys. Rev. Lett., 82, 3835 (1999)
[124]	Matheron, G.; de Marsily, G., Water Res. Res., 16, 901 (1980)
[125]	Zumofen, G.; Klafter, J.; Blumen, A., J. Stat. Phys., 65, 991 (1991)
[126]	Batchelor, G. K., Quart. J. Roy. Meteor. Soc., 76, 133 (1950)
[127]	Shlesinger, M. F.; West, B. J.; Klafter, J., Phys. Rev. Lett., 58, 1100 (1987)
[128]	P. Tabeling, A.E. Hansen, J. Paret, in [55]; P. Tabeling, A.E. Hansen, J. Paret, in [55]
[129]	Sokolov, I.; Blumen, A.; Klafter, J., Europhys. Lett., 47, 152 (1999)
[130]	Bychuk, O. V.; O’Shaugnessy, B., Phys. Rev. Lett., 74, 1795 (1994)
[131]	Stapf, S.; Kimmich, R.; Seitter, R.-O., Phys. Rev. Lett., 75, 2855 (1995)
[132]	Bodurka, J.; R. Kimmich, R.-O. Seitter; Gutsze, A., J. Chem. Phys., 107, 5621 (1997)
[133]	Klafter, J.; Blumen, A.; Zumofen, G.; Shlesinger, M. F., Physica, 168A, 637 (1990)
[134]	Ott, A.; Bouchaud, J.-P.; Langevin, D.; Urbach, W., Phys. Rev. Lett., 65, 2201 (1990)
[135]	Sahimi, M., Phys. Rep., 306, 214 (1998)
[136]	Schaufler, S.; Schleich, W. P.; Yakovlev, V. P., Europhys. Lett., 39, 383 (1997)
[137]	Schaufler, S.; Schleich, W. P.; Yakovlev, V. P., Phys. Rev. Lett., 83, 3162 (1999)
[138]	Zumofen, G.; Klafter, J., Chem. Phys. Lett., 219, 303 (1994)
[139]	Barkai, E.; Silbey, R., Chem. Phys. Lett., 310, 287 (1999)
[140]	Balescu, R., Phys. Rev. E, 51, 4807 (1995)
[141]	Klafter, J.; White, B. S.; Levandowsky, M.; Alt, W.; Hoffmann, G., Biological Motion, Lecture Notes in Biomathematics, Vol. 89 (1990), Springer: Springer Berlin · Zbl 0734.92006
[142]	Levandowsky, M.; White, B. S.; Schuster, F. L., Acta Protozool., 36, 237 (1997)
[143]	Nossal, R., J. Stat. Phys., 30, 391 (1983)
[144]	H. Linder, Biologie, J.B. Metzler, Stuttgart, 1984.; H. Linder, Biologie, J.B. Metzler, Stuttgart, 1984.
[145]	Matthews, C. K.; van Holde, K. E., Biochemistry (1996), Benjamin/Cummings: Benjamin/Cummings Menlo Park, CA
[146]	Viswanathan, G. M.; Afanasyev, V.; Buldyrev, S. V.; Murphy, E. J.; Prince, P. A.; Stanley, H. E., Nature, 381, 413 (1996)
[147]	Mandelbrot, B. B.; van Ness, J. W., SIAM Rev., 10, 422 (1968) · Zbl 0179.47801
[148]	O’Shaugnessy, B.; Procaccia, I., Phys. Rev. Lett., 54, 455 (1985)
[149]	Scher, H.; Lax, M., Phys. Rev. B, 7, 4491, 4502 (1973)
[150]	Klafter, J.; Silbey, R., Phys. Rev. Lett., 44, 55 (1980)
[151]	Klafter, J.; Blumen, A.; Shlesinger, M. F., Phys. Rev. A, 35, 3081 (1987)
[152]	E. Barkai, J. Klafter, in [55]; E. Barkai, J. Klafter, in [55]
[153]	E. Barkai, V. Fleurov, Phys. Rev. E 56, 6355.; E. Barkai, V. Fleurov, Phys. Rev. E 56, 6355.
