Fractional calculus view of complexity: A tutorial - Physical Review ...

REVIEWS OF MODERN PHYSICS, VOLUME 86, OCTOBER–DECEMBER 2014

Colloquium: Fractional calculus view of complexity: A tutorial Bruce J. West* Mathematics and Information Sciences Directorate, U.S. Army Research Office, Research Triangle Park, North Carolina, 27709, USA

(published 9 October 2014) 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. DOI: 10.1103/RevModPhys.86.1169

PACS numbers: 01.30.Rr, 02.50.−r, 64.60.aq, 47.27.E−

necessary and sufficient to provide a mechanical description of the physical world. On the other hand, experiment indicates that a broad range of physical, biological, and social phenomena cannot be understood using the analytic functions we have come to rely on in physics. These functions do not capture the complex dynamics of such common physical phenomena as earthquakes and hurricanes (Sornette, 2003); everyday social phenomena including group consensus (Turalska et al., 2009), market crashes (Mantegna and Stanley, 2000), and healthcare networks (Sturmberg and Martin, 2013); or the familiar psychological activity of cognition and habituation (West and Grigolini, 2010). The inherent complexity of these phenomena is beyond the scope of the familiar 19th century analysis that forms the mathematical foundation of physics and engineering since that time. Understanding complexity as an extended class of problems with common properties requires a new way of modeling and consequently more innovative ways of thinking. Phenomena that require the notion of noninteger derivatives and integrals were believed to be interesting curiosities outside mainstream science. However, the increased sensitivity of experimental tools, enhanced data processing techniques, and ever increasing computational capabilities have all contributed to the expansion of science in such a way that those phenomena once thought to be outliers are now center stage. These curious processes are now described as exotic scaling phenomena, but as we discuss in this Colloquium in order to form a basic understanding of them requires a new mathematical perspective, one which might well be provided by the fractional calculus. An apparently different strategy, that of network science, has recently been used to model physical, social, and life science phenomena in part because the complexity of phenomena in these disciplines is manifest as emergent from the network dynamics. Complexity may be broadly partitioned

CONTENTS I. Introduction A. New ways of thinking B. Dynamics and fractals 1. Weierstrass function 2. Nondifferentiablity can be physical II. Fractional Dynamics A. Fractional difference equation B. Some fractional operators C. Fractional rate equations 1. Distribution of rates 2. Viscoelastic material experiments 3. New fractional Brownian motion D. Subordination and networks 1. Fractional Langevin equation 2. The average person III. Fractional Diffusion and Probability A. Fractional turbulence B. Fractional Bloch equation C. Lévy foraging D. Phase space fractional equations IV. Conclusion Acknowledgments References

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I. INTRODUCTION

The implementation of the fractional calculus in the physical, social, and life sciences has languished in part because until recently the larger scientific community did not acknowledge a need for it. The calculus of Newton and the analytic functions that solve the differential equations resulting from his force laws have historically been seen as *

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© 2014 American Physical Society

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Bruce J. West: Colloquium: Fractional calculus view of …

into topology that relates to the ways in which the elements of a network are interconnected in a scale-free distribution, and chronology that relates to the timing of significant events within network dynamics that also scale. The scaling topology of networks has been widely adopted by the network science community as a measure of complexity and is the topic of discussion in texts (Newman, 2010; West and Grigolini, 2011), whereas temporal complexity has only recently been identified as an important measure of complex network dynamics (Turalska et al., 2009). The scaling observed in numerical calculations of large dynamic complex networks will be shown to dovetail with the fractional calculus. Exemplars of complex phenomena used in this tutorial to highlight various aspects of the fractional calculus are homogeneous turbulence and foraging, with their non-Gaussian statistics in time; the dynamics of viscoelastic materials, with their non-Newtonian dynamical properties; and the phase transitions in complex social networks. The non-Gaussian statistics, non-Newtonian dynamics, and phase transitions are each in their different ways connected to the fractional calculus. A. New ways of thinking

This Colloquium focuses on the different ways the fractional calculus prompts us to think about complexity in physical phenomena and elsewhere. We explore the more obvious reasons why the traditional calculus including differential equations is not sufficient to capture the full range of dynamics found in natural and man-made processes and events. Specifically, the complexity of nonlinear dynamic phenomena demands that we extend our horizons beyond analytic functions and analysis suggests that the functions of interest lack traditional equations of motion. To explore this lack we introduce fractional thinking, which is a kind of inbetween thinking; between the integer-order moments, such as the mean and variance, there are fractional moments required when empirical integer moments fail to converge; between the integer dimensions there are the fractal dimensions that are important when data have no characteristic scale length; and between the integer-valued operators that are local in space and time, are the noninteger operators necessary to describe dynamics that have long-time memory and spatial heterogeneity. Complex phenomena require new ways of thinking and the fractional calculus provides one framework for that thinking (West, Bologna, and Grigolini, 2003). Recall that Alice on the other side of the looking glass was confused by the apparent lack of rules governing the world in which she found herself (see Fig. 1). Things that were relatively simple back home seemed unnecessarily complicated in Wonderland. However, after some time she understood that rules did exist; they were just different from those that determined the world she had left behind. Much like Wonderland it is not that the quantitative reasoning discussed in this Colloquium does not have rules; it is just that the rules of quantification are very different from those with which most of us are familiar. This Colloquium is about how the rules have changed and what that change implies about the phenomena they describe and how we are to understand them. Why is the fractional calculus entailed by complexity? The short answer is that the fractional calculus is probably not Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

FIG. 1. In this classic view of the tea party, when the Mad

Hatter, White Rabbit, and Door Mouse discuss a fractal dimensional Seripinski gasket, Alice sees a Euclidean cube.

required to understand any specific instance of complexity. However, we interpret this lack of uniqueness in the same way we do that of Newton not using his then newly formulated fluxions in his discussion of mechanical motion in the Principia. He confined his often remarkable arguments to geometry, the mathematical and scientific language of the day, but eventually it was seen by subsequent generations of scientists that those arguments lent themselves more readily to the calculus. In an analogous way I believe that many of the complex phenomena that have required often tortuous explanations using traditional methods will eventually be shown to be more naturally described using the fractional calculus. This is a statement of a research vision that has been only partially realized (Magin, 2006; West, Turalska, and Grigolini, 2014). We leave complexity explicitly undefined and consider phenomena or structures to be complex when traditional analytic functions are not able to capture their richness in space and/or time. It was believed for a long time that physical theories such as classical mechanics could be used to describe with absolute certainty the dynamics of highly idealized systems using limiting concepts abstracted from the real world. However, that particular bubble was burst by Poincaré (1888), even though it took over half a century for his discovery of the significance of nonlinear dynamics and the nonpredictability of nonintegrable Hamiltonian systems to penetrate the insulation of mainstream physics. Historically it was statistical physics that restored to the mechanical description of complex systems the uncertainty observed in actual measurements and the associated probability density function (PDF) as a measure of that uncertainty. Probability theory provided the first universally accepted systematic treatment of physical complexity and was the mathematical foundation of kinetic theory. Ludwig Boltzmann, one of the architects of the kinetic theory of gases, spent a great deal of time thinking about the discontinuous changes in particle velocity that occur in collisions and wondering about their proper mathematical representation (Boltzmann, 1904). He believed that such microscopic dynamics should be described by continuous


