Newtonian physics beganwith an attempt to

make precise predictionsabout natural phenomena,predictions that could be ac-curately checked by observa-tion and experiment. Thegoal was to understand na-ture as a deterministic,“clockwork” universe. Theapplication of probability dis-tributions to physics devel-oped much more slowly. Early uses of probability argu-ments focused on distributions with well-defined meansand variances. The prime example was the Gaussian lawof errors, in which the mean traditionally represented themost probable value from a series of repeated measure-ments of a fixed quantity, and the variance was related tothe uncertainty of those measurements.

But when we come to the Maxwell–Boltzmann distri-bution or the Planck distribution, the whole distributionhas physical meaning. Being away from the mean is nolonger an error, and a large variance is no longer an indi-cator of poor measurement accuracy. In fact the whole dis-tribution is the prediction. That was a major conceptualadvance.

In this article we take that idea to its extreme limitand investigate probability distributions, called Lévy dis-tributions, with infinite variances, and sometimes even in-finite means. These distributions are intimately con-nected with fractal random-walk trajectories, called Lévyflights, that are composed of self-similar jumps.1 Lévyflights are as widely applied in nonlinear, fractal, chaoticand turbulent systems as Brownian motion is in simplersystems.

Brownian motionThe observation of Brownian motion was first reported in1785, by the Dutch physician Jan Ingenhausz. He waslooking at powdered charcoal on an alcohol surface. Butthe phenomenon was later named for Robert Brown, whopublished in 1828 his investigation of the movements offine particles, including pollen, dust and soot, on a watersurface. Albert Einstein eventually explained Brownianmotion in 1905, his annus mirabilis, in terms of random

thermal motions of fluidmolecules striking the mi-croscopic particle and caus-ing it to undergo a randomwalk.

Einstein’s famous paperwas entitled “Über die vonder molekularkinetischenTheorie der Wärmegeforderte Bewegung von inruhenden Flüssigkeitensuspendierten Teilchen,”

that is to say, “On the motion, required by the molecu-lar-kinetic theory of heat, of particles suspended in fluidsat rest.” Einstein was primarily exploiting molecular mo-tion to derive an equation with which one could measureAvogadro’s number. Apparently he had never actually seenBrown’s original papers, which were published in thePhilosophical Magazine. “It is possible,” wrote Einstein,“that the motions discussed here are identical with theso-called Brownian molecular motion. But the referencesaccessible to me on the latter subject are so imprecise thatI could not form an opinion about that.”2 Einstein’s predic-tion for the mean squared displacement of the randomwalk of the Brownian particle was a linear growth withtime multiplied by a factor that involved Avogadro’s num-ber. This result was promptly used by Jean Perrin to mea-sure Avogadro’s number and thus bolster the case for theexistence of atoms. That work won Perrin the 1926 NobelPrize in Physics.

Less well known is the fact that Louis Bachelier, astudent of Henri Poincaré, developed a theory of Brownianmotion in his 1900 thesis. Because Bachelier’s work wasin the context of stock market fluctuations, it did not at-tract the attention of physicists. He introduced what is to-day known as the Chapman–Kolmogorov chain equation.Having derived a diffusion equation for random processes,he pointed out that probability could diffuse in the samemanner as heat. (See PHYSICS TODAY, May 1995, page 55.)

Bachelier’s work did not lead to any direct advancesin the physics of Brownian motion. In the economic con-text of his work there was no friction, no place for Stokes’law nor any appearance of Avogadro’s number. But Ein-stein did employ all of these ingredients in his theory. Per-haps that illustrates the difference between a mathemati-cal approach and one laden with physical insight.

The mathematics of Brownian motion is actually deepand subtle. Bachelier erred in defining a constant velocityv for a Brownian trajectory by taking the limit x t/ forsmall displacement x and time interval t. The properlimit involves forming the diffusion constant D x t= 2 / asboth x and t go to zero. In other words, because the ran-dom-walk displacement in Brownian motion grows only asthe square root of time, velocity scales like t− 1

2 and there-

BEYOND BROWNIAN

MOTIONFractal generalizations of Brownian

motion have proven to be a rich field inprobability theory, statistical physics and

chaotic dynamics.