[154]	Kutner, R.; Maass, P., J. Phys. A, 31, 2603 (1998) · Zbl 0921.60089
[155]	Bernasconi, J.; Schneider, W. R.; Wyss, W., Z. Phys. B, 37, 175 (1980)
[156]	Seshadri, V.; West, B. J., Proc. Natl. Acad. Sci. USA, 79, 4501 (1982) · Zbl 0488.60079
[157]	West, B. J.; Seshadri, V., Phsyica A, 113, 203 (1982)
[158]	Peseckis, F., Phys. Rev. A, 36, 892 (1987)
[159]	Fogedby, H. C., Phys. Rev. Lett., 73, 2517 (1994)
[160]	Fogedby, H. C., Phys. Rev. E, 50, 1657 (1994)
[161]	Muralidhar, R.; Ramkrishna, D.; Nakanishi, H.; Jacobs, D., Physica A, 167, 539 (1990)
[162]	Wang, K. G.; Dong, L. K.; Wu, X. F.; Zhu, F. W.; Ko, T., Physica A, 203, 53 (1994)
[163]	Wang, K. G.; Tokuyama, M., Physica A, 265, 341 (1999)
[164]	Kenkre, V. M.; Montroll, E. W.; Shlesinger, M. F., J. Stat. Phys., 9, 45 (1973)
[165]	Kenkre, V. M., Phys. Lett. A, 65, 391 (1978)
[166]	Bedeaux, D.; Lakatos, K.; Shuler, K., J. Math. Phys., 12, 2116 (1971) · Zbl 0236.60045
[167]	Oppenheim, I.; Shuler, K. E.; Weiss (Eds.), G. H., Stochastic Processes in Chemical Physics: The Master Equation (1977), MIT Press: MIT Press Cambridge, Massachusetts
[168]	Tsallis, C.; Levy, S. V.F.; Souza, A. M.C.; Maynard, R., Phys. Rev. Lett., 75, 3589 (1995) · Zbl 1020.82593
[169]	Tsallis, C.; Bukman, D. J., Phys. Rev. E, 54, R2197 (1996)
[170]	Borland, L., Phys. Rev. E, 57, 6634 (1998)
[171]	Compte, A.; Jou, D., J. Phys. A, 29, 4321 (1996) · Zbl 0900.82066
[172]	Zanette, D. H.; Alemany, P. A., Phys. Rev. Lett., 75, 366 (1995)
[173]	Balakrishnan, V., Physica A, 132, 569 (1985) · Zbl 0654.60065
[174]	Wyss, W., J. Math. Phys., 27, 2782 (1986) · Zbl 0632.35031
[175]	Schneider, W. R.; Wyss, W., J. Math. Phys., 30, 134 (1989) · Zbl 0692.45004
[176]	R.L. Bagley, P.J. Torvik, J. Rheol. 27 (1983) 201, J. Rheol. 30 (1986) 133.; R.L. Bagley, P.J. Torvik, J. Rheol. 27 (1983) 201, J. Rheol. 30 (1986) 133.
[177]	Koeller, R. C., J. Appl. Mech., 51, 299 (1984) · Zbl 0544.73052
[178]	Nigmatullin, R. R., Phys. Stat. Sol. B, 133, 425 (1986)
[179]	C. Friedrich, Rheol. Acta 30 (1991) 151; Phil. Mag. Lett. 66 (1992) 287.; C. Friedrich, Rheol. Acta 30 (1991) 151; Phil. Mag. Lett. 66 (1992) 287.
