Colloquium: Fractional calculus view of complexity: A tutorial

The fractional calculus has been part of the mathematics and science literature for 310 years. However, it is only in the past decade or so that it has drawn the attention of mainstream science as a way to describe the dynamics of complex phenomena with long-term memory, spatial heterogeneity, along with nonstationary and nonergodic statistics. The most recent application encompasses complex networks, which require new ways of thinking about the world. Part of the new cognition is provided by the fractional calculus description of temporal and topological complexity. Consequently, this Colloquium is not so much a tutorial on the mathematics of the fractional calculus as it is an exploration of how complex phenomena in the physical, social, and life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions such as differentiability are discarded and the ordinary calculus is replaced with the fractional calculus. Exemplars considered include the fractional differential equations describing the dynamics of viscoelastic materials, turbulence, foraging, and phase transitions in complex social networks.

[1]  D. Chialvo.  Emergent complex neural dynamics , 2010, 1010.2530.

[2]  Stefan Mihalas,et al.  Self-organized criticality occurs in non-conservative neuronal networks during Up states , 2010, Nature physics.

[3]  H. Stanley,et al.  Optimizing the success of random searches , 1999, Nature.

[4]  T. Plutchak,et al.  One, two, three.... , 2001, Bulletin of the Medical Library Association.

[5]  J. Peinke,et al.  On the Statistics of Wind Gusts , 2001, physics/0112063.

[6]  Didier Sornette,et al.  Critical Market Crashes , 2003, cond-mat/0301543.

[7]  G. Parisi,et al.  Scale-free correlations in starling flocks , 2009, Proceedings of the National Academy of Sciences.

[8]  I. Couzin,et al.  Inferring the structure and dynamics of interactions in schooling fish , 2011, Proceedings of the National Academy of Sciences.

[9]  Allan Widom,et al.  Fractal Brownian motion and nuclear spin echoes , 1995 .

[10]  Iu.N. Rabotnov.  Elements of hereditary solid mechanics , 1980 .

[11]  Francesco Mainardi,et al.  The Fractional Langevin Equation: Brownian Motion Revisited , 2008, 0806.1010.

[12]  Clifford T. Brown,et al.  Lévy Flights in Dobe Ju/’hoansi Foraging Patterns , 2007 .

[13]  Filippo Radicchi,et al.  Levy flights in human behavior and cognition , 2013, 1306.6533.

[14]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[15]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[16]  Bruce J. West,et al.  Fractal physiology for physicists: Lévy statistics , 1994 .

[17]  Ralf Metzler,et al.  Lévy strategies in intermittent search processes are advantageous , 2008, Proceedings of the National Academy of Sciences.

[18]  A. M. Edwards,et al.  Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer , 2007, Nature.

[19]  Bruce J. West,et al.  Complexity and synchronization. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  T. Nonnenmacher,et al.  Fractional integral operators and Fox functions in the theory of viscoelasticity , 1991 .

[21]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[22]  Filippo Radicchi,et al.  Evolution of optimal Lévy-flight strategies in human mental searches , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  A. Caspi,et al.  Enhanced diffusion in active intracellular transport. , 2000, Physical review letters.

[24]  I. Podlubny.  Fractional differential equations , 1998 .

[25]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[26]  Filippo Radicchi,et al.  Rationality, Irrationality and Escalating Behavior in Lowest Unique Bid Auctions , 2011, PloS one.

[27]  D. Wacks,et al.  Nonclassical velocity statistics in a turbulent atomic Bose-Einstein condensate. , 2009, Physical review letters.

[28]  H. Stanley,et al.  Introduction to Phase Transitions and Critical Phenomena , 1972 .

[29]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[30]  Tang,et al.  Self-Organized Criticality: An Explanation of 1/f Noise , 2011 .

[31]  L. Richardson,et al.  Atmospheric Diffusion Shown on a Distance-Neighbour Graph , 1926 .

[32]  A. Einstein.  On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heart , 1905 .

[33]  Lineshape theory and photon counting statistics for blinking quantum dots: a Lévy walk process , 2002, cond-mat/0204378.

[34]  Jeffrey S. Guasto,et al.  Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms. , 2009, Physical review letters.

[35]  John R. Anderson,et al.  Reflections of the Environment in Memory Form of the Memory Functions , 2022 .

[36]  Xiaohong Joe Zhou,et al.  Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. , 2008, Journal of magnetic resonance.

[37]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[38]  K. Müllen,et al.  Exponential and Power-Law Kinetics in Single-Molecule Fluorescence Intermittency , 2004 .

[39]  Gopikrishnan,et al.  Economic fluctuations and anomalous diffusion , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[40]  Bruce J. West,et al.  Complex webs : anticipating the improbable , 2010 .

[41]  Bruce J. West,et al.  Linear systems with Lévy fluctuations , 1982 .

[42]  M. Moreau,et al.  Intermittent search strategies , 2011, 1104.0639.

