By Rainer Klages, Günter Radons, Igor M. Sokolov
This multi-author reference paintings presents a distinct advent to the presently rising, hugely interdisciplinary box of these shipping approaches that can't be defined through the use of average equipment 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 alternative disciplines whereas supplying up to date insurance of this new box of analysis, contemplating such different purposes as plasma physics, glassy fabric, telephone technology, and socio-economic elements. The booklet could be of curiosity to either theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic strategies. It additionally kinds a fantastic start line for graduate scholars entering into this sector. 18 chapters written through across the world famous specialists during this box supply in-depth introductions to basic points of anomalous shipping.
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Extra info for Anomalous transport: foundations and applications
During this time, I was thankful to Radu’s kindness, enthusiasm, and support. ” Although with his characteristic modesty Radu referred to them as preliminary, the notes contained not only a valuable summary of fractional calculus, but also a very interesting discussion on the connection between particle dynamics with random velocity (V-Langevin equation), the CTRW model, and fractional diffusion. ” At this point Radu was fully engaged in the study of fractional diffusion. The year was 2006 and I had received an invitation to participate in the summer workshop on “Anomalous Transport: Experimental Results and Theoretical Challenges” in Bad Honnef, Germany.
For the deﬁnition of Lebesgue spaces see Appendix B. If f ∈ L1 ([ a, b]) then (Iαa+ f ) ∈ L1 ([ a, b]) and (Iαa+ f )( x ) is ﬁnite for almost all x. If f ∈ L p ([ a, b]) with 1 ≤ p ≤ ∞ and α > 1/p then (Iαa+ f )( x ) is ﬁnite for all x ∈ [ a, b]. Analogous statements hold for (Iαb− f )( x ) . A short table of Riemann–Liouville fractional integrals is given in Appendix A. For a more extensive list of fractional integrals see . 1 is well suited for fractional integration of power series, but not for functions deﬁned by Fourier series.
52) can be used as a deﬁnition for the convolution of distributions provided that the right-hand side has meaning. This is not always the case as the counterexample K = f = 1 shows. In general, the convolution product is not associative (see Eq. 113)). However, associative and commutative convolution algebras exist . 52) is always meaningful when supp K or supp f is compact . Another case is when K and f have support in R + . This will be assumed in the following. 4 Let f be a distribution f ∈ C0∞ (R ) with supp f ⊂ R + .
Anomalous transport: foundations and applications by Rainer Klages, Günter Radons, Igor M. Sokolov