Fractional Kinetics
FRACTIONAL KINETICS It isn’t the calculus we knew: Equations built on fractional derivatives describe the anomalously slow diffusion observed in systems with a broad distribution of relaxation times.
Igor M. Sokolov, Joseph Klafter, and Alexander Blumen
ntil about 10 years ago, expressions involving frac- Pierre-Simon Laplace, Bernhard Riemann, Joseph Liou- Utional derivatives and integrals were pretty much re- ville, Oliver Heaviside, Arthur Erdélyi, and many others. stricted to the realm of mathematics. But over the past One way to formally introduce fractional derivatives decade, many physicists have discovered that a number of proceeds from the repeated differentiation of an integral systems—particularly those exhibiting anomalously slow power: n diffusion, or subdiffusion—are usefully described by frac- d m! ⊗ x m ⊂ x mn. (1) tional calculus. Those systems include charge transport in dxn ()!mn⊗ amorphous semiconductors, the spread of contaminants in underground water, relaxation in polymer systems, and For an arbitrary power m, repeated differentiation gives n tracer dynamics in polymer networks and in arrays of con- d m Gm(1)⊕ m⊗ x ⊂ x n, (2) vection rolls. dxn Gm(1)⊗⊕n Fractional diffusion equations generalize Fick’s sec- ond law and the Fokker–Planck equation by taking into with gamma functions replacing the factorials. The account memory effects such as the stretching of polymers gamma functions allow for a generalization to an arbitrary under external fields and the occupation of deep traps by order of differentiation a, a charge carriers in amorphous semiconductors. Such gen- d m Gm(1)⊕ ma⊗ x ⊂ x . (3) eralized diffusion equations allow physicists to describe dxa Gm(1)⊗⊕ a complex systems with anomalous behavior in much the same way as simpler systems. The extension defined by equation 3 corresponds to the Riemann–Liouville derivative.2 It is sufficient for handling Fractional calculus functions that can be expanded in Taylor series. Physicists are all familiar with the high-school calculus A second, more elegant and general way to introduce that introduces students to derivatives of integer order n, fractional derivatives uses the fact that the nth derivative dny/dxn. Those derivatives and their inverse operations— is an operation inverse to an n-fold repeated integration. integrations—provide the language for formulating and Basic is the integral identity analyzing many laws of physics. But physicists generally xy1 yn1⊗ aren’t taught about fractional-order derivatives, which . . . f()d y y . . . d y might be formally expressed as, for example, d1/2y/dx1/2. Is nn 1 aa a the fractional calculus all that difficult? In fact, the calculus of fractional integrals and deriv- x 1 n⊗1 atives is almost as old as calculus itself. As early as 1695, ⊂ ()xy⊗ fy()d y. (4) (n ⊗ 1)! Gottfried von Leibnitz, in a reply to Guillaume de l’Hôpi- a tal, wrote, “Thus it follows that d1/2x will be equal to Clearly, the equality is satisfied at x ⊂ a, and it is not x=dx : x, . . . from which one day useful consequences will difficult to see iteratively that the derivatives of both sides be drawn.” About 300 years had to pass before what is now of the equality are equal. A generalization of the expres- known as fractional calculus was slowly accepted as a sion allows one to define a fractional integral of arbitrary practical instrument in physics.1 Before that acceptance, order a via fractional calculus had to be more rigorously formulated. x ⊗a 1 ⊗ Important contributions to that end stem from the work of D fx()⊂ ()xy⊗ a 1 fy()d( y xa ). (5) a x Ga() a IGOR SOKOLOV is a professor of physics at Humboldt University in A fractional derivative of an arbitrary order is defined Berlin, Germany. YOSSI KLAFTER is a professor of chemistry at Tel through fractional integration and successive ordinary dif- Aviv University in Israel. ALEX BLUMEN is a professor of physics at the University of Freiburg in Germany. ferentiation. For additional elaboration, see box 1, which also discusses the relation between the value of the inte-