but nondifferentiable functions such as the one developed by Weierstrass and which is used as an exemplar below. This theme of using continuous nonanalytic functions to describe complex physical processes was taken up by Jean Perrin who received the Noble Prize in Physics for his diffusion experiments determining Avogadro’s number (Perrin, 1913). Perrin stated that curves without tangents (derivatives) are more common in the physical world than those special ones like the circle that have tangents. He was adamant in his arguments emphasizing the importance of nonanalytic functions to describe complex physical phenomena such as molecular diffusion (Perrin, 1913). The backbone of equilibrium statistical mechanics is the Boltzmann distribution, all of whose moments are finite. However, empirical probability distributions are often inverse power law with diverging first and second moments. For example, in complex networks the time between events is often given to a good approximation over some domain by the waiting-time distribution function ψðtÞ ∝ 1=tμ rather than the more familiar exponential form for a Poisson process. Empirical exemplars of such complex phenomena are the time intervals between earthquakes of a given magnitude (Omori, 1894), the waiting time between solar flares (Grigolini, Leddon, and Scafetta, 2002), the time from one breath to another (Szeto et al., 1992), the interevent times in electroencephalogram (Gong, Nikolaev, and van Leeuwen, 2007), and the decrease in human memory with time (Anderson and Schooler, 1991). In many, if not all, such empirical distributions the power-law index is μ < 2 and consequently the average time between events, the first moment, diverges. Consequently, there is no characteristic time scale for the process and the traditional assumption that time averages and ensemble averages coincide is violated, that is, the statistics in such complex phenomena are nonergodic. Figure 2 displays the probability densities for the size of neuronal quakes (number of neurons involved in the discharge event) as first observed by Beggs and Plenz (2003) and the time interval between neuronal spikes. These are exemplars of topological complexity and temporal complexity associated with the same physiologic phenomenon. However, they are also representative of the two types of complexity evident in complex networks in other contexts.

Left: Probability of a given number of neurons involved in a neuronal avalanche. From Millman et al., 2010. Right: The time interval between neuronal spikes. From Liu et al., 2011. Both are inverse power laws. Note that such “laws” are valid over a limited domain and the mechanism by which they are truncated is often of interest. FIG. 2 (color online).

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B. Dynamics and fractals

The first scientific application of the fractional calculus facilitated the understanding of the dynamics of viscoelastic materials such as taffy. The equations of motion for such materials fall outside traditional analytic dynamics and fluid dynamics because the material properties are neither those of solids nor fluids (Rabotnov, 1977). Historically the viscoelastic equations of motion are of an integro-differential form that were shown to have an equivalent interpretation in terms of the fractional calculus (Scott Blair, Veinoglou, and Caffyn, 1947; Glöckle and Nonnenmacher, 1991). To understand the reason for this equivalence we examine the notion of dimension. Over two millennia ago Euclid organized the understanding of structure within the physical world into classical geometry, giving us the metrics of points, lines, planes, and other surfaces. Two millennia later Mandelbrot (1977) pointed out that “the emperor had no clothes,” which is to say that lightning does not move in straight lines, clouds are not spheres, and most physical phenomena violate the underlying assumptions of Euclidean geometry. Pursuing his observation to their logical conclusions Mandelbrot introduced the idea of fractional or fractal dimension into the scientific lexicon and proceeded to catalog the myriad of physical, social, and biological phenomena that ought to be described by his fractal geometry and fractal statistics. So what is a fractal dimension? Technically the dimension of an object is determined by how it is measured or more pedantically how it is covered. For example, given a ruler of unit length η the length of a curve L can be determined by laying the ruler end to end N times to obtain L ¼ Nη. If L ¼ 1 then N ¼ 1=η. The area A of a surface can be determined by placing a unit of area η2 over the surface N times until it is completely covered to obtain A ¼ Nη2 . If A ¼ 1 then N ¼ 1=η2 . The procedure is clear: to determine the “covering” of an object of dimension D, N unit intervals ηD are required. Consequently, the dimension of an object of dimension D is defined by D ¼ log N= log ð1=ηÞ in the limit η → 0. The limit is taken so that concepts like the length of a curve or the area of a surface are independent of the measuring instrument and are therefore objective. Note, however, that this definition does not require that the dimension be integer. In fact, when the dimension D is not integer more and more structure emerges as η → 0 and the traditional measure of length and area cease to have meaning. The existence of noninteger or fractal dimensions led to the development of fractal geometry and fractal statistics. The decades of the 1980s and 1990s witnessed an explosion of applications of fractal geometry in disconnected fields of study from statistical physics (Meakin, 1998) to the convoluted foldings of the surface of the brain and the branching network of the mammalian lung (Weibel, 2000), to the growth patterns of cities (Batty and Longley, 1994), and to intermittent search strategies (Bénichou et al., 2011). In these and many other studies it became apparent that not just static structures are fractal, but the dynamics of complex phenomena are fractal as well, including their statistical fluctuations (West and Grigolini, 2011). We consider how fractals are tied to the fractional calculus and to do this requires a brief digression into the scaling properties of fractals.


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Fractals give rise to a number of scientific curiosities. For example, it is possible to construct a continuous line connecting two points a finite Euclidean distance apart and show that line to be infinitely long. Such a fractal curve has the property of being self-similar at all scales, that is, in the neighborhood of any point along the curve there is variability. In fact, the curve has such infinite variability that it is not possible to draw a tangent to the curve at any point. As the limit is taken in the vicinity of any point more and more structure of the curve is revealed and the derivative of the curve becomes ill defined as anticipated by Perrin (1913) in his study of diffusion; see Mandelbrot (1977) and Feder (1988) for a complete inventory of fractal properties. 1. Weierstrass function

The 19th century mathematician Weierstrass was able to construct a Fourier series that had the property of being continuous everywhere but being nowhere differentiable. A century later Mandelbrot (1977) generalized the Weierstrass function (GWF) to the form WðtÞ ≡

∞ X 1 n n ½1 − cos ðb ω0 tÞ a n¼−∞

ð1Þ

with the parameter values b > a > 1. The continuous curve depicted in Fig. 3 is generated by the GWF. It is straightforward to show that multiplying time in the GWF by the parameter b results in the renormalization group equation WðbtÞ ¼ aWðtÞ obtained by reindexing terms in the infinite series. The solution to the renormalization group equation is of the scaling form (Stanley, 1979) WðtÞ ¼ AðtÞtμ ;

μ ¼ log a= log b;

the time derivative of the GWF can be shown to diverge for b > a (Mandelbrot, 1977). Consequently, this function is not the solution to any traditional equation of motion and therefore was considered to be a mathematical curiosity and not to describe any “real” time-dependent process. 2. Nondifferentiablity can be physical

The potential physical significance of the Weierstrass function was first recognized by Richardson (1926) who had measured the increasing span of plumes of smoke ejected from chimneys driven by fluctuating wind fields; see Fig. 4. From his observations he speculated that the turbulent air speed, which was known to be nondifferentiable, could be characterized by a Weierstrass function. This was motivated in part by the observation that the span of the plume increased as tβ with β ≥ 3, a value inconsistent with molecular diffusion for which β ¼ 1. Half a century later Mandelbrot established that turbulent velocity fields are fractal statistical processes and that the eddy cascade model of turbulence invented by Kolmogorov (1941) was, in fact, a dynamic fractal so that turbulence has no characteristic space or time scale. The details of the fractal description of turbulence are of interest because this story has been repeated again and again in multiple disciplines over the past 40 years, not the specifics but the notion that complex phenomenon are self-similar and therefore do not possess characteristic scales in space and/or time. This recognition of self-similarity in turn led to the reinterpretation of existing data sets in terms of geometric, dynamic, or statistical fractal processes. The crucial feature of the process is the scaling of the data, which appears in multiple forms, reaching back 500 years to Leonardo da Vinci’s branching relation for the limbs of trees and the tributaries of streams (Richter, 1970), jumping forward

ð2Þ

where AðtÞ is a periodic function in log t with period log b. The scaling in Eq. (2) is the analytic manifestation of the selfsimilarity property of the GWF. It is clear from the figure that if any segment of the curve is magnified the entire curve is again revealed due to the property of self-similarity. Finally

The curve is a graph of the GWF using the series Eq. (1). The function depicted by the curve obeys a scaling relation (2) characterized by a fractional dimension.

