By Rainer Klages, Günter Radons, Igor M. Sokolov
This multi-author reference paintings presents a distinct creation to the at present rising, hugely interdisciplinary box of these delivery approaches that can not be defined by utilizing regular tools of statistical mechanics. It comprehensively summarizes themes 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 strategies from many alternative disciplines whereas delivering up to date insurance of this new box of analysis, contemplating such assorted functions as plasma physics, glassy fabric, cellphone technological know-how, and socio-economic facets. The booklet may be of curiosity to either theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic techniques. It additionally kinds an excellent place to begin for graduate scholars stepping into this sector. 18 chapters written by means of the world over well-known specialists during this box offer in-depth introductions to primary elements of anomalous shipping.
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Extra resources for Anomalous Transport: Foundations and Applications (Wiley 2008)
If K ∈ L p (R ) and f ∈ Lq (R ) then K ∗ f ∈ Lr (R ) and Young’s inequality K ∗ f r ≤ K p f q holds. 2 Mathematical Introduction to Fractional Derivatives 1 ≤ p ≤ q ≤ ∞ and K ∈ Lr (R ) with 1/r = 1 + (1/q) − (1/p). The Hardy– α although Littlewood theorem states that these estimates remain valid for K± p these kernels do not belong to any L (R )-space [37, 38]. The theorem was generalized to higher dimensions by Sobolev in 1938, and is also known as the Hardy–Littlewood–Sobolev inequality (see [37, 38, 63, 113]).
29a) a for x > a. 29b) x for x < b. 30) completes the deﬁnition. The deﬁnition may be generalized to α ∈ C with Re α > 0. 29a) appears in [96, p. 363] with a > −∞ and in [76, p. 8] with a = −∞. The notation is not standardized. Leibniz, Lagrange, and Liouville α α x [22, 73, 76], Grünwald wrote [· · · dx α ] xx= used the symbol = a , while Rieα −α −α . 29) is mann used ∂− x  and Most wrote da /dx that of [52, 54, 98, 99]. 2 a D x [85, 91, 102], or d /d( x − a ) a+ The fractional integral operators Iαa+ , Iαb− are commonly called Riemann– Liouville fractional integrals [94, 98, 99] although sometimes this name is reserved for the case a = 0 .
This conﬂuence of common interests triggered a very intense collaboration between Radu Balescu and us during the last 2 years of his life. We started exchanging numerous e-mails with notes, drafts of papers, calculations, comments, questions, doubts, criticisms, and what not. We discussed details of what, to the best of my knowledge, was the last physics problem Radu Balescu attacked: the derivation of superdiffusive fractional differential equations (FDEs) from what he called the V-Langevin kinetic equation .
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