48 NOVEMBER 2002 PHYSICS TODAY © 2002 American Institute of Physics, S-0031-9228-0211-030-1 Box 1. Definitions and Examples ⊗a he fractional integration operator a Dx is defined by equa- Ttion 5. The a-th fractional derivative is then2 n d ⊗ Da ⊂ Da n . a⊂1 a x dxn a x Examples of Fractional Calculus with /2 The number of additional differentiations n is Semi-integralFunction Semi-derivative equal to [a] ⊕ 1, where [a] is the whole part of a. –1 1 From the above definition it follows that –1 /2 1 /2 /2 ⊂ d /2 ⊂ d 0D x f() x fx() fx() 0D x f() x fx() –1/ 1/ Gm(1)⊕ ⊗ dx 2 dx 2 Dax m ⊂ x ma, 0 x Gm(1)⊗⊕ a 2C=+++x/p C, any constant C/=+++px as envisaged in equation 3. Note that here the lower limit of integration is zero. An interesting =++p 1 / =++x 0 consequence of the rule for differentiating powers is x=+p /2 =++x =+p /2 a ⊂ 1 ⊗ a 0 Dx 1 x . Ga(1⊗ ) 3 4/3x /2 =++p x 2=+++x/p That is, the derivative of a constant vanishes only ⊕ ⊕ if the order of the derivative is integer, in which Gm(1) ⊕1 Gm(1) 1 ⊗ x m /2 x m,>m –1 x m– /2 case G(1 a) diverges. Another interesting result Gm(⊕ 3/2) Gm(⊕ 1/2) holds for the derivative of the exponential func- tion: exp(x ) erf(=++ x ) exp(x ) 1/=+++px ⊕ exp(x ) erf(=++ x ) a x ⊂ x ga(– ,)x 0 Dx e e , p/x px Ga(– ) 2=+++[ln(4x ) – 2] ln x ln(4x ) /=+++ where g(⊗a, x) is the incomplete g function. The practical use of fractional calculus is un- derlined by the fact that, under Laplace transform, the opera- From Fick to fractional diffusion tor D⊗a has the simple form The foundations of kinetics were established more than 150 0 t years after the prophecy of Leibnitz, without the use of frac- ⇒⊂⊗a ⊗a ⇒ {()}{()}.0 Dftt u ft tional calculus. In 1855, the young Adolf Fick, a pathologist at the University of Zürich, wrote a work entitled “Über The result for the differentiation of an exponential may Diffusion” (“On Diffusion”). The work was published in seem disappointing. But if one chooses the lower limit of in- Poggendorf’s Annalen der Physik, the Physical Review Let- tegration to be a ⊂⊗Fin equation 5, the resulting Weyl de- a x x ters of that time. Fick started by observing that “diffusion rivative satisfies ⊗FD e ⊂ e . Moreover, the Weyl definition x in water confined by membranes is not only one of the basic reproduces the familiar properties of Fourier-transformed in- factors of organic life, but it is also an extremely interest- tegrals and derivatives: ing physical process and, as such, should attract much more ↓⊂a a {()}()().–FDftt iww f attention from physicists than it has so far.” Diffusion a a processes such as those considered by Fick, and processes We see that there are several ways to interpret “d /dx ,” all of that are described by fractional calculus, continue to fasci- a which coincide with the usual differentiation if is an integer. nate physicists and others. That may have been one of the reasons for the late acceptance Fick was an experimental physiologist, but his work of fractional calculus as a tool to describe physical phenom- on diffusion was theoretical, and his approach would today ena. In fact, the freedom of definition is an advantage that al- be called a phenomenological linear-response theory ap- lows one to take additional physical information (such as plied to diffusion. In brief, the result of diffusion is known whether a force acting on a system is always applied or is to