[180]	Glöckle, W. G.; Nonnenmacher, T. F., J. Stat. Phys., 71, 755 (1993)
[181]	Metzler, R.; Schick, W.; Kilian, H.-G.; Nonnenmacher, T. F., J. Chem. Phys., 103, 7180 (1995)
[182]	Schiessel, H.; Metzler, R.; Blumen, A.; Nonnenmacher, T. F., J. Phys. A, 28, 6567 (1995) · Zbl 0921.73154
[183]	Giona, M.; Roman, H. E., Physica A, 185, 87 (1992)
[184]	Metzler, R.; Glöckle, W. G.; Nonnenmacher, T. F., Physica A, 211, 13 (1994)
[185]	Roman, H. E.; Alemany, P. A., J. Phys. A, 27, 3407 (1994) · Zbl 0827.60057
[186]	Vlad, M. O., Chaos, Solitons & Fractals, 4, 191 (1994) · Zbl 0810.35171
[187]	Zaslavsky, G. M., Chaos, 4, 25 (1994) · Zbl 1055.82525
[188]	Alemany, P. A., Chaos, Solitons & Fractals, 6, 7 (1995) · Zbl 0900.60083
[189]	Hilfer, R.; Anton, L., Phys. Rev. E, 51, R848 (1995)
[190]	Hilfer, R., Fractals, 3, 211 (1995) · Zbl 0881.60066
[191]	Mainardi, F., Chaos, Solitons & Fractals, 7, 1461 (1996) · Zbl 1080.26505
[192]	Compte, A., Phys. Rev. E, 53, 4191 (1996)
[193]	Saichev, A. I.; Zaslavsky, G. M., Chaos, 7, 753 (1997) · Zbl 0933.37029
[194]	El-Sayed, A. M.A., Int. J. Th. Phys., 35, 311 (1996) · Zbl 0846.35001
[195]	West, B. J.; Grigolini, P.; Metzler, R.; Nonnenmacher, T. F., Phys. Rev. E, 55, 99 (1997)
[196]	Zanette, D. H., Physica A, 252, 159 (1998)
[197]	Chaves, A. S., Phys. Lett. A, 239, 13 (1998) · Zbl 1026.82524
[198]	Metzler, R.; Nonnenmacher, T. F., Phys. Rev. E, 57, 6409 (1998)
[199]	Grigolini, P.; Rocco, A.; West, B. J., Phys. Rev. E, 59, 2603 (1999)
[200]	Metzler, R.; Klafter, J., Physica A, 278, 107 (2000)
[201]	Zavada, T.; Südland, N.; Kimmich, R.; Nonnenmacher, T. F., Phys. Rev. E, 60, 1292 (1999)
[202]	Metzler, R.; Klafter, J.; Sokolov, I., Phys. Rev. E, 58, 1621 (1998)
[203]	Metzler, R.; Barkai, E.; Klafter, J., Physica A, 266, 343 (1999)
[204]	Compte, A.; Càceres, M. O., Phys. Rev. Lett., 81, 3140 (1998)
[205]	Compte, A., Phys. Rev. E, 55, 6821 (1997)
[206]	Compte, A.; Metzler, R.; Camacho, J., Phys. Rev. E, 56, 1445 (1997)
[207]	Metzler, R.; Compte, A., J. Phys. Chem. B, 104, 3858 (2000)
[208]	Meerschaert, M. M.; Benson, D. A.; Baumer, B., Phys. Rev. E, 59, 5026 (1999)
[209]	Anderson, P. L.; Meerschaert, M. M., Water Resour. Res., 34, 2271 (1998)
[210]	Honkonen, J., Phys. Rev. E, 53, 327 (1996)
[211]	Zaslavsky, G. M.; Edelman, M.; Niyazov, B. A., Chaos, 7, 159 (1997) · Zbl 0933.37024
[212]	Metzler, R.; Barkai, E.; Klafter, J., Phys. Rev. Lett., 82, 3563 (1999)
[213]	Metzler, R.; Barkai, E.; Klafter, J., Europhys. Lett., 46, 431 (1999)
[214]	R. Metzler, submitted.; R. Metzler, submitted.