[43]  Theo F. Nonnenmacher,et al.  Fox function representation of non-debye relaxation processes , 1993 .

[44]  Rosario N. Mantegna,et al.  An Introduction to Econophysics: Contents , 1999 .

[45]  E. Montroll,et al.  Anomalous transit-time dispersion in amorphous solids , 1975 .

[46]  Cees van Leeuwen,et al.  Intermittent dynamics underlying the intrinsic fluctuations of the collective synchronization patterns in electrocortical activity. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Kapral,et al.  Periodic dichotomous-noise-induced transitions and stochastic coherence. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[48]  V. E. Lynch,et al.  Critical points and transitions in an electric power transmission model for cascading failure blackouts. , 2002, Chaos.

[49]  Michael T. Turvey,et al.  Human memory retrieval as Lévy foraging , 2007 .

[50]  Non-normal tracer diffusion from stirring by swimming microorganisms , 2012, The European physical journal. E, Soft matter.

[51]  J. Hyde,et al.  Water diffusion heterogeneity index in the human brain is insensitive to the orientation of applied magnetic field gradients , 2006, Magnetic resonance in medicine.

[52]  Nicolas E. Humphries,et al.  Foraging success of biological Lévy flights recorded in situ , 2012, Proceedings of the National Academy of Sciences.

[53]  P. A. Prince,et al.  Lévy flight search patterns of wandering albatrosses , 1996, Nature.

[54]  Christian A. Yates,et al.  Inherent noise can facilitate coherence in collective swarm motion , 2009, Proceedings of the National Academy of Sciences.

[55]  David B. Goldstein,et al.  Genomics: Understanding human diversity , 2005, Nature.

[56]  Bruce J. West,et al.  Fractional Calculus and the Evolution of Fractal Phenomena , 1999 .

[57]  John M. Beggs,et al.  Neuronal Avalanches in Neocortical Circuits , 2003, The Journal of Neuroscience.

[58]  D. Turcotte,et al.  Fractality and Self-Organized Criticality of Wars , 1998 .

[59]  Joachim P. Sturmberg,et al.  Handbook of Systems and Complexity in Health , 2013 .

[60]  Ewald R. Weibel,et al.  Symmorphosis: On Form and Function in Shaping Life , 2000 .

[61]  P. Grigolini,et al.  Fractional calculus as a macroscopic manifestation of randomness , 1999 .

[62]  M Dahan,et al.  Statistical aging and nonergodicity in the fluorescence of single nanocrystals. , 2002, Physical review letters.

[63]  Ercília Sousa.  Numerical solution of a Model for turbulent Diffusion , 2013, Int. J. Bifurc. Chaos.

[64]  J. Doob.  Stochastic processes , 1953 .

[65]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[66]  G. Jug.  Theory of NMR field-gradient spectroscopy for anomalous diffusion in fractal networks , 1986 .

[67]  T F Nonnenmacher,et al.  A fractional calculus approach to self-similar protein dynamics. , 1995, Biophysical journal.

[68]  J. Hyde,et al.  Characterization of continuously distributed cortical water diffusion rates with a stretched‐exponential model , 2003, Magnetic resonance in medicine.

[69]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[70]  Bruce J. West,et al.  Lévy dynamics of enhanced diffusion: Application to turbulence. , 1987, Physical review letters.

[71]  P. Liang,et al.  Spikes with short inter-spike intervals in frog retinal ganglion cells are more correlated with their adjacent neurons’ activities , 2011, Protein & Cell.

[72]  Non-Markov noise in barrier-fluctuation model , 1997 .

[73]  Injong Rhee,et al.  On the levy-walk nature of human mobility , 2011, TNET.

[74]  Bruce J. West,et al.  Fractal dimensionality of Lévy processes. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[75]  D. Chialvo,et al.  Ising-like dynamics in large-scale functional brain networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[76]  D. Sornette.  Discrete scale invariance and complex dimensions , 1997, cond-mat/9707012.

[77]  L. Boltzmann.  Lectures on the Principles of Mechanics , 1974 .

[78]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[79]  Rainer Kimmich,et al.  Strange kinetics, porous media, and NMR , 2002 .

[80]  A. Einstein.  Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [AdP 17, 549 (1905)] , 2005, Annalen der Physik.

[81]  Diffusion entropy and waiting time statistics of hard-x-ray solar flares. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[82]  G. W. Blair,et al.  Limitations of the Newtonian time scale in relation to non-equilibrium rheological states and a theory of quasi-properties , 1947, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[83]  J. Dunkel,et al.  Lévy fluctuations and mixing in dilute suspensions of algae and bacteria , 2010, Journal of The Royal Society Interface.

[84]  A. B. Basset,et al.  Treatise on Hydrodynamics , 1889, Nature.

[85]  Robert Brown,et al.  XXIV. Additional remarks on active molecules , 1829 .

[86]  L. Vinci.  The notebooks of Leonardo da Vinci. , 1952 .