be the equilibration of concentrations. Thus, particle turned on at a specific time) directly into account. current has to flow against the concentration gradient. In analogy with Ohm’s law for electric current, or with Fourier’s law for heat flow, Fick assumed that the current j is proportional to the concentration gradient, so that gration limit a and definitions for fractional differentiation jr(,)tct⊂⊗k ∇ (,), r (7) based on the Laplace transform (when a ⊂ 0) and the Fourier transform (when a ⊂⊗F). The table above pres- an equation now known as Fick’s first law. Here, k is the ents (for a ⊂ 0) several examples of semi-integrals and diffusion coefficient with dimension of L2/T and c is the semi-derivatives, operations for which a ⊂1/2. concentration. If, in addition, particles are neither created In this article, we are generally concerned with frac- nor destroyed, then, according to the continuity equation, tional time derivatives, and we set a ⊂ 0, in effect choos- ]ct(,)r (jr(,).t (8 ؒ∇⊗⊃ ⊃ ing t 0 as the beginning of the system’s time evolution. ]t In particular, we note that the operator Combining Fick’s first law with the continuity equation 1⊗a ⊂ d ⊗a gives Fick’s second law, also known as the diffusion 0Dt 0 Dt , (6) dt equation: ]ct(,)r 2 with 0 < a < 1, plays a central role in generalized diffusion ⊂∇k ct(,),r (9) equations. ]t http://www.physicstoday.org NOVEMBER 2002 PHYSICS TODAY 49 tt
ond order in the spatial coordinates and 33first order in time: Changing the spatial scale by a factor of 3 corresponds to chang- 0.5 0.5 ing the time scale by a factor of 9. In a variety of physical systems, how- ever, the simple scaling pertinent to Fick- 22ian diffusion is violated.3,4 The mean ∀ 2¬ a (,) (,) squared displacement grows as r } t Pxt Pxt with the exponent aÞ1. A consistent gen- eralization of the diffusion equation could still be second order in the spatial coordi- 11nate and have a fractional-order temporal derivative—for example, ]a Pt(,)r ⊂∇k 2 (12) ]ta a Pt(,),r xx where the dimension of the fractional dif- 2 a –4–2002244 –4 –2 fusion coefficient ka is [L /T ]. Equation 12 looks rather unusual. Does it make any sense? It does, as we now proceed to show. FIGURE 1. CONTINUOUS-TIME RANDOM WALKS (CTRWs) do not cover ground as quickly as simple random walks. The black Continuous-time random walks lines indicate a specific realization of a simple random walk Random walks and diffusion serve as an interface between (left) and a CTRW (right). Note that CTRW steps occur very kinetics on one hand and derivatives and integrals of frac- irregularly; most of the time the walker doesn’t move at all. As tional order on the other. The simplest model leading to a consequence, the mean square displacement in CTRWs grows normal diffusion is the random walk. Related models were considerably slower than in simple random walks. The blue introduced in Lord Rayleigh’s studies of isoperiodic vibra- and the red curves indicate the typical behavior of the displace- tions (1880) and in Louis Bachelier’s analysis of stock- ment: ∀x2¬1/2 } t1/2 for the simple random walk and ∀x2¬1/2 } t1/4 ⊂ market fluctuations (1900). If a random walker strolling for a CTRW with power-law parameter (see text) a 1/2. The in one dimension moves a step of length a in either direc- filled yellow curves show the probability distributions P(x,t) at tion precisely when each unit time elapses, then the dis- t = 1. For regular diffusion, which corresponds to the simple placement after a large number of steps (that is, after a random walk, the distribution is Gaussian. For diffusion gov- long time) will be distributed according to the Gaussian, erned by CTRWs, the distribution satisfies the fractional diffu- equation 