Joseph Klafter, Michael F. Shlesinger,Gert Zumofen

© 1996 American Institute of Physics, S-0031-9228-9602-030-3 FEBRUARY 1996 PHYSICS TODAY 33

JOSEPH KLAFTER is a professor in the School of Chemistry atTel Aviv University in Israel. MICHAEL SHLESINGER is chiefscientist for nonlinear science at the Office of Naval Research, inArlington, Virginia. GERT ZUMOFEN is a lecturer and seniorresearch scientist at the Laboratory for Physical Chemistry of theEidgenössische Technische Hochschule in Zurich, Switzerland.

fore is not defined in the small-t limit.A Brownian trajectory does not possess a well-defined

derivative at any point. Norbert Wiener developed amathematical measure theory to handle this complication.He proved that the Brownian trajectory is continuous, butof infinite length between any two points. The Browniantrajectory wiggles so much that it is actuallytwo-dimensional. Therefore an area measure is more ap-propriate than a length measure. Lévy flights have a di-mension somewhere between zero and two.

Among the methods that have been explored to go be-yond Einstein’s Brownian motion is fractal Brownian mo-tion,1 which incorporates self-similarity and produces atrajectory with a mean squared displacement that growswith time raised to a power between zero and two. It is acontinuous process without identifiable jumps, and it hasbeen used to model phenomena as diverse as price fluctua-tions and the water level of the Nile. Two other venturesbeyond traditional Brownian motion are fractal distribu-tions of waiting times between random-walk steps andrandom walks on fractal structures such as percolationlattices. These two examples lead to slower-than-lineargrowth of the mean squared displacement with time. Butin this article we focus on random walk processes that pro-duce faster-than-linear growth of the mean squared dis-placement.

Another approach to generalizing Brownian motion isto view it as a special member of the class of Lévy-flightrandom walks. Here we explore applications of Lévyflights in physics.3 The interest in this area has grownwith the advent of personal computers and the realizationthat Lévy flights can be created and analyzed experimen-tally. The concept of Lévy flight can usefully be applied toa wide range of physics issues, including chaotic phase dif-fusion in Josephson junctions,4 turbulent diffusion,5,6 mi-celle dynamics,7 vortex dynamics,8 anomalous diffusion inrotating flows,9 molecular spectral fluctuations,10 trajec-tories in nonlinear Hamiltonian systems,11–17 moleculardiffusion at liquid–solid interfaces,18, transport in turbu-lent plasma,19 sharpening of blurred images20 and nega-tive Hall resistance in anti-dot lattices.21 In these com-plex systems, Lévy flights seem to be as prevalent asdiffusion is in simpler systems.

Lévy flightsThe basic idea of Brownian motion is that of a randomwalk, and the basic result is a Gaussian probability distri-bution for the position of the random walker after a time t,with the variance (square of the standard deviation) pro-portional to t. Consider an N-step random walk in one di-mension, with each step of random length x governed bythe same probability distribution p x( ), with zero mean.

LÉVY FLIGHT RANDOM WALK of 1000 steps in twodimensions. For clarity, a dot is shown directly below eachturning point. Limited resolution of this plot makes itdifficult to discern indivdual turning points. They tend tocluster in self-similar patterns characteristic of fractals.Occasional long flight segments initiate new clusters. Thelonger the step, the less likely is its occurence. But Lévyflights have no characterisic length. (See the box on page 35.)FIGURE 1

ROTATING-ANNULUS APPARATUS withwhich Harry Swinney’s group at theUniversity of Texas investigate Lévy

flights and anomalous transport inliquids. The flow is established by

pumping liquid into the rotating annulusthrough holes in its bottom. Showing

off the equipment are (clockwise)graduate student Eric Weeks and postdoc

Jeff Urbach. FIGURE 2

34 FEBRUARY 1996 PHYSICS TODAY

The French mathematician Paul Lévy (1886–1971) posedthe question: When does the probability P XN ( ) for thesum of N steps X X X XN= + + +1 2 ... \ have the same dis-tribution p x( ) (up to a scale factor) as the individual steps?This is basically the question of fractals, of when does thewhole look like its parts. The standard answer is that p x( )should be a Gaussian, because a sum of N Gaussians isagain a Gaussian, but with N times the variance of theoriginal. But Lévy proved that there exist other solutionsto his question. All the other solutions, however, involverandom variables with infinite variances.