[215]	Barkai, E.; Metzler, R.; Klafter, J., Phys. Rev. E, 61, 132 (2000)
[216]	Fogedby, H. C., Phys. Rev. E, 58, 1690 (1998)
[217]	Jespersen, S.; Metzler, R.; Fogedby, H. C., Phys. Rev. E, 59, 2736 (1999)
[218]	Kolwankar, K. M.; Gangal, A. D., Phys. Rev. Lett., 80, 214 (1998) · Zbl 0945.82005
[219]	Metzler, R.; Klafter, J., J. Phys. Chem. B, 104, 3851 (2000)
[220]	Metzler, R.; Klafter, J., Phys. Rev. E, 61, 6308 (2000)
[221]	Kusnezov, D.; Bulgac, A.; Do Dang, G., Phys. Rev. Lett., 82, 1136 (1999)
[222]	Guyer, R. A., Phys. Rev. A, 29, 2751 (1984)
[223]	Klafter, J.; Zumofen, G.; Blumen, A., J. Phys. A, 25, 4835 (1991)
[224]	Roman, H. E., Phys. Rev. E, 51, 5422 (1995)
[225]	Metzler, R.; Nonnenmacher, T. F., J. Phys. A, 30, 1089 (1997) · Zbl 1001.82538
[226]	Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives - Theory and Applications (1993), Gordon and Breach: Gordon and Breach New York · Zbl 0818.26003
[227]	Bouchaud, J.-P.; Georges, A., Phys. Rev. A, 41, 1156 (1990)
[228]	Klafter, J.; Blumen, A.; Shlesinger, M. F.; Zumofen, G., Phys. Rev. A, 41, 1158 (1990)
[229]	Wolf, K. B., Integral Transforms in Science and Engineering (1979), Plenum Press: Plenum Press New York · Zbl 0409.44001
[230]	Smirnow, W. I., Lehrgang der höheren Mathematik, Vols. III/2 and IV/1 (1988), VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin · Zbl 0997.00510
[231]	Jumarie, G., J. Math. Phys., 33, 3536 (1992) · Zbl 0761.60071
[232]	Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[233]	Ross (Ed.), B., Fractional Calculus and its Applications, Lecture Notes in Mathematics, Vol. 457 (1975), Springer: Springer Berlin · Zbl 0293.00010
[234]	Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[235]	Hilfer (Ed.), R., Applications of Fractional Calculus in Physics (1999), World Scientific: World Scientific Singapore · Zbl 0998.26002
[236]	Abramowitz, M.; Stegun, I., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0543.33001
[237]	Srivastava, H. M.; Gupta, K. C.; Goyal, S. P., The H-Functions of One and Two Variables with Applications (1982), South Asian Publishers: South Asian Publishers New Delhi · Zbl 0506.33007
[238]	Zumofen, G.; Klafter, J., Phys. Rev. E, 47, 851 (1993)
[239]	Klafter, J.; Zumofen, G., J. Phys. Chem., 98, 7366 (1994)
[240]	Fox, C., Trans. Amer. Math. Soc., 98, 395 (1961) · Zbl 0096.30804
[241]	Braaksma, B. L.J., Compos. Math., 15, 239 (1964)
[242]	Mathai, A. M.; Saxena, R. K., The \(H\)-Function with Applications in Statistics and Other Disciplines (1978), Wiley Eastern: Wiley Eastern New Delhi · Zbl 0382.33001
[243]	Srivastava, H. M.; Kashyap, B. R.K., Special Functions in Queuing Theory and Related Stochastic Processes (1982), Academic Press: Academic Press New York · Zbl 0492.60089
[244]	Wolfram, S., The Mathematica Book (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0924.65002
[245]	Barkai, E.; Fleurov, V. N., J. Chem. Phys., 212, 69 (1996)
[246]	Onuki, A., Prog. Theor. Phys., 74, 1155 (1985)
[247]	Morse, P. M.; Feshbach, H., Methods of Theoretical Physics (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.40603
[248]	Jackson, J. D., Classical Electrodynamics (1975), Wiley: Wiley New York · Zbl 0114.42903
[249]	Bluman, G. W.; Cole, J. D., Similarity Methods for Differential Equations (1974), Springer: Springer New York · Zbl 0292.35001
[250]	Damion, R. A.; Packer, K. J., Proc. Roy. Soc. Lond. A, 453, 205 (1997) · Zbl 0870.60080
[251]	M. Köpf, R. Metzler, O. Haferkamp, T.F. Nonnenmacher, in [252]; M. Köpf, R. Metzler, O. Haferkamp, T.F. Nonnenmacher, in [252]