10. sion equation. Its characteristic tentlike form displays a cusp at ⊂ In continuous-time random walks (CTRWs)—intro- x 0. duced in physics by Elliot Montroll and George Weiss—the condition that the steps occur at fixed times is relaxed.5 Rather, the time intervals between consecutive steps are governed by a waiting-time distribution c(t). In describing transport, the c(t) distributions may stem from possible which is a closed equation for the temporal evolution of the obstacles and traps that delay the particle’s jumps and thus introduce memory effects into the motion. If the mean concentration. ⊂ Fick’s phenomenology missed the probabilistic point waiting time between consecutive steps is finite, t ∫t c(t)dt < F, the CTRW is described by Fick’s diffusion of view central to statistical mechanics. It was Albert Ein- 2 stein who, 50 years after “On Diffusion,” first derived the equation, with the diffusion coefficient k equal to a /2t. diffusion equation from the postulates of molecular theory, The situation changes drastically if the mean waiting in which particles move independently under the influence time diverges, as is the case for power-law waiting-time of thermal agitation. In his picture, the concentration of distributions of the form ⊕ particles c(r, t) at some point r is proportional to the prob- ct(tt )} 1/(1⊕ / )1 a , (13) ability P(r, t) of finding a particle there. Thus, the diffu- 6 sion equation holds when probabilities are substituted for with 0 < a < 1. (See the article by Harvey Scher, Michael concentrations. F. Shlesinger, and John T. Bendler, PHYSICS TODAY, Janu- If, for example, a particle is initially placed at the ori- ary 1991, page 26.) Figure 1 compares the displacements gin of coordinates in d-dimensional space, then its evolu- of walkers undergoing simple random walks with those fol- tion, according to equation 9, is given by lowing CTRWs. The behavior for a CTRW is subdiffusive. The mean square displacement grows as 1 r2 Pt(,)r ⊂ exp⊗ . (10) ∀¬xt2 () } ta. (14) (4pkt ) d/2 4kt Transport phenomena in systems exhibiting subdiffusion The mean squared displacement of the particle is thus have a < 1, whereas systems that exhibit superdiffusion have a > 1. ∀¬⊂rt2 () rP2 ()r ,td3 r ⊂ 2dkt. (11) Fractional modifications of the commonly used diffusion and Fokker–Planck equations generate the sca- Note that the scaling form ∀r2¬ } t follows directly from the ling behavior seen in subdiffusive systems. Consider, for structure of the diffusion equation. That equation is sec- example, the fractional equation first introduced
50 NOVEMBER 2002 PHYSICS TODAY http://www.physicstoday.org Pxt( , =0) 0.8 Pxt( , =0.25) 0.8
0.6 0.6 FIGURE 2. PROBABILITY DISTRIBUTIONS EVOLVE as particles governed by Fokker–Planck equations with 0.4 0.4 harmonic potentials relax toward Boltzmann equilib- rium. The graphs at left (with position x and time t in arbitrary units) show the evolution of the probability 0.2 0.2 distribution for a particle collection initially prepared x x at x ⊂ 1. For both the regular (blue) and a ⊂ 1/2 frac- tional Fokker–Planck (red) equations, the distribu- –3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3 tions asymptotically approach Gaussians with a mean value tending to zero. For the regular equation, the distribution is always a Gaussian. The behavior of the solution in the fractional case is strikingly different. Pxt( , =1) 0.8 Pxt( , =4) 0.8 It’s not just that the speed of relaxation is consider- ably slower. The form of the distribution is character- 0.6 0.6 istic of subdiffusive systems, showing a cusp singular- ity, at the initial value of x, that remains visible even at very long times. 0.4 0.4