Augustine Cauchy, in 1853, was the first to realizethat other solutions to the N-step addition of random vari-ables existed. He found the form for the probability whenit is transformed from real x space to Fourier k space:

~ ( ) ( | | )p k exp N kN = −b

(1)

Cauchy’s famous example is the case β =1, which, whentransformed back into x space, has the form

p xN

Nx N N

p x NN ( )( / )

( / )= + =1 1 11π

(2)

This is now known as the Cauchy distribution. It showsexplicitly the connection between a one-step and an N-stepdistribution. Exhibiting this scaling is more importantthan trying to describe the Cauchy distribution in terms ofsome pseudo-variance, as if it were a Gaussian.

Lévy showed that b in equation 1 must lie between 0and 2 if p x( ) is to be nonnegative for all x, which is re-quired for a probability. Nowadays the probabilities repre-sented in equation 1 are named after Lévy. When the ab-solute value of x is large, p x( ) is approximately | |x

− −1 b,which implies that the second moment of p x( ) is infinitewhen b is less than 2. This means that there is no charac-teristic size for the random walk jumps, except in theGaussian case of b =2. It is just this absence of a charac-teristic scale that makes Lévy random walks (flights)scale-invariant fractals. The box at right makes this moreevident with an illustrative one-dimensional Lévy-flightprobability function, and figure 1 is a plot of 1000 steps ofa similar Lévy flight in two dimensions.

Despite the beauty of Lévy flights, the subject has

FEBRUARY 1996 PHYSICS TODAY 35

OBSERVING FLUID FLOW with tracerparticles in the rotating-annulusapparatus shown in figure 2 revealsa: a state of six stable vortices thatfrequently trap the particle as it circlesaround the annulus. b: In anotherobserved flow state, the particle spendsless time trapped in vortices. c: Plottingazimuthal displacement of severalparticle trajectories against time showsthat the b trajectory (purple), exhibitinglong flights between vortex captures,gets around the annulus much fasterthan the a trajectory (red). (Courtesy ofH. Swinney.) FIGURE 3

A simple Lévy-flight random walk

Consider a one-dimensional random walk designed to illus-trate the self similarity of Lévy flights. Start with the dis-

crete jump-probability distribution

p x x b x bj

j j j( ) [ ( , ) ( , )]= − + + −=

∞−∑l

ll d d

1

2 0

where b > >l 1 are parameters that characterize the distribu-tion, and d(x, y) is the Kronecker delta, which equals 1 whenx y= and otherwise vanishes.

This distribution function allows jumps of length 1, b, b2,b3 . . . . But note that whenever the length of the jump in-creases by an order of magnitude (in base b), its probability ofoccurring decreases by an order of magnitude (in base l).Typically one gets a cluster of l jumps roughly of length 1 be-fore there is a jump of length b. About l such clusters separatedby lengths of order b are formed before one sees a jump of orderb 2. And so it goes on, forming a hierarchy of clusters withinclusters.

Figure 1 shows a two-dimensional example of this kind ofrandom walk.

The Fourier transform of p(x) is

~( ) ( )p k b kj

j j= −

=

∞−∑l

ll

1

2 0

cos

which is the famous self-similar Weierstrass function. So wecould call this the Weierstrass random walk. Theself-similarity of the Weierstrass curve appears explicitlythrough the equation

~( ) ~( ) ( )p k p bk k= + −1 1

ll

lcos

which has a small-k solution

~( ) ( | | )p k k≈ −exp b

with b = log log( )λ/ b . That is the form of the Lévy probabilityin equation 1 of the main text.

