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.