BEYOND BROWNIAN MOTION ewtonian physics began Fractal generalizations of Brownian thermal motions of fluid Nwith an attempt to molecules striking the micro- make precise predictions motion have proven to be a rich field in scopic particle and causing it about natural phenomena, probability theory, statistical physics and to undergo a random walk. predictions that could be ac- Einstein's famous paper curately checked by observa- chaotic dynamics. was entitled "Uber die von tion and experiment. The der molekularkinetischen goal was to understand na- Theorie der Warme gefor- ture as a deterministic, Joseph Klafter, Michael F. Shlesinger, derte Bewegung von in "clockwork" universe. The Gert Zumofen ruhenden Flussigkeiten sus- application of probability dis- pendierten Teilchen," that is tributions to physics devel- to say, "On the motion, re- oped much more slowly. Early uses of probability argu- quired by the molecular-kinetic theory of heat, of particles ments focused on distributions with well-defined means suspended in fluids at rest." Einstein was primarily and variances. The prime example was the Gaussian law exploiting molecular motion to derive an equation with of errors, in which the mean traditionally represented the which one could measure Avogadro's number. Apparently most probable value from a series of repeated measure- he had never actually seen Brown's original papers, which ments of a fixed quantity, and the variance was related were published in the Philosophical Magazine. "It is to the uncertainty of those measurements. possible," wrote Einstein, "that the motions discussed here But when we come to the Maxwell-Boltzmann distri- are identical with the so-called Brownian molecular mo- bution or the Planck distribution, the whole distribution tion. But the references accessible to me on the latter has physical meaning. Being away from the mean is no subject are so imprecise that I could not form an opinion longer an error, and a large variance is no longer an about that."2 Einstein's prediction for the mean squared indicator of poor measurement accuracy. In fact the whole displacement of the random walk of the Brownian particle distribution is the prediction. That was a major concep- was a linear growth with time multiplied by a factor that tual advance. involved Avogadro's number. This result was promptly In this article we take that idea to its extreme limit used by Jean Perrin to measure Avogadro's number and and investigate probability distributions, called Levy dis- thus bolster the case for the existence of atoms. That tributions, with infinite variances, and sometimes even work won Perrin the 1926 Nobel Prize in Physics. infinite means. These distributions are intimately con- Less well known is the fact that Louis Bachelier, a nected with fractal random-walk trajectories, called Levy student of Henri Poincare, developed a theory of Brownian flights, that are composed of self-similar jumps.1 Levy motion in his 1900 thesis. Because Bachelier's work was flights are as widely applied in nonlinear, fractal, chaotic in the context of stock market fluctuations, it did not and turbulent systems as Brownian motion is in simpler attract the attention of physicists. He introduced what systems. is today known as the Chapman—Kolmogorov chain equa- tion. Having derived a diffusion equation for random Brownian motion processes, he pointed out that probability could diffuse in The observation of Brownian motion was first reported in the same manner as heat. (See PHYSICS TODAY, May 1995, 1785, by the Dutch physician Jan Ingenhausz. He was page 55.) looking at powdered charcoal on an alcohol surface. But Bachelier's work did not lead to any direct advances the phenomenon was later named for Robert Brown, who in the physics of Brownian motion. In the economic published in 1828 his investigation of the movements of context of his work there was no friction, no place for fine particles, including pollen, dust and soot, on a water Stokes' law nor any appearance of Avogadro's number. But surface. Albert Einstein eventually explained Brownian Einstein did employ all of these ingredients in his theory. motion in 1905, his annus mirabilis, in terms of random Perhaps that illustrates the difference between a mathe- matical approach and one laden with physical insight. The mathematics of Brownian motion is actually deep JOSEPH KLAFTER is a professor in the School of Chemistry at and subtle. Bachelier erred in defining a constant velocity Tel Aviv University in Israel. MICHAEL SHLESINGER is chief v for a Brownian trajectory by taking the limit x/t for scientist for nonlinear science at the Office of Naval Research, small displacement x and time interval t. The proper in Arlington, Virginia. GERT ZUMOFEN is a lecturer and limit involves forming the diffusion constant D =x2lt as senior research scientist at the Laboratory for