been largely ignored in the physics literature, mostly be-cause the distributions have infinite moments. The firstpoint we wish to make is that one should focus on the scal-ing properties of Lévy flights rather than on the infinitemoments. The divergence of the moments can be tamedby associating a velocity with each flight segment. Onethen asks how far a Lévy walk has wandered from itsstarting point in time t, rather than what is the meansquared length of a completed jump. The answer to thefirst question will be a well-behaved time-dependent mo-ment of the probability distribution, while the answer tothe second is infinity. Specifically, a Lévy random walkermoving with a velocity v, but with an infinite mean dis-placement per jump, can have a mean squared displace-ment from the origin that varies as v2 t2. (See the box onpage 37.) Even faster motion is possible if the walker ac-celerates, as we shall see when we come to the phenome-non of turbulent diffusion.

Lévy walks in turbulenceTo employ Lévy flight for trajectories, one introduces ve-locity through a coupled spatial–temporal probability den-sity ψ(r t, ) for a random walker to undergo a displacementr in a time t. We write

ψ( ψ(r t t r p r, ) | ) ( )= (3)

The factor p r( ) is just the probability function, discussedabove, for a single jump. The ψ(t r| ) factor is the probabil-ity density that the jump takes a time t, given that itslength is r. Let us, for simplicity, make ψ(r t, ) the Diracdelta function δ\ ( | |/ ( ))t r v r− , which ensures that r vt= .Such random walks, with explicit velocities, visit all pointsof the jump on the path between 0 and r. They are calledLévy walks as distinguished from Lévy flights, which visitonly the two endpoints of a jump.

The velocity v need not be a constant; it can depend onthe size of the jump. A most interesting case is turbulentdiffusion. In 1926 Lewis Fry Richardson published hisdiscovery that the mean square of the separation r be-tween two particles in a turbulent flow grows like t3. Di-mensional analysis of Brownian motion tells us that< > =r t Dt2 ( ) . This means that the diffusion constant Dcan be endowed with a specific space or time depend-ence—for example, D r r( ) ≈

43 or D t t( ) ≈ 2—to produce tur-

bulent diffusion.Richardson chose the D r r( ) /≈ 4 3 route. Note that a

36 FEBRUARY 1996 PHYSICS TODAY

FRACTAL COMPLEXITY OF THE CANTORI-ISLAND STRUCTURE

around a period-3 orbit emerges with successive magnificationsof a plot of the standard map (equation 5) with K = 1.1. a: Anorbit with a 3-step period emerges when periodic boundaryconditions are imposed in the x direction. Points near thisorbit stick close by for some time before coming under theinfluence of substructure. b: A tenfold magnification of thered island loop in 4a reveals 7 subislands. c: Furtherexpanding the rightmost loop in 4b reveals 10 sub-subislands.FIGURE 4

diffusion constant has dimensions of [rv]. Therefore Rich-ardson’s 4/3 power law dimensionally implies that v r2 ( )scales like r2 3/ . Then the Fourier transform, v k2 ( ), mustscale like k−5 3/ .

This last result, first stated in 1941, is Kolmogorov’swell-known inertial-range turbulence spectrum. Althoughit looks as if Richardson could have predicted theKolmogorov spectrum, the two scaling laws do not neces-sarily imply each other. Only if the Kolmogorov scaling(v r r( ) /≈ 1 3) is combined with Lévy flights (p r r( ) ≈ −1 β), sothat the mean absolute value of r is infinite, does one re-cover Richardson’s result: < > =r t t2 3( ) . If p(r) decays fastenough so that all of its moments are finite, then one re-covers the standard Brownian law, even with Kolmogorovscaling. The main point here is that the Lévy walk withKolmogorov scaling describes aspects of turbulent diffu-sion. The box on this page shows an example of the possi-ble scaling laws for the constant-velocity Lévy randomwalks found in dynamical systems.

An important feature of the Lévy-walk approach toturbulent diffusion is that it provides a method for simu-lating trajectories of turbulent particles. Fernand Hayot6has used this procedure to describe turbulent flow in pipesby means of Lévy-walk trajectories with Kolmogorov veloc-ity scaling in a lattice gas simulation.

