Fractional Models of Anomalous Transport

Abstract

In the framework of the probabilistic approach to transport in random flows, the probability density $\rho (\vec{r},t)$ plays a central role. This is the probability to find a random walker at time $t$ at distance $r$ from its starting point. In a random system, $\rho (\vec{r},t)$ contains information on both static disorder and the dynamical process. In homogenous systems, the probability density is Gaussian and does not depend on the configuration considered

$$ \rho (\vec{r},t) = \frac{{\delta {r^d}}}{{{{(4\pi {D_0}t)}^{d/2}}}}\exp \left( { - \frac{{{r^2}}}{{4{D_0}t}}} \right). $$

(11.1.1)

Further Reading

1.1 Levy Flights

• R. Balescu, Statistical Dynamics (Imperial College, London, 1997)

• R. Botet, M. Poszajczak, Universal Fluctuations (World Scientific, Singapore, 2002)

• G.P. Bouchaud, A. Georges, Physics Reports 195(132–292), 1990 (1990)

• J. Bricmont et al., Probabilities in Physics (Springer, Berlin, 2001)

• R. Klages, G. Radons, I. Sokolov (eds.), Anomalous Transport, Foundations and Applications. (Wiley, New York, 2008)

• A. Pekalski, K. Sznajd-Weron (eds.), Anomalous Diffusion. From Basics to Applications (Springer, Berlin, 1999)

• M.F. Shiesinger, G.M. Zaslavsky, Levy Flights and Related Topics in Physics (Springer, Berlin, 1995)

• G.M. Zaslavsky, Physics Reports 371, 461–580 (2002)

• Ya.B. Zeldovich, A.A. Ruzmaikin, D.D. Sokoloff, The Almighty Chance (World Scientific, Singapore, 1990)

1.2 Continuous Time Random Walk and Scaling

• D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 1996)

• J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987)

• R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

• E.W. Montroll, M.F. Shlesinger, On the Wonderful World of Random Walks. Studies in Statistical mechanics, vol. 11 (Elsevier, Amsterdam, 1984), p. 1

• E.W. Montroll, B.J. West, On an Enriches Collection of Stochastic Processes, in Fluctuation Phenomena, ed. by E.W. Montroll, J.L. Lebowitz (Elsevier, Amsterdam, 1979)

• V.V. Uchaikin, V.M. Zolotarev, Chance and Stability Stable Distributions and Their Applications (VSP, Utrecht, 1999)

1.3 Fractal Operators

• K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition (Springer, Berlin, 2010)

• L. Pietronero, Fractals' Physical Origin and Properties (Plenum, New York, 1988)

• D. Sornette, Critical Phenomena in Natural Sciences (Springer, Berlin, 2006)

• B.J. West, M. Bologna, P. Grigolini, Physics of Fractal Operators (Springer, New York, 2003)

• G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford, 2005)

1.4 Vortex Structures and Trapping

• A. Crisanti, M. Falcioni, A. Vulpiani, Rivista Del Nuovo Cimento 14, 1–80 (1991)

• E. Guyon, J.-P. Nadal, Y. Pomeau (eds.), Disorder and Mixing (Kluwer, Dordrecht, 1988)

• P.J. Holmes, J.L. Lumley, G. Berkooz, J.C. Mattingly, R.W. Wittenberg, Physics Reports 287, 337–384 (1997)

• A. Maurel, P. Petitjeans (eds.), Vortex Structure and Dynamics Workshop (Springer, Berlin, 2000)