Physical both x and t go to zero. In other words, because the Chemistry of the Eidgenossische Technische Hochschule in random-walk displacement in Brownian motion grows Zurich, Switzerland. only as the square root of time, velocity scales like r' • • 1996 American Institute of Physics, S-0031-9228-96024)30-3 FEBRUARY 1996 PHYSICS TODAY 33 LEVY FLIGHT RANDOM WALK of 1000 steps in two dimensions. For clarity, a dot is shown directly below each turning point. Limited resolution of this plot makes it difficult to discern indivdual turning points. They tend to cluster in self-similar patterns characteristic of fractals. Occasional long flight segments initiate new clusters. The longer the step, the less likely is its occurence. But Levy flights have no charactensic length. (See the box on page 35.) FIGURE 1 and therefore is not defined in the small-? limit. squared displacement. A Brownian trajectory does not possess a well-defined Another approach to generalizing Brownian motion is derivative at any point. Norbert Wiener developed a to view it as a special member of the class of Levy-flight mathematical measure theory to handle this complication. random walks. Here we explore applications of Levy He proved that the Brownian trajectory is continuous, but flights in physics.3 The interest in this area has grown of infinite length between any two points. The Brownian with the advent of personal computers and the realization trajectory wiggles so much that it is actually two-dimen- that Levy flights can be created and analyzed experimen- sional. Therefore an area measure is more appropriate tally. The concept of Levy flight can usefully be applied than a length measure. Levy flights have a dimension to a wide range of physics issues, including chaotic phase somewhere between zero and two. diffusion in Josephson junctions,4 turbulent diffusion,56 Among the methods that have been explored to go micelle dynamics,7 vortex dynamics,8 anomalous diffusion beyond Einstein's Brownian motion is fractal Brownian in rotating flows,9 molecular spectral fluctuations,10 tra- motion,1 which incorporates self-similarity and produces jectories in nonlinear Hamiltonian systems,11"17 molecular a trajectory with a mean squared displacement that grows diffusion at liquid-solid interfaces,18, transport in turbu- with time raised to a power between zero and two. It is lent plasma,19 sharpening of blurred images20 and negative a continuous process without identifiable jumps, and it Hall resistance in anti-dot lattices.21 In these complex has been used to model phenomena as diverse as price systems, Levy flights seem to be as prevalent as diffusion fluctuations and the water level of the Nile. Two other is in simpler systems. ventures beyond traditional Brownian motion are fractal distributions of waiting times between random-walk steps Levy flights and random walks on fractal structures such as percola- The basic idea of Brownian motion is that of a random tion lattices. These two examples lead to slower-than-lin- walk, and the basic result is a Gaussian probability ear growth of the mean squared displacement with time. distribution for the position of the random walker after a But in this article we focus on random walk processes time t, with the variance (square of the standard deviation) that produce faster-than-linear growth of the mean proportional to t. Consider an ./V-step random walk in one ROTATING-ANNULUS APPARATUS with which Harry Swinney's group at the University of Texas investigate Levy flights and anomalous transport in liquids. The flow is established by pumping liquid into the rotating annulus through 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 OBSERVING FLUID FLOW with tracer particles in the rotating-annulus apparatus shown in figure 2 reveals a: a state of six stable vortices that frequently trap the particle as it circles around the annulus. b: In another observed flow state, the particle spends less time trapped in vortices, c: Plotting azimuthal displacement of several particle trajectories against time shows that the b trajectory (purple), exhibiting long flights between vortex captures, 400 800 1200 gets around the annulus much faster TIME (seconds) than the a trajectory (red). (Courtesy of H. Swinney.) FIGURE 3 dimension, with each step of random length x governed by the same probability distribution p(x), with zero mean. A simple Levy-flight random walk The French mathematician Paul Levy (1886-1971) posed onsider a one-dimensional random walk designed to illus- the question: When does the probability PN(X) for the trate the self similarity of Levy flights. Start with the sum of N steps X = Xx + X2 + . + XN have the same dis- tribution p{x) (up to a scale factor) as the individual steps? discrete jump-probability distribution This is basically the question of fractals, of when does the whole look like its parts.