Two-dimensional fluid flowIn two-dimensional computer simulations of fluid flow,James Viecelli8 has found Lévy walks and enhanced diffu-sion with mean squared displacement scaling like tg withg =167. for point vortices all spinning in the same sense athigh temperatures. At low temperatures the vortices forma rotating triangular lattice. As the temperature is raised,the motion of the vortices eventually becomes turbulent.The vortices cluster and rotate around a common center,but now with Lévy-walk paths. If both clockwise andcounterclockwise vortices are present, the scaling expo-nent g becomes 2.6. Oppositely spinning vortices pair andthe pairs move in straight-line Lévy walks, changing di-rection when they collide with other pairs.

These results suggest that two-dimensional rotatingflows are fertile territory for seeking anomalous diffusion.They might serve to approximate large-scale global atmo-spheric and oceanic flows. We predict that Lévy walks willbecome increasingly important for understanding global

environmental questions of transport and mixing in theatmosphere and the oceans.

Harry Swinney and coworkers at the University ofTexas have been investigating quasi-two-dimensional fluidflow in a rotating laboratory vessel.9 (See figure 2.) Intheir experiment, a fluid-filled annulus rotates as a rigidbody. The flow is established by pumping fluid throughholes in the bottom of the annulus. In this nearly

FEBRUARY 1996 PHYSICS TODAY 37

DISTRUBUTION OF TIME it takes astandard-map trajectory (with K = 1.03)to leave the Cantori structure around aperiod-5 orbit, plotted as a function ofthe trajectory’s initial ( ,x q) position.The time distribution exhibits acomplex fractal hierarchy of stickingregions. The black areas mark islands ofstability inside the period-5 orbit, fromwhich a trajectory will never leave.FIGURE 5

Mean squared displacement

The mean squared displacement in simple Brownian mo-tion in one dimension has a linear dependence on time

< > ≈x t Dt2 2( )

where D is the diffusion constant and < >x t2( ) is the secondmoment of the Gaussian distribution that governs the proba-bility of being at site x at time t. The Gaussian is the hallmarkof Brownian motion. In the case of Lévy flights, by contrast,the mean squared displacement diverges. Therefore we con-sider the more complicated random walk described by theprobability distribution of equation 3, which accounts for thevelocity of the Lévy walk. For the Kolmogorov turbulencecase, the velocity depended on the jump size. But here we con-sider the simpler constant-velocity case with

ψ( βr t t, ) ( )≈ + +1 1 1

being the probability density for time spent in a flightsegment. For various values of b, that leads to the followingdifferent time dependences of the mean square displacement:

< >≈

< <=

< <−r t

t for

t t for

t for

t t for

2

2

2

3

0 1

1

1 2( )

/ ln

ln

βββ

β

β

=>

2

t for forβ 2

The enhanced diffusion observed in various dynamical sys-tems, such as the standard map (equation 5 in the main text),corresponds to the case 1 2< <b . Similar results, with timedependences ranging from t 3 down to t, are found in turbulentdiffusion.

two-dimensional flow, neutrally buoyant tracer particlescan be carried along in Lévy walks even though the veloc-ity field is laminar. In fact, Swinney and company wereable to make direct measurements of Lévy walks and en-hanced diffusion in this experimental system. (See figure3.) An instability of the axisymmetrically pumped fluidled to a stable chain of six vortices, as shown in figure 3a.Tracer particles were followed to produce probability dis-tributions for the distance and time a particle can travelbefore it gets trapped in a vortex

We use n, the number of vortices passed by a tracerparticle, as a convenient measure of the travel distance r.When a particle leaves a vortex and then passes n vorticesin a counterclockwise (or clockwise) fashion before it istrapped again, we count that as a random walker jumpingn units to the right (or left) at a constant velocity. InLévy-walknotation,taking ψ ( δ(

walkt r t n| ) | |)= − , the Uni-

versity of Texas experimenters found

p n r nwalk ( ) | | | |.≈ =− −2 3 1 β (4)

and they found that the mean time the particle spendstrapped in a vortex is finite. Under these conditionsLévy-walk theory implies that the variance< > − < >r t r t2 2( ) ( ) should scale like t3 − b. (See the box onpage 37.) That turns out to be quite consistent with theexperimental data, and it gives us a beautiful example ofenhanced diffusion.

