By Rainer Klages, Günter Radons, Igor M. Sokolov
This multi-author reference paintings offers a distinct creation to the presently rising, hugely interdisciplinary box of these shipping strategies that can not be defined by utilizing common equipment of statistical mechanics. It comprehensively summarizes issues starting from mathematical foundations of anomalous dynamics to the latest experiments during this box. In so doing, this monograph extracts and emphasizes universal rules and techniques from many alternative disciplines whereas offering updated insurance of this new box of analysis, contemplating such various purposes as plasma physics, glassy fabric, telephone technological know-how, and socio-economic features. The ebook may be of curiosity to either theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic techniques. It additionally varieties an awesome start line for graduate scholars entering into this region. 18 chapters written by way of across the world well-known specialists during this box supply in-depth introductions to primary facets of anomalous shipping.
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Extra info for Anomalous Transport: Foundations and Applications (Wiley 2008)
For this reason the Riemann–Liouville 25 26 2 Threefold Introduction to Fractional Derivatives fractional integrals with limits ±∞, Iα+ f = Iα(−∞)+ f , and Iα− f = Iα∞− f are often called Weyl fractional integrals [24, 85, 94, 99]. 39) for α > 0. 40) is the Heaviside unit step function, and x α = exp α log x with the convention that log x is real for x > 0. 2) in Appendix C. Note that K± loc for α > 0. 4 Riesz Fractional Integrals Riemann–Liouville and Weyl fractional integrals have upper or lower limits of integration, and are sometimes called left-sided resp.
Franc. Phys. 57, 56 (1985). 4 M. Pettini, A. H. Misguich, M. de Leener, J. Orban, and R. Balescu, “Chaotic diffusion across a magnetic ﬁeld in a model of electrostatic turbulent plasma”, Phys. Rev. A 38, 344 (1988). 5 R. Balescu, Transport Processes in Plasma: Vol. 1: Classical Transport. Vol. 2: Neoclassical Transport (North-Holland, Amsterdam, 1988). 11 12 References 6 F. Mainardi, Yu. Luchko, and G. Pagnini, “The fundamental solution of the spacetime fractional diffusion equation”, Fract. Calc.
Under special assumptions this equation can be reduced to a FDE. In the present work we use a quite different approach, based on an analysis of the various types of propagators appearing in the treatment of the HKE. We introduce a nonlocal extension of an approximation similar to the Corrsin factorization assumption of turbulence theory. The result is a nonMarkovian and nonlocal, formally linear equation, in which the rate of change of the density proﬁle n( x, t) at point x and time t is related to the values of this function at neighboring points x + r and at past times t − T.