[252]	Losa, G. A.; Merlini, D.; Nonnenmacher, T. F.; Weibel (Eds.), E. R., Fractals in Biology and Medicine, Vol. 2 (1998), Birkhäuser: Birkhäuser Basel
[253]	W.G. Glöckle, T. Mattfeld, T.F. Nonnenmacher, in T.F. Nonnenmacher, G.A. Losa, E.R. Weibl (Eds.), Fractals in Biology and Medicine, Vol. I, Birkhäuser, Basel, 1993.; W.G. Glöckle, T. Mattfeld, T.F. Nonnenmacher, in T.F. Nonnenmacher, G.A. Losa, E.R. Weibl (Eds.), Fractals in Biology and Medicine, Vol. I, Birkhäuser, Basel, 1993. · Zbl 0797.92007
[254]	Nonnenmacher, T. F.; Losa, G. A.; Weibl (Eds.), E. R., Fractals in Biology and Medicine, Vol. II (1993), Birkhäuser: Birkhäuser Basel
[255]	Sokolov, I. M.; Mai, J.; Blumen, A., Phys. Rev. Lett., 79, 857 (1997)
[256]	Geisel, T.; Nierwetberg, J.; Zacherl, A., Phys. Rev. Lett., 54, 616 (1985)
[257]	A.E. Hansen, M.C. Jullien, J. Paret, P. Tabeling, in [53]; A.E. Hansen, M.C. Jullien, J. Paret, P. Tabeling, in [53]
[258]	Chaikovsky, D.; Zaslavsky, G., Chaos, 1, 463 (1991) · Zbl 0902.60086
[259]	Geisel, T.; Zacherl, A.; Radons, G., Z. Phys. B, 71, 117 (1988)
[260]	Zumofen, G.; Klafter, J., Phys. Rev. E, 51, 2805 (1995)
[261]	Drysdale, P. M.; Robinson, P. A., Phys. Rev. E, 58, 5382 (1998)
[262]	Zumofen, G.; Klafter, J., Phys. Rev. E, 51, 1818 (1995)
[263]	Yarmola, E.; Calabrese, P.; Chrambach, A.; Weiss, G. H., J. Phys. Chem., 101, 2381 (1997)
[264]	Moss, F.; McClintock, P. V.E., Noise in Nonlinear Dynamical Systems, Vol. 1, Theory of continuous Fokker-Planck systems (1989), Cambridge University Press: Cambridge University Press Cambridge
[265]	Moss, F.; McClintock, P. V.E., Noise in Nonlinear Dynamical Systems, Vol. 2, Theory of Noise Induced Processes in Special Applications (1989), Cambridge University Press: Cambridge University Press Cambridge
[266]	Moss, F.; McClintock, P. V.E., Noise in Nonlinear Dynamical Systems, Vol. 3, Experiments and Simulations (1989), Cambridge University Press: Cambridge University Press Cambridge
[267]	Barkai, E.; Fleurov, V. N., Phys. Rev. E, 58, 1296 (1998)
[268]	Hilfer, R., Phys. Rev. E, 48, 2466 (1993)
[269]	Hilfer, R., Chaos, Solit. & Fract., 5, 1475 (1995) · Zbl 0907.58039
[270]	Weron, K.; Kotulski, M., Physica, A 232, 180 (1996)
[271]	Cole, K. S.; Cole, R. H., J. Chem. Phys., 9, 341 (1941)
[272]	Barkai, E.; Silbey, R., J. Phys. Chem. B, 104, 3866 (2000)
[273]	Metzler, R.; Klafter, J., Chem. Phys. Lett., 321, 238 (2000)
[274]	Arrhenius, S., Z. Phys. Chem., 4, 226 (1889)
[275]	Weiss, G. H., J. Chem. Phys., 80, 2880 (1984)
[276]	Zwanzig, R., J. Chem. Phys., 97, 3587 (1992)
[277]	A. Blumen, J. Klafter, G. Zumofen in: L. Pietronero, E. Tosati (Eds.), Fractals in Physics.; A. Blumen, J. Klafter, G. Zumofen in: L. Pietronero, E. Tosati (Eds.), Fractals in Physics. · Zbl 0652.92023
[278]	S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1; reprinted in [24]; S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1; reprinted in [24] · Zbl 0061.46403
[279]	M.Ch. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323; reprinted in [24]; M.Ch. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323; reprinted in [24]
[280]	G. Zumofen, J. Klafter, Phys. Rev. E 49 (1995) 4873. L. Kuznetsov, G.M. Zaslavsky, Phys. Rev. E 58 (1998) 7330. S. Benkadda, S. Kassibrakis, R.B. White, G.M. Zaslavsky, Phys. Rev. E 55 (1997) 4909.; G. Zumofen, J. Klafter, Phys. Rev. E 49 (1995) 4873. L. Kuznetsov, G.M. Zaslavsky, Phys. Rev. E 58 (1998) 7330. S. Benkadda, S. Kassibrakis, R.B. White, G.M. Zaslavsky, Phys. Rev. E 55 (1997) 4909.
