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Time-space fabric underlying anomalous diffusion. (English) Zbl 1098.60078
Summary: This study unveils the time-space transforms underlying anomalous diffusion process. Based on this finding, we present the two hypotheses concerning the effect of fractal time-space fabric on physical behaviors and accordingly derive fractional quantum relationships between energy and frequency, momentum and wavenumber which further give rise to fractional Schrödinger equation. As an alternative modeling approach to the standard fractional derivatives, we introduce the concept of the Hausdorff derivative underlying the Hausdorff dimensions of metric spacetime. And in terms of the proposed hypotheses, the Hausdorff derivative is used to derive a linear anomalous transport-diffusion equation underlying anomalous diffusion process. Its Green’s function solution turns out to be a stretched Gaussian distribution and is compared with that from the Richardson’s turbulence diffusion equation.

60J60Diffusion processes
Full Text: DOI arXiv
[1] Gorenflo, R.; Mainardi, F.; Moretti, D.; Pagnini, G.; Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem phys 284, No. 1/2, 521-541 (2002) · Zbl 0986.82037
[2] Herrchen MP. Stochastic modeling of dispersive diffusion by non-Gaussian noise. Ph.D. thesis, Switzerland: ETH, 2000.
[3] Li X. Fractional calculus, fractal geometry, and stochastic processes. Ph.D. thesis, Canada: University of Western Ontario, 2003.
[4] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys rep 339, 1-77 (2000) · Zbl 0984.82032
[5] Scher, H.; Montroll, E. W.: Anomalous transit-time dispersion in amorphous solids. Phys rev B 12, 2455-2477 (1975)
[6] Szabo, T. L.; Wu, J.: A model for longitudinal and shear wave propagation in viscoelastic media. J acoust soc am 107, No. 5, 2437-2446 (2000)
[7] Del-Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E.: Front dynamics in reaction-diffusion systems with Lévy flights: a fractional diffusion approach. Phys rev lett 91, No. 1, 018301-018304 (2003)
[8] Amblard, F.; Maggs, A. C.; Yurke, B.; Pargellis, A. N.; Leibler, S.: Subdiffusion and anomalous local viscoelasticity in acting networks. Phys rev lett 77, No. 21, 4470-4473 (1996)
[9] Sokolov, I. M.; Klafter, J.; Blumen, A.: Ballistic vs. Diffusive pair--dispersion in the Richardson regime. Phys rev E 61, 2717-2722 (2000)
[10] Klappauf, B. G.; Oskay, W. H.; Steck, D. A.; Raizen, M. G.: Observation of atomic momentum transfer in a regime of classical anomalous transport. Phys rev lett 81, 4044-4047 (1998)
[11] Shlesinger, M. F.: Asymptotic solutions of continuous-time random walks. J stat phys 10, 421-434 (1974)
[12] Weiss, G. H.; Rubin, R. J.: Random walks: theory and selected applications. Adv chem phys 52, 363-505 (1983)
[13] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[14] Chen, W.; Holm, S.: Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency. J acoust soc am 115, 1424-1430 (2004)
[15] Saichev, A.; Zaslavsky, G. M.: Fractional kinetic equations: solutions and applications. Chaos, solitons & fractals 7, No. 4, 753-764 (1997) · Zbl 0933.37029
[16] Jespersen, S.; Metzler, R.; Fogedby, H. C.: Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations, and their solutions. Phys rev E 59, 2736-2745 (1999)
[17] Mandelbrot, B. B.: Multifractals and 1 over f noise: wild self-affinity in physics (1963-1976). (1998)
[18] Hoffmann, K.; Essex, C.; Schulzky, C.: J non-equilib thermodyn. 23, 166-175 (1998)
[19] Nottale L. Non-differentiable space-time and scale relativity. In: Flament D, editor. Proceedings of the international colloquium geometrie au XXe siecle, Paris, 24-29 September 2001.
[20] Laskin, N.: Fractional Schrödinger equation. Phys rev E 66, 056108 (2002)
[21] Brockmann, D.; Geisel, T.: Lévy flights in inhomogeneous media. Phys rev lett 90, No. 17, 170601 (2003) · Zbl 1267.82097
[22] Woon MSC. New applications of operators of non-integer order. Ph.D. dissertation, University of Cambridge, 1998.
[23] Kim, K.; Kong, Y. S.: Fractional dynamical behavior in quantum Brownian motion. Bull am phys soc 47, No. 1, 462 (2002)
[24] Weitzner, H.; Zaslavsky, G. M.: Some applications of fractional equations. Commun nonlin sci numer simul 8, No. 3, 273-281 (2003) · Zbl 1041.35073
[25] Mayou, D.: Introduction to the theory of electronic properties of quasicrystals. Lectures on quasicrystals (1994)
[26] Bardou, F.; Bouchaud, J. P.; Aspect, A.; Cohen-Tannoudji, C.: Lévy statistics and laser cooling--how rare events bring atoms to rest. (2002) · Zbl 1058.81800
[27] Goldfain, E.: Fractional dynamics, Cantorian space-time and the gauge hierarchy problem. Chaos solitons & fractals 22, No. 3, 513-520 (2004) · Zbl 1068.81633
[28] Goldfain, E.: Complex dynamics and the high-energy regime of quantum field theory. Int J nonlin sci numer simul 6, No. 3, 223-234 (2005)
[29] Martienssen, W.: Mohamed el naschie and the geometrical interpretation of quantum physics. Chaos, solitons & fractals 25, No. 4, 805-806 (2005) · Zbl 1073.83526
[30] El Naschie, M. S.: A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, solitons & fractals 25, No. 5, 955-964 (2005) · Zbl 1071.81503
[31] El Naschie, M. S.: Einstein in a complex time--some very personal thoughts about E-infinity theory and modern physics. Int J nonlin sci numer simul 6, No. 3, 331-333 (2005)
[32] Feynman, R. P.; Hibbs, A. R.: Quantum mechanics and path integrals. (1965) · Zbl 0176.54902
[33] Abbott, L. F.; Wise, M. B.: Dimension of a quantum-mechanical path. Am J phys 49, 37-39 (1981)
[34] Vlad, M. O.; Metzler, R.; Nonnenmacher, T. F.; Mackey, M. C.: Universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder: dynamical generalizations of stretched exponential. J math phys 37, 2279-2306 (1996) · Zbl 0872.60097
[35] Tschoegl, N. W.: The phenomenological theory of linear viscoelastic behavior. (1989) · Zbl 0681.73022
[36] Lutz, E.: Anomalous diffusion and Tsallis statistics in an optical lattice. Phys rev A 67, 051402 (2003)
[37] Porta, A. L.; Voth, G. A.; Crawford, A. M.; Alexander, J.; Bodenschatz, E.: Fluid particle accelerations in fully developed turbulence. Nature 409, 1017-1019 (2001) · Zbl 1152.76315