Lévy walks in nonlinear dynamics: MapsThe Lévy-walk case of dynamic scale-invariant circularmotion with constant speed was introduced in 1985 byTheo Geisel and coworkers at the University ofRegensburg, in Germany.4 Their investigations involvedthe study of chaotic phase diffusion in a Josephson junc-

tion by means of a dynamical map, that is to say, a simplerule for generating the nth discrete value of a dynamicalvariable from its predecessor. With a constant voltageacross the junction, the phase rotates at a fixed rate, but itcan change direction intermittently. N complete clockwisevoltage-phase rotations correspond to a random walk witha jump of N units to the right. Analysis of experimentaldata led Geisel and his colleagues to a consideration of thenonlinear map x x axt t t

z+ = + + −1 1 1( )ε , with e small. Aver-

aging over initial conditions, they found that

< >≈

>< <

=

< <

x t

t for zt for z

t t for z

t for z

2

2 23

2 23

21 3

2

( )ln

γ

where g = − − −3 1 1( )z . This scaling behavior of the meansquare displacement corresponds to constant-velocity Lévywalk with a flight-time distribution between reversalsgiven by ψ ( )t t≈ − −1 b, where b = −1 1/ ( )z .

In the above example, one can go directly from z, theexponent of the map, to the temporal exponentb = −1 1\

\ ( )z for the mean-square displacement. In mostnonlinear dynamical systems, however, the connection be-tween the equations and the kinetics is not obvious or sim-ple. For example, the so-called standard map:

x x Ksinn n n+ = +1 2( )πθ (5)

θ θn n nx+ += +1 1

has no explicit exponent in sight. But when one plots theprogression of points x, q, noninteger exponents abound inthe description of the orbit kinetics, hinting at complex dy-namics.

Plotting successive (x versus q) points, at first onemight see simple, nearly closed circular orbits, eventuallygiving way to thoroughly chaotic wandering. But beforechaos sets in one can see orbits that exhibit hierarchicalperiodicities characteristic of the laminar segments ofLévy walks. The structures that emerge depend sensi-tively on the parameter K. With K =11. , for example, wesee (in figure 4a) a period-3 orbit. The heirarchical, fractalnature of its three loops (called Cantori islands, because

38 FEBRUARY 1996 PHYSICS TODAY

LÉVY FLIGHT BEHAVIOR is seen in the trajectory of acomputer-generated Zaslavsky map with fourfold symmetry.13

The random walk starts near the origin and the color ischanged every 1000 steps. FIGURE 6

they form something like a Cantor set of tori) is revealedin the successively magnified plots of figures 4b and 4c.This fractal regime, describing complex kinetics with Lévystatistics, has been dubbed “strange kinetics.”16

Figure 5 shows the distribution of the time a stan-dard-map x-versus-q trajectory spends around a period-5Cantori orbit that emerges when K = 1.03. The time dis-tribution exhibits a complex fractal hierarchy of stickingregions in the phase space. The probability densities forspending time in laminar states near period-3 and pe-riod-5 orbits have, respectively, the Lévy scaling behaviorst−2 2. and t−2 8. .

Another example of Lévy walks in dynamical systemsis provided by the orbits in a map devised by GeorgeZaslavsky of the Courant Institute.12,13 These walks forman intricate fractal web throughout a two-dimensionalphase space. The Zaslavsky map has only simple trigono-metric nonlinearities but, like the standard map, it re-quires noninteger exponents for the characterization of itstrajectories. Figure 6 shows a Lévy-like trajectory in a4-fold symmetric Zaslavsky map.