FIG. 3 (color online).


FIG. 4 (color online). The smoke plume from chimneys is seen to expand with distance from the source. Richardson (1926) speculated that this could be described by a Weierstrass function since the turbulent velocity field of the wind was not differentiable.


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400 years to the distribution of income by Pareto (1897), up to the contemporary probability of the occurrence of wars of a given size (Roberts and Turcotte, 1998) and the scale-free character of complex networks (Albert and Barabási, 2002). The oscillatory coefficient in the scaling solution of the GWF given in Eq. (2) has been shown to be a consequence of the underlying process having a complex dimension (Shlesinger and West, 1991; Sornette, 1998). Such complex dimensions have been observed in the architecture of the human lung (West, Bargarva, and Goldberger, 1986) as well as in earthquakes, turbulence, and financial crashes (West and Grigolini, 2011). The most important point here is that once a scale-free process has been identified as being described by a fractal function we know that the traditional calculus will not be available to determine the dynamics of the process the fractal function represents because the integer derivatives of fractal functions diverge. This is where fractional operators enter the story. II. FRACTIONAL DYNAMICS

Calculus is the formal method by which we carry out quantitative reasoning. In complex phenomena, particularly those that change over time, the traditional guideposts such as identifying causality and making predictions become problematic. One way to anticipate the improbable and prepare for unintended consequences is through the systematic handling of fractal operators using the fractional calculus and fractional differential equations. But we also need to incorporate the more traditional notions of complexity such as randomness into the discussion. Therefore we start our tour with random walks using fractional difference equations. A. Fractional difference equation

Most physical scientists are introduced to statistical processes through the study of diffusion or radioactive decay. Here we consider how we might generalize both. One way to think differently about scale-free processes is to replace integer difference equations with fractional difference equations. We do this for a discrete random walk and determine the meaning of a fractional difference. This exercise provides the first insight into fractional thinking. A familiar derivation of the Gaussian distribution starts from a simple random walk process as first recognized by Lord Rayleigh (1905), the same year Einstein explained molecular diffusion (Einstein, 1905). The random walk argument involves updating the displacement of a walker, as depicted in Fig. 5, at step n given by Qn by adding a discrete random number ξn to obtain Qnþ1 ¼ Qn þ ξn . Introducing the downshift operator B the random walk equation can be written ð1 − BÞQnþ1 ¼ ξn . The total displacement after N steps is QN ¼ ξ1 þ ξ2 þ þ ξN and the statistical properties are known to be given by a Gaussian distribution with a variance that increases linearly with the number of steps, that is, linearly in time (Einstein, 1905; Rayleigh, 1905). Hosking (1981), who was interested in economic time series, generalized the simple random walk to the fractional difference equation Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

FIG. 5. Cartoon of a two-dimensional random walk or drunkard’s walk. From Gamow, 1955.

ð1 − BÞα Qnþ1 ¼ ξn ;

ð3Þ

where the index α is a noninteger. He was able to establish that the operator on the left side of the equation has an inverse given by a binomial expansion to obtain for the total displacement of the walker after n steps: Qn ¼

∞ X

Γðk þ αÞ ξ : Γðk þ 1ÞΓðαÞ n−k k¼0

ð4Þ

Consequently, the total displacement after n steps is influenced by random impulses stretching infinitely far back in time with their relative impact determined by the ratio of gamma functions. As the step index (discrete time) k becomes large the ratio of gamma functions becomes proportional to kα−1 , which is an inverse power law since jαj < 1=2 in the analysis. The statistics of the displacement remain Gaussian in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the continuous limit exp ½−x2 =σ 2 ðtÞ= 4πσ 2 ðtÞ, because the relation of the total displacement to the random impulses is linear but the variance σ 2 ðtÞ no longer increases linearly in time but rather as an inverse power law t2α−1. This is our first indication of the connection between fractional dynamics and temporal complexity, albeit the dynamics are discrete. The fractional random walk generates a random process with memory. A new random walk ð1 − BÞXnþ1 ¼ Qn uses the above process to generate random impulses with memory. In this case the walk has a solution that in the continuum limit has the stationary autocorrelation function CðτÞ ¼ hXðt þ τÞXðtÞi ∝ τ2H . The scaling exponent is often called the Hurst exponent after Mandelbrot (1977) associated it with the civil engineer who first used this scaling in the study of time series. An extended discussion of the Hurst exponent and the properties of the underlying processes are found in


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Feder’s book (Feder, 1988) on fractals. In particular, random walks with H > 1=2 have a positive correlation, with long (short) intervals more likely to be followed by long (short) intervals and with H < 1=2 a negative correlation, with long (short) intervals more likely to be followed by short (long) intervals. The power-law index is given by H ¼ α þ 1=2, so that since 0 ≤ H ≤ 1, the fractional index lies in the interval −1=2 ≤ α ≤ 1=2. This new process has an inverse powerlaw spectrum. We stress that when the Hurst exponent H ≠ 1=2 the system has long-time memory as measured by the autocorrelation function. Such memory has been observed consistently in the financial time series since Hosking first introduced fractional difference equations into the study of the economic time series in 1981. In the physical sciences the generalization of simple random walks to complex phenomena was made by Montroll and Weiss (1965). They argued that in complex materials individual lattices sites have complicated structures resulting in a walker waiting at such sites for time intervals of random length before taking steps of random size. The transition PDF for taking fluctuating steps in space and time was incorporated into an integral equation for the PDF for the total displacement of the particle in a given time resulting in the continuous time random walk (CTRW) model. This model is important for the present discussion because it provided the first systematic theory that connected the local statistics of the random walker to an integral equation that under certain general conditions reduces to fractional derivatives in both space and time. The resulting fractional diffusion equation provided the first explanation of anomalous diffusion (Seshadri and West, 1982; Montroll and West, 1987). The simple random walk process has a continuous analog given by a Langevin equation (Langevin, 1908) that provided the first stochastic model of particle dynamics. Analogously the discrete fractional random walk has a continuous fractional Langevin equation. Here the connection between temporal complexity and the fractional derivative in time should be evident. Before exploring the fractional generalization of the dynamics we review some of the more straightforward formal properties of the fractional calculus. B. Some fractional operators

The question of what is a fractional derivative was first raised by de l’Hôpital in a letter to Leibniz in 1695. He asked what one would obtain from the nth order derivative of a function when n is not an integer, in particular, when n ¼ 1=2. Before giving the algebraic result of such an operation Leibniz responded with what has become one of the most remarkable understatements of all time: “It will lead to a paradox from which one day useful consequences will be drawn.” Starting from the properties of gamma functions extended into the complex plane it is possible to write the derivative of an algebraic function as follows: dα β Γðβ þ 1Þ β−α t : ½t ¼ dtα Γðβ þ 1 − αÞ

