Self-similar anomalous diffusion and Levy-stable laws

Stochastic principles for constructing the process of anomalous diffusion are considered, and corresponding models of random processes are reviewed. The self-similarity and the independent-increments principles are used to extend the notion of diffusion process to the class of Levy-stable processes. Replacing the independent-increments principle with the renewal principle allows us to take the next step in generalizing the notion of diffusion, which results in fractional-order partial space-time differential equations of diffusion. Fundamental solutions to these equations are represented in terms of stable laws, and their relationship to the fractality and memory of the medium is discussed. A new class of distributions, called fractional stable distributions, is introduced.

PACS: 02.50.−r, 05.40.Fb, 05.40.Jc (all)
DOI: 10.1070/PU2003v046n08ABEH001324
URL: https://ufn.ru/en/articles/2003/8/c/
Web of Science

000187205200003
Citation: Uchaikin V V "Self-similar anomalous diffusion and Levy-stable laws" Phys. Usp. 46 821–849 (2003)

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Оригинал: Учайкин В В «Автомодельная аномальная диффузия и устойчивые законы» УФН 173 847–876 (2003); DOI: 10.3367/UFNr.0173.200308c.0847

References (76) Cited by (155) Similar articles (20) ↓

L.M. Zelenyi, A.V. Milovanov “Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics” Phys. Usp. 47 749–788 (2004)
R.T. Sibatov, V.V. Uchaikin “Fractional differential approach to dispersive transport in semiconductors” Phys. Usp. 52 1019–1043 (2009)
A.I. Olemskoi, A.Ya. Flat “Application of fractals in condensed-matter physics” Phys. Usp. 36 (12) 1087–1128 (1993)
V.V. Uchaikin “Fractional phenomenology of cosmic ray anomalous diffusion” Phys. Usp. 56 1074–1119 (2013)
V.V. Slezov, V.V. Sagalovich “Diffusive decomposition of solid solutions” Sov. Phys. Usp. 30 23–45 (1987)
A.V. Gapontsev, V.V. Kondrat’ev “Hydrogen diffusion in disordered metals and alloys” Phys. Usp. 46 1077–1098 (2003)
E.G. Berezhko, G.F. Krymskii “Acceleration of cosmic rays by shock waves” Sov. Phys. Usp. 31 27–51 (1988)
V.V. Zosimov, L.M. Lyamshev “Fractals in wave processes” Phys. Usp. 38 347–384 (1995)
V.V. Uchaikin, A.D. Erlykin, R.T. Sibatov “Nonlocal (fractional-differential) model of cosmic ray transport in the interstellar medium” Phys. Usp. 66 221–262 (2023)
E.G. Berezhko, G.F. Krymskii “Acceleration of cosmic rays by shock waves” Sov. Phys. Usp. 31 27–51 (1988)
V.P. Budaev, S.P. Savin, L.M. Zelenyi “Investigation of intermittency and generalized self-similarity of turbulent boundary layers in laboratory and magnetospheric plasmas: towards a quantitative definition of plasma transport features” Phys. Usp. 54 875–918 (2011)
G.M. Zaslavskii, B.V. Chirikov “Stochastic instability of non-linear oscillations” Sov. Phys. Usp. 14 549–568 (1972)
I.L. Fabelinskii “Macroscopic and molecular shear viscosity” Phys. Usp. 40 689–700 (1997)
K.P. Zybin, V.A. Sirota “Stretching vortex filaments model and the grounds of statistical theory of turbulence” Phys. Usp. 58 556–573 (2015)
V.I. Klyatskin “Clustering and diffusion of particles and passive tracer density in random hydrodynamic flows” Phys. Usp. 46 667–688 (2003)
V.I. Klyatskin, V.I. Tatarskii “Diffusive random process approximation in certain nonstationary statistical problems of physics” Sov. Phys. Usp. 16 494–511 (1974)
V.I. Tikhonov “Peaks of random processes” Sov. Phys. Usp. 5 594–611 (1963)
O.G. Bakunin “Stochastic instability and turbulent transport. Characteristic scales, increments, diffusion coefficients” Phys. Usp. 58 252–285 (2015)
V.S. Anishchenko, T.E. Vadivasova et al “Statistical properties of dynamical chaos” Phys. Usp. 48 151–166 (2005)
A. Loskutov “Fascination of chaos” Phys. Usp. 53 1257–1280 (2010)

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