Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

We examine the detrended fluctuation analysis (DFA), which is a well-established method for the detection of long-range correlations in time series. We show that deviations from scaling which appear at small time scales become stronger in higher orders of DFA, and suggest a modified DFA method to remove them. The improvement is necessary especially for short records that are affected by non-stationarities. Furthermore, we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of DFA.

Detecting long-range correlations with detrended fluctuation analysis

LeVY flights are a special class of random walks whose step lengths are not constant but rather are chosen from a probability distribution with a power-law tail. Realizations of Levy flights in physical phenomena are very diverse, examples including fluid dynamics, dynamical systems, and micelles1,2. This diversity raises the possibility that Levy flights may be found in biological systems. A decade ago, it was proposed that Levy flights may be observed in the behaviour of foraging ants3. Recently, it was argued that Drosophila might perform Levy flights4, but the hypothesis that foraging animals in natural environments perform Levy flights has not been tested. Here we study the foraging behaviour of the wandering albatross Diomedea exulans, and find a power-law distribution of flight-time intervals. We interpret our finding of temporal scale invariance in terms of a scale-invariant spatial distribution of food on the ocean surface. Finally, we examine the significance of our finding in relation to the basis of scale-invariant phenomena observed in biological systems.

Lévy flight search patterns of wandering albatrosses

The nature of fractals and the use of fractals instead of classical scaling concepts to describe the irregular surfaces, structures, and processes exhibited by physiological systems are described. The mathematical development of fractals is reviewed, and examples of natural fractals are cited. Relationships among power laws, noise, and fractal time signals are examined. >

Fractal Physiology

By preparing this book Chris Barton and Paul La Pointe have earned the gratitude of all geologists and students of fractals. I continue to belong to this second group, and Chris and Paul clearly have put me in a very special debt to them.

E. H. Lloyd

Papers

Long-Term Storage: An Experimental Study.