ð5Þ

This is not a particularly surprising equation until we stipulate that the order of the derivative α is not an integer. Consider the Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

case of interest to de l’Hôpital α ¼ 1=2 for various values of β: d1=2 −1=2 ½t ¼ 0; dt1=2

d1=2 1 ½1 ¼ pffiffiffiffiffi ; 1=2 dt πt

d1=2 ½t ¼ dt1=2

rffiffiffi t : π ð6Þ

The result on the left is strange and is a consequence of the divergence of the gamma function in the denominator in Eq. (5). The middle result is perhaps even stranger yielding a time-dependent function for the fractional derivative of a constant. Of course, neither of these curious findings is consistent with the ordinary calculus and has to do with the nonlocal nature of fractional derivatives. Finally, the expression on the right is Leibniz’s answer to the de l’Hôpital question. These three fractional derivatives alert us to the fact that we have entered a world in which the rules for quantitative analysis are different from what we have always believed, but they are not arbitrary. It remains to be seen if this mathematical world can explain the complexity of the physical, biological, and social worlds in which we live. For a mathematician such a question is not always of interest and the development of the fractional calculus proceeded in the mathematical diaspore for over 300 years, independently of the mathematics being developed for the physical sciences. Only recently have fractional operators attracted the attention of physical scientists (Podlubny, 1999; West, Bologna, and Grigolini, 2003; Magin, 2006). Note that there is no single fractional calculus, just as there is no single geometry. Different definitions of fractional operators, differential and integral, have been constructed to satisfy various constraints. There are a number of excellent texts that review the mathematics of the fractional calculi (Oldham and Spanier, 1974; Miller and Ross, 1993; Samko, Kilbas, and Marichev, 1993), others that emphasize the engineering applications of the fractional operators (Podlubny, 1999; Magin, 2006), and still others that provide physical interpretations of the fractal operators (Grigolini, Rocco, and West, 1999; West, Bologna, and Grigolini, 2003). In other words the literature is much too vast to cover here. However, many insights into complexity can be made by judiciously choosing various forms of the fractional operators that have been used in specific applications. To facilitate subsequent discussion we denote the Laplace ˆ transform of a time-dependent function QðtÞ by QðuÞ: ˆ QðuÞ ≡

Z∞

e−ut QðtÞdt ¼ LTfQðtÞ; ug

ð7Þ

0

and write the Laplace transform of the Caputo fractional derivative (Caputo, 2001) for α < 1: LT

dα ˆ QðtÞ; u ≡ uα QðuÞ − uα−1 Qð0Þ; dtα

ð8Þ

where Qð0Þ is the initial value of the dynamic variable. There are a number of definitions of fractional integrals and


derivations, each depending on a given set of assumptions. The Riemann-Liouville (RL) fractional operators are particularly useful (Podlubny, 1999; West, Bologna, and Grigolini, 2003; Magin, 2006). The RL fractional integral is defined D−α t ½QðtÞ

1 ≡ ΓðαÞ

Zt 0

Qðt0 Þdt0 ; ðt − t0 Þ1−α

ð9Þ

the RL fractional derivative is defined Dαt ½QðtÞ ≡ Dnt Dtα−n ½QðtÞ;

ð10Þ

and the operator index is in the range n − 1 ≤ α ≤ n for integer n. The Laplace transform of the RL fractional derivative for α < 1 is ˆ LT½Dαt ½QðtÞ; u ≡ uα QðuÞ:

ð11Þ

It is now possible to return to the GWF and examine its RL fractional derivative using Eq. (11) in which we replace QðtÞ with WðtÞ from Eq. (1). The inverse of the Laplace transform of the RL fractional derivative of the Weierstrass function we ˆ t. The fractional derivative denote as W α ðtÞ ¼ LT −1 ½uα WðuÞ; of the Weierstrass function can then be shown to satisfy the following renormalization group relation: W α ðbtÞ ¼

a α W ðtÞ bα

ð12Þ

which is solved just as we did previously to obtain W α ðtÞ ¼ AðtÞtγ ;

γ¼

log a − α: log b

ð13Þ

Comparing the scaling index in Eq. (2) with that in Eq. (13) we see that taking the fractional derivative changes the scaling parameter by a factor α and the resulting function can be shown to converge (Rocco and West, 1999). A fractional operator of order α acting on a fractal function of fractional dimension D consequently yields another fractal function with fractal dimension D jαj. The fractional operator can be either a derivative or an integral; the former increases the fractal dimension, thereby making the function more erratic, whereas the latter decreases the fractal dimension, thereby making the function smoother. The fact that the fractional operator acting on a fractal function converges supports the conjecture that the fractional calculus can provide an appropriate description of the dynamics for fractal phenomena (Rocco and West, 1999).

The simplest dynamic process is described by the relaxation rate equation for the dynamic variable QðtÞ: ð14Þ

whose solution is given by the exponential relaxation from the initial condition Qð0Þ to zero QðtÞ ¼ Qð0Þe−λt . This is the unique solution to the rate equation and provides everything Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

we can know about the system. We can also interpret QðtÞ as the probability of the occurrence of an event, such as the decay of a radioactive particle by interpreting the initial condition as Qð0Þ ¼ λ. Consequently, the rate equation describes the generation of decay events by a Poisson process. 1. Distribution of rates

Of course, the relaxations of disturbances in complex materials such as taffy are not described by Eq. (14) and are found to require a fractional relaxation equation. One derivation of the fractional calculus representation of relaxation is based on the notion of self-similar dynamics as manifest through renormalization behavior. Glöckle and Nonnenmacher (1991) argued that the renormalization concept may be applied to the rate equation by assuming the existence of many conformational substrates separated by energy barriers. They assume a dichotomous stochastic process in which the relaxation between two states is not given by a single rate λ but by a distribution of rates such that the relaxation function is Z∞ QðtÞ ¼

ρðλÞ exp ½−λtdλ;

ð15Þ

0

where ρðλÞ is the distribution of rates that represent the reaction kinetics and relaxation. However, there are many other phenomena that can be modeled in this way including thermally activated escape processes (Chvosta and Reineker, 1997), intermittent fluorescence of single molecules (Haase et al., 2004) and nanocrystals (Brokmann et al., 2003), stochastic resonance (Fraser and Kapral, 1992), and blinking quantum dots (Jung, Barkai, and Silbey, 2002) to name but a few. Glöckle and Nonnenmacher (1995) introduced a fractal scaling model for the distribution of reaction rates from which they were able to derive the fractional differential relaxation equation: Dαt ½QðtÞ −

t−α Qð0Þ ¼ −λ0 QðtÞ; Γð1 − αÞ

ð16Þ

using the RL fractional operators. Note that the fractional relaxation equation (16) reduces to the ordinary one Eq. (14) when α ¼ 1, since the initial value term vanishes due to the divergence of the gamma function at zero argument. The solution to the fractional rate equation (16) was obtained by the mathematician Mittag-Leffler (1903) at the opening of the twentieth century: QðtÞ ¼ Qð0ÞEα ð−λ0 tα Þ

C. Fractional rate equations

dQðtÞ ¼ −λQðtÞ; dt

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ð17Þ

in terms of the infinite series that now bears his name Eα ð−λtα Þ ¼

∞ X ð−λ0 tα Þk : Γðkα þ 1Þ k¼0

ð18Þ

It is clear that the exponential simplicity of radioactive decay is here replaced by a more complex decay process, but the exponential simplicity is regained when α ¼ 1.