0.2 0.2 where ma is the fractional mobility. x x In equilibrium, the current j must vanish. After expressing the force in terms of a potential –3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3 function U(r), one may write the equilibrium prob- ability distribution that satisfies equation 18 in the ⊗ form P(r) } exp( maU(r)/ka). Now, for independent by Venkataraman Balakrishnan and by W. R. Schneider particles, the equilibrium probability is a Boltzmann dis- 6 ⊂ and W. Wyss: tribution, so ka /ma kBT, a generalization of Einstein’s re- ⊂ lation k/m kBT. ] 1⊗a Pxt(,)⊂∇ Dk 2 Pxt (,). (15) For example, consider a constant external force acting ]t 0 t a ∀ ¬ ⊂ in the x direction. The force leads to a mean drift, x(t) f a ⊕ Equation 15 can be derived from the CTRW scheme along ma ft / G(1 a), which is related to the force-free mean 7 ∀ 2 ¬ ⊂ ⊕ the lines used in Einstein’s work on Brownian motion, and square displacement x (t) f ⊂ 0 2kata / G(1 a) through thus applies to all situations discussed adequately by ∀¬xt2 () ∀¬⊂1 f⊂0 CTRW. It treats systems that behave anomalously in a xt()f f . (19) framework very much like the framework used for systems 2 kTB with normal diffusion, so that known solutions of the sim- The expression above is the fluctuation-dissipation theo- ple case can be easily generalized to the anomalous case.8 rem, which holds for subdiffusion in the fractional Fokker–Planck framework. Fractional Fokker–Planck equation Ornstein–Uhlenbeck processes in one dimension pro- The standard diffusion equation accounts for a particle’s vide a second example. Such processes involve diffusion in motion due to uncorrelated molecular impacts. In many the harmonic potential U(x) ⊂ bx2/2 so that the force is cases, an external deterministic force is imposed on a sys- f ⊂⊗bx. The corresponding fractional Fokker–Planck tem in addition to such random impacts. The equation is Fokker–Planck equation considers both contributions.9 It can be derived by combining Fick’s first law—expressed in ]Pxt(,) ] ]2 ⊂ D1–a m b [(,)]xP x t ⊕ k Pxt(,). (20) terms of probability current and taking into account the ]t 0 t a ]x a ]x2 external force—with the continuity equation. The proba- bility current is Figure 2 shows snapshots of the time-dependent probabil- ity distribution for particles originally at a well-defined lo- jr(,)tPttPt⊂⊗km∇ (,) r ⊕ fr (,) (,), r (16) cation, according to both the fractional and conventional Fokker–Planck equations with harmonic potentials. In where f is the external force acting on the particle and m both cases, the asymptotic forms of the distribution are is the particle’s mobility. When the continuity equation is Gaussian, because the Fokker–Planck equations describe applied to the probability current, the Fokker–Planck the relaxation of thermodynamic systems to equilibrium. equation follows: In the fractional case, though, the distribution at finite ]Pt(,)r times has a cusp characteristic of broad CTRW distribu- .mkfrPt ( , ) ⊕ ∇ Pt ( r , )). (17) tions⊗ )ؒ∇⊃ ]t ⊂ In thermodynamic applications, one is mostly inter- In the absence of an external force, f 0, the ested in mean values—for example, the mean particle po- Fokker–Planck equation reduces to the standard diffusion sition
log10 t /t
–2
–4
x
∀¬x
FIGURE 3. THE MITTAG–LEFFLER FUNCTION describes the relaxation toward equilibrium of particles governed by the fractional Fokker–Planck equation. For t close to zero, the ⊗ a function behaves like a stretched exponential, Ea ( (t/t) ) exp (⊗(t/t)a) /G(1 ⊕ a). For large t, the function approaches a ⊗ a ⊗a ⊗ power-law, Ea ( (t/t) ) (t/t) /G(1 a). The red curve shows the Mittag–Leffler function for a ⊂ 1/2. The dashed lines show the short-time (blue) and the long-time (black) forms. Note the double logarithmic scales.