PotentialsOne might expect that motion in a simple periodic potentialshould look simple. But closer inspection of thephase-space structure of trajectories in a simpletwo-dimensional egg-crate potential reveals a behavior asrich as what we get from the standard and Zaslavskymaps11–13,15. It turns out that Lévy walks emerge quitenaturally in such potentials, and one finds enhanced diffu-sion. See figure 7. Although the trajectories possess longwalks with Lévy scaling in the x and y directions of theegg-crate array, the motion is complicated by intermittenttrapping in the potential wells. This trapping is related tothe observed vortex sticking of tracer particles in theSwinney group’s experiment. In neither case is the distri-bution of such “localizing” events broad enough to affectthe predicted enhanced-diffusion exponent.

Lévy walks also appear for time-dependent potentials.Igor Aranson and colleagues at the Institute for AppliedPhysics in Nizhny Novgorod (formerly Gorky) in Russia,investigated a potential that varies sinusoidally in timeand found that it produces Lévy walks similar to those onegets with static egg-crate potentials. All of these examplesindicate that we are exploring an exciting, wide-open field

when we venture beyond the traditional confines ofBrownian motion.

We thank Harry Swinney for providing his experimental resultsand for sharing his insights into Lévy walks in fluids. We alsothank George Zaslavsky for illuminating discussions and for sug-gesting the Lévy walk shown in figure 6.

References1. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San

Francisco (1982).2. A. Einstein, Annalen der Physik 17, 549 (1905).3. Lévy Flights and Related Topics in Physics, M. Shlesinger, G.

Zaslavsky, U. Frisch, Eds., Springer, Berlin (1995).4. T. Geisel, J. Nierwetberg, A. Zacherl, Phys. Rev. Lett. 54, 616

(1985).5. M. Shlesinger, B. West, J. Klafter, Phys. Rev. Lett. 58, 1100

(1987). M. Shlesinger, J. Klafter, Y. M. Wong, J. Stat. Phys. 27,499 (1982).

6. F. Hayot, Phys. Rev. A. 43, 806 (1991).7. A. Ott, J. Bouchaud, D. Langevin, and W. Urbach, Phys. Rev.

Lett. 65, 2201 (1990). J. Bouchaud, A. Georges, Phys. Reports195, 127 (1980).

8. J. Viecelli, Phys. Fluids A 5, 2484 (1993).9. T. Solomon, E. Weeks, H. Swinney, Phys. Rev. Lett. 71, 3975

(1993).10. G. Zumofen, J. Klafter, Chem. Phys. Lett. 219 303 (1994).11. T. Geisel, A. Zacherl, G. Radons, Z. Phys. B. 71, 117 (1988).12. A. Chernikov, B. Petrovichev, A. Rogalsky, R. Sagdeev, G.

Zaslavsky, Phys. Lett. A 144, 127 (1990).13. D. Chaikovsky, G. Zaslavsky, Chaos 1, 463 (1991).14. I. Aranson, M. Rabinovich, L. Tsimring, Phys. Lett. A 151, 523

(1990).15. J. Klafter, G. Zumofen, Phys. Rev. E 49, 4873 (1994).16. M.Shlesinger,G.Zaslavsky,J.Klafter,Nature363,31 (1993).17. R. Ramashanker, D. Berlin, J. Gollub, Phys. Fluids A 2, 1955

(1980).18. O. Baychuk, B. O’Shaughnessy, Phys. Rev. Lett 74, 1795 (1985).

S. Stapf, R. Kimmich, R. Seitter, Phys. Rev. Lett. 75, 2855(1995).

19. G. Zimbardo, P. Veltrei, G. Basile, S. Principato, Phys. Plasma2, 2653 (1995). R. Balescu, Phys. Rev. E 51, 4807 (1995).

20. A. Carasso, in Mathematical Methods in Medical Imaging II,SPIE vol. 2035 (1993), p. 255.

21. R. Fleischmann, T. Geisel, R. Ketzmerick, Europhys. Lett. 25,219(1994).

�

FEBRUARY 1996 PHYSICS TODAY 39

THROUGH AN EGG-CRATE potential in the x y, plane, acomputer-generated 105-step random walk exhibits Lévy flight.The second moment of the step-length distribution is infinite.Between long flights, the trajectory gets trapped intermittentlyin individual potential wells. FIGURE 7