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2. Viscoelastic material experiments

The time dependence of the Mittag-Leffler function (MLF) is depicted in Fig. 6. At early times the MLF has the analytic form of a stretched exponential exp½−λ0 tα =Γð1 − αÞ as indicated by the long-dashed line in the figure, which deviates from the exact MLF solution at long times. In rheology the dashed curve is the Kohlrausch-Williams-Watts law for stress relaxation. Asymptotically in time the MLF yields an inverse power law t−α as shown by the short-dashed line in Fig. 6, which deviates from the exact MLF solution at early times. The short-dashed line corresponds to the Nutting law of stress relaxation. The relation of the fractional relaxation equation and its solution, the MLF, to these two empirical laws for stress relaxation was explored by Glöckle and Nonnenmacher (1993). Note that the MLF smoothly joins these two empirical regions with a single analytic function. So analytic functions are still useful, they just do not appear as solutions to familiar differential equations. This fractional generalization of the stress relaxation equations is phenomenological rather than fundamental. The empirical law that stress is proportional to strain for solids was provided by Hooke. For fluids Newton proposed that stress is proportional to the first derivative of strain. Scott Blair, Veinoglou, and Caffyn (1947) suggested that a material with properties intermediate to that of a solid and a fluid, for example, a polymer, should be modeled by a fractional derivative of the strain. The conjecture of Scott Blair is vindicated in Fig. 6 where data from stress relaxation experiments using polyisobutylene are shown to be well fit by the MLF. Glöckle and Nonnenmacher (1991) also compared the theoretical results with experimental data sets obtained by stress-strain experiments carried out on polyisobutylene and natural rubber and found agreement over more than 10 orders of magnitude. They

also successfully modeled self-similar protein dynamics in myoglobin (Glöckle and Nonnenmacher, 1991) and to formulate slow diffusion processes in biological tissue (Köpf et al., 1998). The asymptotic form of the Mittag-Leffler function is an inverse power law as noted above. This asymptotic behavior suggests that perhaps many of the data sets that have been modeled strictly in terms of an inverse power law may in fact be more faithfully modeled using an MLF when examined more carefully. 3. New fractional Brownian motion

As mentioned earlier one of the first successful phenomenological treatments of the dynamics of complex physical systems was given by Langevin (1908). In the Langevin equation model the time scale separation between the microscopic and macroscopic worlds enabled smoothing over the microscopic degrees of freedom resulting in a stochastic differential equation of motion over macroscopic time scales. This is how the force equation for a heavy particle in a fluid of lighter particles is usually constructed to obtain the rate equation for a Brownian particle. In physics randomness and the central limit theorem are often introduced through the elegant phenomenon of molecular diffusion. In this way uncertainty in the position of the diffusing particle that is buffeted by the lighter particles of the fluid in which it is embedded was first successfully described in terms of the PDF by Einstein (1905) five years after Bachelier (1900) had solved the same problem in terms of the diffusion of profit in the French stock market. In a subsequent paper Einstein conjectured that the observations made through a microscopic of the erratic path of a pollen mote in water by the botanist Robert Brown (1829) might be a diffusive phenomenon. This off-hand comment was sufficient to insure Brown’s scientific immortality. What neither Einstein, Langevin, nor multiple generations of scientists after them knew was that the force equation used to describe Brownian motion is not exact. The derivation of the exact equation for the motion of a heavy spherical particle in a fluid had been derived by Basset (1888). A recent discussion of the Brownian particle’s dynamics including the Basset force was given by Mainardi and Pironi (1996) in terms of the RL fractional derivative: dVðtÞ þ λDt1=2 ½VðtÞ ¼ −γVðtÞ þ fðtÞ; dt

Data from stress relaxation experiments using polyisobutylene in units of dynes=mm2 at constant strain for two different initial conditions are indicated by the discrete data points. Upper: The solid curve is the MLF with index α ¼ 0.60. The short-dashed curve is the inverse power law, and the longdashed curve is the stretched exponential. Lower: The dashed curve is the MLF and the boxes are data. The two fits use the MLF with different parameter values. Adapted from Glöckle and Nonnenmacher, 1991. FIG. 6.


ð19Þ

where the parameters λ and γ are known functions of the fluid viscosity, the particle masses, the radius of the Brownian particle, and fðtÞ is a stochastic force produced by random collisions of the ambient fluid with the diffusing particle. The fractional derivative in Eq. (19) is the result of the backreaction onto the spherical particle by the ambient fluid flowing around it. Of course, the solution to this equation is dominated by Stokes’s dissipation due to viscosity at long times, which accounts for the success of the usual description of Brownian motion without the inclusion of the fractional derivative term. However, when the background fluid is not


homogeneous the derivation of the rate equations needs to be reexamined. Leptos et al. (2009) studied experiments on the motion of Brownian particles (tracers) suspended in a fluid of swimming Eukaryotic microorganisms of varying concentrations. The interplay between the inanimate tracer particles and the advection by flows from the swimming microorganisms results in their displacement having a self-similar PDF with a Gaussian core and exponential tails. Eckhardt and Zammert (2012) reanalyzed these data and obtained an excellent fit to a MLF PDF based on the CTRW model. A theoretical study of a simplified tracer-swimmer interaction by Zaid, Dunkel, and Yeomans (2011) showed that the non-Gaussian effect of the tails of the PDF arise from a combination of a truncated Lévy statistics for the velocity field and the inverse power-law decay of correlations in the ambient fluid. They showed that the dynamics of the PDF leading to the truncated Lévy statistics is given by a fractional diffusion equation, which we discuss subsequently. It is evident that rigorous modeling of Brownian motion in heterogeneous fluids such as microbial suspensions in marine ecologies would potentially benefit from applications of the fractional calculus. D. Subordination and networks

Another aspect of complexity is revealed through cooperative behavior in complex networks. The flocking of birds (Cavagna et al., 2010), the schooling of fish (Katz et al., 2011), the swarming of insects (Yates et al., 2009), the epidemic spreading of diseases (Barrat, Barthelemy, and Vespignani, 2008), the spatiotemporal activity of the brain (Beggs and Plenz, 2003; Fraiman et al., 2009; Chialvo, 2010), the flow of highway traffic (Bak, Tang, and Wiesenfeld, 1987), and the cascades of load shedding on power grids (Carreras et al., 2002) all demonstrate collective behavior reminiscent of the physics of particle dynamics near a critical point, where a dynamic system undergoes a phase transition. Thus, the macroscopic fluctuations observed in complex networks display emergent properties of spatial and/or temporal scale invariance, manifest in inverse power laws of connectivity and waiting-time distributions. These inverse power laws cannot be inferred from the equations describing the nonlinear dynamics of the individual elements of the network. Despite the advances made by the renormalization group and self-organized criticality theories that have shown how scale-free phenomena emerge at critical points, the issue of determining how the emergent properties influence the microscopic dynamics at criticality is only partly resolved (West, Turalska, and Grigolini, 2014). Herein the utility of the fractional calculus is demonstrated by capturing the dynamics of the individual elements within a complex network from the information quantifying that network’s global behavior. The phase transitions of complex networks suggest the wisdom of using a generic model from the Ising universality class to characterize the network dynamics. West, Turalska, and Grigolini (2014) demonstrated, using a subordination argument, that the individual trajectory response to the collective motion of the network is described by a linear fractional differential equation. The solution to Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

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these linear fractional equations retains the full influence of the nonlinear network dynamics on the individual. 1. Fractional Langevin equation