t
It follows that ∀x(t)¬ decays exponentially toward equilib- FIGURE 4. A PARTICLE DIFFUSING in a harmonic potential rium: ∀x(t)¬⊂∀x(0)¬ exp(⊗t/t). exemplifies thermodynamic relaxation. For regular diffusion, In the case of diffusion governed by the fractional the mean value of the particle’s position decays exponentially Fokker–Planck equation, the mean displacement obeys with time (blue). In the fractional, subdiffusive case, the relax- ation is described by a Mittag–Leffler function. The particular ∀¬ d()xt ⊂⊗ ⊗a 1–a ∀¬ Mittag–Leffler function leading to the relaxation illustrated in t 0 Dxt(), (22) dt t red has as its fractional parameter a ⊂ 1/2. ⊗a ⊂ with t bma, as is readily verified by multiplication of equation 20 by x and integration, followed by an integra- tion by parts of the right-hand side. The solution of equa- tion 22 can be expressed in terms of the Mittag–Leffler ⊂ ⊗ a function Ea via
52 NOVEMBER 2002 PHYSICS TODAY http://www.physicstoday.org Box 2. From Oliver Heaviside to Stress Moduli of Polymers ractional derivatives often emerge in the Fdescription of self-similar, hierarchically R organized systems. Consider a discrete model for a transatlantic telegraph cable, as proposed C by Lord Kelvin. The model consists of identi- cal resistances R and identical capacitors C, il- lustrated in the figure at right. The response of the cable to a voltage V(t) applied at, say, its right end may be related to the impedance Z of the system. The relation between impedance, current I, and the voltage is most simply ex- pressed in terms of the Fourier-transformed functions: ZI()()ww⊂ V (). w simultaneously—for example, when the monomers of a net- work are randomly charged and exposed to an external volt- Calculating Z(w) is a standard problem in the theory of electri- age. In such cases, one still observes scaling, but with values of cal circuits. For the transatlantic cable model, the solution is ⊗ a that depend on the distribution of the charges on the poly- Z(w) ⊂ R ⊕ [iwC ⊕ 1/Z(w)] 1, or Z(w) ⊂ R/2 ⊕ (R2/4 ⊕ mer. Many complicated systems can be modeled using frac- R/iwC )1/2. In the limit in which both R and C tend to zero tional differential equations whose parameters depend on a. (both depend on the subdivision length of the cable), but the ⊗ For example, the figure below presents measurements for the quotient R/C ⊂ z2 stays constant, one obtains Z(w) O z(iw) 1/2. mechanical storage (circles) and mechanical loss (squares) mod- Hence uli for two ethane-co-1-butenecopolymer compounds respond- 1 IV()ww⊂ ()/Zi()ww⊂ ( )/2 V(),w ing to a harmonic external strain field. The horizontal axis gives the field’s frequency and the vertical axis the resulting where, for convenience, we have set z ⊂ 1. In terms of the cor- stress moduli, both in appropriately normalized units. (For ⊂ 1/2 responding functions of time, I(t) ⊗FDt V(t), involving further details of the measurements by Christian Friedrich of semi-differentiation, as pointed out by Oliver Heaviside. The the University of Freiburg in Germany, see ref. 1, p. 331.) The response of the cable to a voltage V(t) ⊂ v(t)q(t) switched on at black curves, derived from the so-called fractional Maxwell t ⊂ 0 is given by model, fit the data well. 1 ⊂ /2 It() 0 Dt V() t . 10 6 As can be confirmed by referring to equation 5, which defines the fractional integral, the above expression for the current de- Tref = 130°C ⊂ ∫t ⊗ Ј Ј Ј scribes a retarded response, I(t) d/dt 0 M(t t )V(t )dt , with the memory function M(t) decaying slowly, as t⊗1/2. Fractional derivatives, as the transatlantic cable model shows, are a natu- 10 5 ral tool for describing the linear response of systems with long, power-law memory. The mechanical equivalent of Kelvin’s model, a chain con- sisting of springs and beads immersed in a viscous fluid, is the EB64 standard Rouse model in polymer dynamics. That model, illus- 10 4 trated below the transatlantic cable, accounts for the fluctuat- ing forces due to solvent molecules, and for viscous friction. ⊂ GЈ If a force f (t) f0q(t) acts on one of the monomers of the STRESS MODULUS chain, then the mean displacement of the monomer satisfies GЉ 3 a ⊂ 10