[281]	Davies, R. W., Phys. Rev., 93, 1169 (1954) · Zbl 0057.35202
[282]	J.W. Strutt, Lord Rayleigh, Phil. Mag. 32 (1891) 424. M.R. Hoare, Adv. Chem. Phys. 20 (1971) 135. J.A. Barker, M.R. Hoare, S. Raval, J. Phys. A: Math. Gen. 14 (1981) 423.; J.W. Strutt, Lord Rayleigh, Phil. Mag. 32 (1891) 424. M.R. Hoare, Adv. Chem. Phys. 20 (1971) 135. J.A. Barker, M.R. Hoare, S. Raval, J. Phys. A: Math. Gen. 14 (1981) 423.
[283]	Cattaneo, G., Atti Sem. Mat. Fis. Univ. Modena, 3, 83 (1948)
[284]	Casas-Vázquez, J.; Jou, D.; Lebon, G., Extended Irreversible Thermodynamics (1996), Springer: Springer Berlin · Zbl 0869.73001
[285]	Compte, A.; Metzler, R., J. Phys. A, 30, 7277 (1997)
[286]	Metzler, R.; Compte, A., Physica, A 268, 454 (1999)
[287]	Shlesinger, M. F.; Klafter, J., J. Phys. Chem., 93, 7023 (1989)
[288]	Lapidus, M. L.; Pomerance, C., Proc. London Math. Soc., 66, 41 (1993) · Zbl 0739.34065
[289]	Lapidus, M. L.; Maier, H., J. London Math. Soc., 52, 15 (1995) · Zbl 0836.11031
[290]	Lapidus, M. L., Fractals, 3, 725 (1995) · Zbl 0870.58063
[291]	Hebert, B.; Sapoval, B.; Russ, S., J. Acc. Soc. Amer., 105, 1567 (1999)
[292]	Haeberle, O.; Sapoval, B.; Menou, K.; Vach, H., Appl. Phys. Lett., 73, 3357 (1998)
[293]	B. Sapoval, in T.F. Nonnenmacher, G.A. Losa, E.R. Weibl (Eds.), Fractals in Biology and Medicine, Vol. II, Birkhäuser, Basel, 1993.; B. Sapoval, in T.F. Nonnenmacher, G.A. Losa, E.R. Weibl (Eds.), Fractals in Biology and Medicine, Vol. II, Birkhäuser, Basel, 1993. · Zbl 0797.92007
[294]	Sabelko, J.; Ervin, J.; Gruebele, M., Proc. Natl. Acad. Sci. U.S.A., 96, 6031 (1999)
[295]	Metzler, R.; Klafter, J.; Jortner, J.; Volk, M., Chem. Phys. Lett., 293, 477 (1998)
[296]	Metzler, R.; Klafter, J.; Jortner, J., Proc. Natl. Acad. Sci. USA, 96, 11085 (1999)
[297]	Kolwankar, K. M.; Gangal, A. D., Chaos, 6, 505 (1996) · Zbl 1055.26504
[298]	Rocco, A.; West, B. J., Physica, A 265, 535 (1999)
[299]	Mittag-Leffler, G., Acta Math., 29, 101 (1905) · JFM 36.0469.02
[300]	Erdélyi (Ed.), A., Tables of Integral Transforms, Bateman Manuscript Project, Vol. I (1954), McGraw-Hill: McGraw-Hill New York · Zbl 0058.34103
[301]	Schneider, W. R.; Albeverio, S.; Casati, G.; Merlini, D., Stochastic Processes in Classical and Quantum Systems, Lecture Notes in Physics, Vol. 262 (1986), Springer: Springer Berlin · Zbl 0598.00012
[302]	Montroll, E. W.; Bendler, J. T., J. Stat. Phys., 34, 129 (1984) · Zbl 0598.60100
[303]	Samorodnitsky, G.; Taqqu, M. S., Stable Non-Gaussian Random Processes (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0925.60027
[304]	Johnson, N.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions (1994), Wiley: Wiley New York · Zbl 0811.62001
[305]	Pólya, G., Math. Ann., 84, 149 (1921) · JFM 48.0603.01
[306]	Suzuki, M., Phys. Lett., 43A, 245 (1973)
[307]	West, B. J., J. Phys. Chem. B, 104, 3830 (2000)
[308]	Rangarajan, G.; Ding, M., Phys. Rev. E, 62, 120 (2000)