A network’s influence on the dynamics of an individual within the network is determined using a subordination argument. Subordination of the dynamics requires the existence of two different notions of time. One is the operational time τ, which is the internal time of a single individual, with an individual generating the ordinary dynamics of a nonfractional system. The other notion is chronological time, the time as measured by the clock of an external observer. Typically in the operational time frame the temporal behavior of an individual is regular and evolves exactly according to the ticks of that individual’s clock. Therefore it is assumed that the opinion of individual i in operational time τ is well defined and given by Qði; τÞ, which is the exponential solution to the rate equation (14) or introducing the discrete time interval Δτ∶Qði; nÞ ¼ ð1 − λ0 ΔτÞn Qði; 0Þ. In the discrete form the operational time τ is replaced by the discrete value n. However, when the element is part of a dynamic network the exponential no longer describes its behavior. Adopting the subordination interpretation the discrete index n is an individual’s operational time that is stochastically connected to the chronological time t, in which the global behavior of the network is observed. So what is the behavior of the individual in chronological time? The answer to that question depends on the network’s dynamics. The decision making model (DMM) is a member of the Ising universality class and in which an isolated individual randomly switches between two states according to the discrete operational time. The subordination argument is applied to the individual to quantify the influence of the network on the individual’s dynamics and the discrete index is tied to the occurrence of fluctuating events within the network. The intermittent statistics of network events, shown by direct numerical integration of the DMM two-state master equation (Turalska et al., 2009) to have an inverse power-law waiting-time distribution, yields a fractional Langevin equation in chronological time (West, Turalska, and Grigolini, 2014): Dαt ½Qði; tÞ −

t−α Qði; 0Þ ¼ −λ0 Qði; tÞ þ ξðtÞ; Γð1 − αÞ

ð20Þ

and here Qði; 0Þ is the initial state of the individual. The fractional index α is determined by the inverse power-law survival probability of the network, how long an observer must wait between the occurrence of successive changes of global opinion. The statistics of the random force ξðtÞ are determined by the fluctuations produced by the finite size of the complex network. 2. The average person

The solution to the noise-free (averaged) fractional Langevin equation (20) is the MLF, and consequently the average opinion of an individual is predicted to change much more slowly than the exponential of the isolated person. The time-dependent average opinion of a randomly chosen

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The dashed curves are the exponentials for the average opinion of an isolated individual. The dotted curves are the fits of the MLF to the erratic network dynamics calculated using the DMM for a network of 104 elements. Left: Subcritical; middle: critical; right: supercritical. From West, Turalska, and Grigolini, 2014.


individual is presented in Fig. 7, where the average is taken over 104 independent realizations of the dynamics. The DMM network dynamics are known to undergo a phase transition as a control parameter is varied and the results of the calculation in the subcritical, critical, and supercritical regimes are shown. The MLF is fit to the numerical calculations of a twodimensional lattice network of 10 000 individuals undergoing nearest neighbor interactions. Note that the influence of the 9999 other members of the highly nonlinear complex dynamic network on the individual of interest is here predicted by the solution of a linear fractional differential equation without linearizing the dynamics. The response of the individual to the group mimics the group’s behavior most closely when the control parameter is equal to or greater than the critical value. In other words an isolated person with Poisson statistic becomes an interactive person with Mittag-Leffler statistics. In this way an individual is transformed by those with whom he or she interacts. III. FRACTIONAL DIFFUSION AND PROBABILITY

Historically the two ways of describing the changing behavior of the complex phenomena is through the dynamics of the Langevin equation or equivalently through the phase space equation for the PDF. We established the utility of the fractional Langevin equation so now we turn our attention to the fractional phase space equations of motion for the PDF. The fractional calculus has proven its value in being able to describe exotic scaling in a variety of nontraditional statistical phenomena (Metzler and Klafter, 2000). A partial list of such anomalous diffusion from the natural to the social sciences includes the advection of passive scalars in the turbulent atmosphere (Richardson, 1926), transport in amorphous solids (Scher and Montroll, 1975; Montroll and West, 1987), magnetic resonance imaging (Magin et al., 2008), microsphere motion in living cells (Caspi, Granek, and Elbaum, 2000), search behavior (Bénichou et al., 2011), and the intermittent fluctuations in the profit of stocks (Plerou et al., 2000). All manner of complex statistical phenomena have had their phase space equations for the PDF replaced by a more appropriate fractional phase space equation (FPSE) Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

sometimes called a fractional Fokker-Planck equation (FFPE) or a fractional kinetic equation. A. Fractional turbulence

We used one of the great mysteries of classical physics, turbulent fluid flow, to establish a connection between intermittent statistics and the phase space equation for the PDF. Although not easily defined turbulent fluid flow has the characteristics of unpredictability, rapid diffusivity, and dissipation of kinetic energy (Sousa, 2013). Almost a century ago Taylor (1921) attempted to adapt the kinetic theory of gases viewpoint to predict the statistical properties of fluid flow by taking velocity measurements at a point in space. The experiments of Richardson (1926) gave rise to the formula for the mean square particle separation of passive scalar particles dhjRðtÞj2 i ∝ hjRðtÞj2 i2=3 ; dt

ð21Þ

where RðtÞ is the displacement vector between two such particles and from which the dispersion was determined to increase in time as hjRðtÞj2 i ∝ t3þ . Here the exponent is slightly larger than 3. As related by Holm (2005) the original experiment on which this equation was based was the simultaneous release of 10 000 balloons at the London Expo on a windy day. Each balloon contained a note asking the finder to call and tell him the location and time when the balloon came to Earth. The implication of the Richardson dispersion law is that the statistics of turbulent flow are nonGaussian and intermittent. An explanation of the statistical intermittence in turbulence was proposed by Shlesinger, West, and Klafter (1987) who showed that this law could be derived by assuming the statistics were Lévy stable. A distribution is stable if it contains all the convolutions of the laws belonging to it, for example, the convolution of an arbitrary number of Lévy distributions is another Lévy distribution. Experiments, observations, and data analysis have vindicated the assumption that turbulent fluctuations are Lévy distributed. An exemplar of observational data is given in Fig. 8, where the probability


Measured probability density of changes of the wind speed over 4 s Pðδuτ =σÞ, obtained from a wind measurement at the German North Sea coastline. The solid curve corresponds to a Gaussian distribution having the same mean and variance as the observational data. The measurements of the velocity deviations from the mean δuτ are normalized to a standard deviation of σ ¼ 0.8 m=s and the data are plotted on a log-linear graph. From Boettcher et al., 2003.


of the dynamic trajectory of the nth particle of the N passive scalar particles is given by Rn ðtÞ and r is the corresponding phase space variable. In Eq. (22) VðtÞ is the fluctuating velocity field of the ambient fluid in which the passive scalar is embedded and is assumed to dominate such influences as chemical reactions and molecular diffusion in the case of particles. The time-dependent source of the passive scalar defines the origin of the coordinate system. The solution to Eq. (22) is obtained by first taking the spatial Fourier transform, then solving the resulting linear rate equation, and finally averaging the solution over the velocity fluctuations. Blackledge, Coyle, and Kearney (2011) did a statistical analysis of wind fields and determined them to be well described by an alpha-stable Lévy process with an average Lévy index of β ¼ 1.5. The averaging of the Fourier transformed solution to Eq. (22) is therefore carried out using the properties of alpha-stable Lévy statistics dLðtÞ to obtain (Doob, 1953) Zt Zt igðt0 ÞdLðt0 Þ ¼ exp −K β jgðt0 Þjβ dt0 ð23Þ exp 0

density for gusts of wind over a short time interval (4 s) is depicted. The wind field distribution shown in the figure is from the wind gusts off the coast of the German North Sea and is seen to have fluctuations that can be significantly greater than that of corresponding Gaussian distribution with the same mean and variance. Note that in the tails of the distributions, for example, at δuτ ¼ 7σ as indicated by the arrows in the figure, the measured probability density of increments in the wind data is approximately 106 times that in the corresponding Gaussian distribution with the same mean and variance. The statistics have an inverse power-law tail, which can be modeled as an alpha-stable Lévy distribution. This explanation of intermittence has subsequently found wide application including the statistics of turbulent quantum fluids (White et al., 2010; Baggaley and Barenghi, 2011) and assessing the quality of wind turbine power (Blackledge, Coyle, and Kearney, 2011). The mystery of turbulence is that hundreds of books have been written concerning the thousands of experiments that have been done, but there is no universally accepted set of equations that describes the full dynamic range of turbulent fluid flow. Therefore any discussion of turbulence begins with a set of simplifying assumptions. Keeping this in mind the dynamics for the number density of a passive scalar (tracer) in a fluctuating velocity field, such as Richardson’s balloons or chimney smoke, can be expressed as (West and Seshadri, 1982) ∂ρðr; tÞ þ VðtÞ · ▿ρðr; tÞ ¼ fðtÞδðrÞ: ∂t