lution have been achieved by Ralf Metzler and Yossi with fractional derivatives. Many other examples of ap- Klafter, both working at Tel Aviv University. plications of fractional calculus to modern physics are pre- Fractional calculus can be applied to many areas of sented in reference 1. physics, other than fractional kinetics. In fact, fractional What about superdiffusion? Fractional generaliza- calculus was introduced in a heuristic manner long ago in tions of the diffusion and Fokker–Planck equations have rheology (for example, by Andrew Gemant in 1936) in an been introduced for superdiffusion as well. Those equa- effort to describe linear viscoelasticity through the exten- tions, which apply spatial fractional derivatives rather sion of the material-dependent constitutive equations used than temporal ones, are intimately related to Lévy in the field.13 Box 2 discusses examples of hierarchical processes in space. George Zaslavsky has advocated using structures whose dynamics are conveniently described such equations to describe chaotic diffusion in Hamilton- http://www.physicstoday.org NOVEMBER 2002 PHYSICS TODAY 53 ian systems, and other applications have been discussed by Hans Fogedby and by Bruce West and Paolo Grigolini.1,14 We hasten to note, however, that superdiffu- sion is far from being completely understood. The classical diffusion laws derived by Fick dominated physicists’ views on diffusion and transport for more than a century. But recent observations have clearly demon- strated that Fick’s laws have exceptions. Those exceptions, which have been termed strange kinetics,3 require a com- pletely fresh view of kinetic processes, based on random- walk approaches and on unconventional distribution func- tions (see the article by Joseph Klafter, Michael F. Shlesinger, and Gert Zumofen, PHYSICS TODAY, February 1996, page 33). Fractional calculus helps formulate the problems of strange kinetics in a simple and elegant way.
We thank our colleagues Eli Barkai, Christian Friedrich, Ralf Metzler and Helmut Schiessel for fruitful discussions.
References 1. R. Hilfer, ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, N. J. (2000). 2. K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, San Diego, Calif. (1974); K. S. Miller, B. Ross, An In- troduction to the Fractional Calculus and Fractional Differ- ential Equations, Wiley, New York (1993). 3. M. F. Shlesinger, G. M. Zaslavsky, J. Klafter, Nature 363, 31 (1993). 4. J.-P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990). 5. E. W. Montroll, G. H. Weiss, J. Math. Phys. 10, 753 (1969); E. W. Montroll, M. F. Shlesinger, in Nonequilibrium Phenom- ena II: From Stochastics to Hydrodynamics, J. L. Lebowitz, E. W. Montroll, eds., North Holland, New York (1984); E. W. Montroll, B. J. West, in Fluctuation Phenomena, E. W. Mon- troll, J. L. Lebowitz, eds., North Holland, New York (1987). 6. V. Balakrishnan, Physica A 132, 569 (1985); W. R. Schneider, W. Wyss, J. Math. Phys. 30, 134 (1989). 7. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000). 8. E. Barkai, Phys. Rev. E 63, 046118 (2001); I. M. Sokolov, Phys. Rev. E 63, 056111 (2001). 9. H. Risken, The Fokker–Planck Equation: Methods of Solution and Applications, Springer-Verlag, New York (1996). 10. S. B. Yuste, K. Lindenberg, Phys. Rev. Lett. 87, 118301 (2001); J. Sung, E. Barkai, R. J. Silbey, S. Lee, J. Chem. Phys. 116, 2338 (2002). 11. J. W. Kirchner, X. Feng, C. Neal, Nature 403, 524 (2000). 12. R. Metzler, J. Klafter, I. M. Sokolov, Phys. Rev. E 58, 1621 (1998). 13. C. Friedrich, H. Schiessel, A. Blumen, in Advances in the Flow and Rheology of Non-Newtonian Fluids, D. A. Siginer, D. De Kee, R. P. Chhabra, eds., Elsevier, New York (1999), p. 429. 14. G. M. Zaslavsky, Chaos 4, 25 (1994); H. C. Fogedby, Phys. Rev. Lett. 73, 2517 (1994).
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