Anomalous Transport: Foundations and Applications by Rainer Klages, Günter Radons, Igor M. Sokolov

By Rainer Klages, Günter Radons, Igor M. Sokolov

This multi-author reference paintings presents a distinct advent to the at present rising, hugely interdisciplinary box of these delivery methods that can't be defined through the use of common tools of statistical mechanics. It comprehensively summarizes subject matters starting from mathematical foundations of anomalous dynamics to the newest experiments during this box. In so doing, this monograph extracts and emphasizes universal ideas and strategies from many various disciplines whereas delivering updated insurance of this new box of analysis, contemplating such varied purposes as plasma physics, glassy fabric, phone technology, and socio-economic aspects.
The e-book can be of curiosity to either theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic techniques. It additionally varieties an amazing start line for graduate scholars stepping into this region. 18 chapters written by way of across the world well-known specialists during this box offer in-depth introductions to basic features of anomalous shipping.

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Plasmas 11, 2272 (2004). 13 R. Balescu, “Anomalous transport in turbulent plasmas and continuous time random walks”, Phys. Rev. E 51, 4807 (1995). 14 R. Balescu, Aspects of Anomalous Transport in Plasmas (IOP, Bristol, 2005). Part I Fractional Calculus and Stochastic Theory 15 Introduction to Part I The opening Part I of this book is devoted to Fractional calculus and stochastic theory. It presents in detail the theoretical and mathematical foundations of the formalism describing anomalous transport, such as fractional differentials and fractional differential equations, and gives a modern outline of random walk approaches to anomalous transport.

In particular, it discussed how the CTRW (Continuous Time Random Walk) formalism was a natural framework to describe this problem, and showed that the probability distribution function of particle displacements can be obtained from the propagator of a space–time fractional diffusion equation. Radu’s response was prompt and enthusiastic; he was “ . . ” Radu’s interest on the application to the CTRW model to study transport in plasmas goes back to 1995 when he proposed for the first time the application of this formalism to study anomalous transport in magnetized plasmas [13].

The formula is known as fractional integration by parts [99]. 61) −∞ holds for Weyl fractional integrals. These formulae provide a second method of generalizing fractional integration to distributions. 62) for a distribution f and a test function ϕ. It shows that right- and left-sided fractional integrals are adjoint operators. The formula may be viewed as a definition of the fractional integral Iαa+ f of a distribution provided that the operator Iαb− maps the test function space into itself. 8 Hardy–Littlewood Theorem The mapping properties of convolutions can be studied with the help of Young’s inequality.

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