ð22Þ

The oversimplification made here is that the fluctuations in the velocity field are not spatially dependent, or alternatively that the velocity field is homogeneous on the spatial scale over which the dispersion of the passive scalar is being tracked. The number density can Pbe written in terms of the phase space distribution ρðr; tÞ ¼ Nn¼1 δ(r − Rn ðtÞ), where the location Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

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0

from which West and Seshadri (1982) determined the homogeneous solution to Eq. (22) in terms of the inverse Fourier transform FT−1 f·; rg to be Pðr; tÞ ¼ FT−1 fexp ½−K β jkjβ tˆρðk; 0Þ; rg:

ð24Þ

The average over the ensemble of trajectories when properly normalized determines the probability density Pðr; tÞ ≡ hρðr; tÞi of being in the interval ðr; r þ drÞ at time t so that the time derivative of Eq. (24) gives the homogeneous fractional diffusion equation ∂Pðr; tÞ ¼ K β ▿β Pðr; tÞ; ∂t

ð25Þ

which is solved subject to the source fðtÞ located at δðrÞ. The fractional nabla operator ▿β ð·Þ in this case is the threedimensional Reisz-Feller fractional derivative (Feller, ~ and K β is a general1968): FTf▿β ½GðrÞ; kg ¼ −jkjβ GðkÞ ized diffusion coefficient. Note that if the velocity fluctuations are Gaussian then β ¼ 2 and Eq. (25) reduces to the turbulent diffusion equation found in a number of standard texts, for example, Monin and Yaglom (1971). In general the self-similar scaling of the velocity in time is inextricably linked to the self-similar variability of the PDF of the passive scalar in space as manifest in the fractional spatial derivative. Introducing the mean energy dissipation rate ϵ the parameters in Eq. (25) can be replaced using K β jkjβ ¼ ϵ1=3 jkj2=3 to obtain for the Fourier representation of the turbulent diffusion equation (Sousa, 2013) ˆ ∂ Pðk; tÞ ˆ tÞ: ¼ −ϵ1=3 jkj2=3 Pðk; ∂t

ð26Þ

The Richardson approximation allows us to introduce the turbulence diffusion coefficient DðkÞk2 ¼ ϵ1=3 jkj2=3 so that DðkÞ ¼ ϵ1=3 jkj−4=3 , which is consistent with the Kolmogorov


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(1941) eddy cascade model of the coupling between scales in homogeneous turbulence, which now has a rationale based on the fractional calculus. B. Fractional Bloch equation

Another application of the fractional calculus appears at the nexus of physics and physiology in the measurement of healthy tissue and tumors. This is where nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) are used to regularly examine complex, porous, and heterogeneous materials for both living and inanimate systems. The motivation to generalize the linear, dissipative, and diffusive phenomenological Bloch equation description of magnetic precession and relaxation of nuclear spins was a new method that Bennett et al. (2003) and Bennett, Hyde, and Schmainda (2006) used to describe diffusion in MRI impleα menting a “stretched exponential” function e−bt to fit experimental data. Rather than interpreting the stretched exponential as a way to improve curve fitting, Magin et al. (2008) saw it as a new way to connect nanoscale fractal models of porous materials (Jug, 1986; Widom and Chen, 1995; Kimmich, 2002) and tissues with observable NMR relaxation and diffusion processes Magin et al. (2008) used fractional order differential operators to generalize the Bloch equation in order to characterize neurodegenerative, malignant, and ischemic diseases. The basis for their generalization was the CTRW to provide the theoretical underpinnings for the introduction of fractional operators in both space and time to capture the complete spin dynamics that has been called “spin turbulence.” The generalized Bloch equation can be written schematically for the transverse component of magnetization ∂ αt ½Mðr; tÞ ¼ λMðr; tÞ þ K β ▿β ½Mðr; tÞ;

ð27Þ

where we used the Caputo derivative in time ∂ αt ½· and the Reisz-Feller nabla operator in three-dimensional space ▿β ½·. The solutions to the new Bloch equation were successfully compared with three kinds of experiments: (1) dextran polymer gel with many small interconnecting pores, (2) human articular cartilage plugs, and (3) diffusion-weighted brain imaging on a healthy human volunteer. Magin et al. (2008) emphasized the fact that the utility of the fractional calculus lies in the encoding of information into the fractional operators about the molecular interactions of spin labeled water that is embedded in the structure of polymers, membranes, and the extracellular matrix of cells and tissue. Finally, they pointed out that clinical applications of these techniques were proposed for assessing the severity of stroke, cancer progression, and spinal injury. C. Lévy foraging

The recognition that the Lévy distribution is a solution to a fractional differential equation (Seshadri and West, 1982) was determined in a search to model and understand anomalous transport using Lévy statistics (Montroll and West, 1987). Once introduced into the physicist’s modeling tool kit the Lévy distribution was observed in all manner of physical Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

Humphries et al. (2012) pointed out the scale-invariant nature of the GPS foraging track of the albatross with the increased resolution of the track from left to right.


(Sokolov, Klafter, and Blumen, 2002), social (Metzler and Klafter, 2000), and biomedical phenomena (West and Deering, 1994), including intermittent search strategies (Bénichou et al., 2011). A characteristic of random walks resulting in Lévy distributions is a scaled clustering of steps in space separated by large walker-free zones as suggested in Fig. 9. The first indication that Lévy random walks could be used as a foraging strategy due to the efficiency with which the walker explores space was given by Shlesinger and Klafter (1986). This suggestion was first applied to a natural system by Viswanathan et al. (1996) for the foraging of a wandering albatross. It was assumed that an albatross flies until prey is spotted and then it lands to feed; the time intervals between landings were recorded and provided a measure of the distance between the locations of prey. A histogram of the flight durations was determined to be inverse power law t−α, that being the tail of a Lévy distribution with α ≈ 2. They interpreted this result to be a consequence of food being fractally distributed on the ocean surface for which a Lévy random walk was an efficient foraging strategy. Subsequently, a torrent of theoretical investigations were undertaken to prove that the Lévy foraging strategy was optimal under a variety of circumstances; see, for example, Lomholt et al. (2008). A decade after the analysis of Viswanathan et al. (1996), Edwards et al. (2007) showed that the original statistical analysis of the data was flawed and called into question the notion that an albatross actually uses a Lévy foraging strategy. They also questioned the inferences from the experimental findings in the search strategies of deer and bumblebees (Viswanathan et al., 1999). The controversy over the evidence of whether animals in ecosystems actually implement the optimal Lévy foraging strategy is well documented by Bénichou et al. (2011), who offered alternative strategies to resolve the disagreement. However this is not the end of the saga. Most recently Humphries et al. (2012) were able to show the success of the Lévy foraging strategy in natural environments. They did this in two ways. One is by reexamining previously studied data sets using new, more robust, statistical techniques and the other is using a more recent global positioning system (GPS) data track of wandering albatross. An example of a GPS track is depicted in Fig. 9, where the different panels have different scales indicating the scaleinvarient property necessary for a Lévy foraging strategy. What they found is that the Lévy pattern did not always occur, but was, in fact, dependent on the environmental context, such as whether or not the prey were sparsely distributed in space,


as had been predicted by theory. When the prey are sparsely distributed the individual albatross is observed to adopt the Lévy foraging strategy. On the other hand, when prey are plentiful the albatross adopts the less efficient simple random walk strategy. Consequently, when the foraging under these different strategies is combined into a single data ensemble, as had been done by previous investigators, the resulting distribution could and did look quite different from the Lévy pattern. Lévy foraging is the ecological application of an optimal search strategy. But this strategy is not only used by animals in search of prey, but it appears that Lévy flights are genuinely intrinsic to human mobility (Rhee et al., 2011) as well as to groups of individuals (Brown, Liebovitch, and Glendon, 2007). In addition evidence is accumulating (Rhodes and Turvey, 2007; Radicchi and Baronchelli, 2012; Radicchi, Baronchelli, and Amaral, 2012) that the human brain employs the same strategy in seeking to put a name to the face of the person we are just about to meet. Baronchelli and Radicchi (2013) noted that Lévy patterns are observed in memory retrieval and they argued that these patterns suggested that the brain regions involved in such activity are old in an evolutionary sense; that is, the regions of the brain engaged in memory searches do not involve the frontal cortex. This most recent vindication of the Lévy search strategy in foraging and recall suggests that the evolution of probability densities can be determined by fractional diffusion equations such as that given by Eq. (25). The identification of the search strategy and human mobility patterns necessary for the correct modeling of social networks with the fractional calculus further suggests that there may be a fundamental approach with which to replace the phenomenological random walk or random flight arguments and that is by fractional variational arguments (Agrawal, 2007; Jarad, Abdeljawad, and Baleanu, 2012). Such an approach would provide a generalization to Noether’s theorem in mechanics to more generic complex phenomena. D. Phase space fractional equations

We restrict the remaining discussion to one space dimension to simplify notation. Consequently the general form of the fractional phase space equation of motion, often called the FFPE, can be written as −α

t P ðxÞ ¼ K β ∂ βjxj ½Pðx; tÞ; Dαt ½Pðx; tÞ − Γð1 − αÞ 0 Dαt ½·

ð28Þ

∂ βjxj ½·

where is the RL fractional derivative in time, is the Reisz-Feller fractional derivative in one space dimension, and P0 ðxÞ is the initial value of the probability density typically taken to be the delta function δðxÞ. The utility of the FFPE is that it enables a modeler to incorporate a potential VðxÞ directly into the fractional equation of motion (Metzler and Klafter, 2000): t−α P ðxÞ Dαt ½Pðx; tÞ − Γð1 − αÞ 0 ∂ ∂V Pðx; tÞ þ K β ∂ βjxj ½Pðx; tÞ: ¼ ∂x ∂x Rev. Mod. Phys., Vol. 86, No. 4, October–December 2014

ð29Þ

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The linear FFPE has a formal solution Pðx; tÞ ¼

∞ X

ϕn ðxÞEα ð−λn tα Þ;

ð30Þ

n¼0

which when α ¼ 1 and the MLF becomes an exponential is the familiar eigenfunction expansion and λn is the nth eigenvalue. Metzler and Klafter (2000) reviewed how to solve the FFPE for a number of standard potentials including the harmonic oscillator when α ≠ 1. The solution to the FFPE when the potential is zero is also of interest. Denoting the Laplace transform of Pðx; tÞ by ˆ uÞ, its Fourier transform by Pðk; ~ tÞ, and its LaplacePðx; ¯ uÞ, the double transform of Eq. (28) Fourier transform by Pðk; yields ¯ uÞ ¼ Pðk;

1 u þ K β u1−α jkjβ

ð31Þ

when P0 ðxÞ ¼ δðxÞ. A general recipe for the Fourier-Laplace transform of the probability density was first obtained by Montroll and Weiss (1965) based on their CTRW theory. They assumed that the distribution of steps and waiting times were independent of one another and this led to the factored form of the transformed variables in Eq. (31). In the CTRW formalism the probability of taking a step of length x would be an inverse power law x−β and the probability of waiting a time t before taking the next step would be the inverse power law t−α, and taken together would yield Eq. (31). However, this particular application was not studied by Montroll and Weiss (1965). The inverse Laplace transform of Eq. (31) yields the complicated looking form of the MLF: ~ tÞ ¼ Eα ð−K β jkjβ tα Þ: Pðk;

ð32Þ

When α ¼ 1 we know that the MLF reduces to the exponential, in which case the solution is the characteristic function for the alpha-stable Lévy distribution in space with Lévy index 0 < β ≤ 2. On the other hand, Eq. (31) can be Fourier inverted for β ¼ 2 to obtain α=2−1

ˆ uÞ ¼ u Pðx;

2

exp ½−K 2 jxjuα=2 ; 0 < α ≤ 1:

ð33Þ

Mainardi (1996) considered the inverse Laplace transform of Eq. (33) and obtained an analytic solution in terms of a Wright function that reduces to the Laplace density exp ½−K 2 jxj=2 pffiffiffiffiffiffiffi when α ¼ 0 and to the Gaussian density exp ½−x2 =4t= 4πt when α ¼ K 2 ¼ 1. It is evident that asymptotically ðu → 0Þ the probability density function given by the inverse Laplace transform of Eq. (33) decays as an inverse power law in time t−α=2 . Thus, we see that the FPSE can generate the two most familiar forms of complexity: topological complexity given by an inverse power law in an extensive variable such as the number of events, for example, where an event can be a new connection within a network (Albert and Barabási, 2002); and temporal complexity given by an inverse power law in time,

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where the time denotes an interval between events, for example, the time between receiving and answering an email or letter (Oliveria and Barabási, 2005). IV. CONCLUSION

Some provisional conclusions we can draw from this Colloquium are as follows: (i) The fractional calculus provides a framework in which to capture the dynamics of scale-free complex networks in which the statistics are non-Poisson, nonstationary, and nonergodic. (ii) The fractal topological measure of complexity is an inverse power-law PDF in an extensive property of a complex phenomenon such as the number, intensity, or order of an event. The PDF can be shown to evolve according to a FPSE with the observable being the dependent phase space variable. (iii) The temporal measure of complexity is an inverse power-law distribution in the time interval between events in a dynamic complex network, such as the time between quakes of the Earth or of the brain. The PDF evolves according to a FPSE in time which may or may not have a corresponding complexity in a network observable. (iv) The optimal search strategy, defined with respect to an objective function, is very often given by an alpha-stable Lévy process whether it is wildlife searching for prey or the human brain accessing an elusive memory. (v) The long sought general principles for complex networks may find their origin in fractional variational principles that generalize the foundational symmetry principles found so useful in understanding generic complex phenomena in the physical sciences. ACKNOWLEDGMENTS

I thank two anonymous referees who made suggestions that improved the paper and found an error in the presentation. I also thank the Army Research Office for support of this research.

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