PRL Top Ten: #9
Diffusion Limited Aggregation, A Kinetic Critical Phenomenon
T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47 (1981) 1400), 2155 citations
When Tom Witten and Len Sander set out to model the growth of clusters in aerosols, one molecule at a time, they had no idea that the emerging patterns that appeared on their pen plotter would eventually ignite a flurry of activity in fields ranging from biology to astroscience. In the past twenty years, Diffusion Limited Aggregation (DLA) has helped to describe the origins of fractal patterns in electrodeposited metals, newly formed river basins, bacterial colonies, blood vessels in the eye, and the initial stages of urban sprawl, to name only a few applications.
As Sander explains it, the algorithms that lead to DLA are generally very simple. Imagine a single particle that serves a nucleation center. If another particle is released from some distant point, it meanders randomly through space, and it may either slip away to infinity or stick to the nucleation center. As subsequent particles wander by, they are more likely to stick to regions that protrude from the aggregate, leading to growing arms that creep forward and occasionally divide. In effect, the arms shield the inner portions of the aggregate, leaving the voids that characterize wispy fractal structures.
"We were doing a poor man's form of solidification," recalls Witten, "and we thought, on the one hand, that we were just going to end up with a solid lump of particles. On the other hand, we thought we had the elements that give you dendritic growth, and dendritic growth leads to these branching, treelike things that are not solid at all. We would have been prepared to see either thing, so we were very happy when it turned out to be tenuous."
Both Sander and Witten were surprised that their simple algorithm seemed to capture the essence of so many other phenomena. "Obviously," says Sander, "bacterial colonies, viscous liquids, and other systems have different origins. But the algorithms are basically the same, under the right circumstances."
Nevertheless, it took a few years before the importance of the DLA was widely realized. The initial hesitation to embrace DLA in other areas had to do in part with the deceptive simplicity of the algorithm. "When I would tell people about it," says Witten, "they would say 'Oh that's kind of interesting. Are you going to put it in a journal of mathematical physics?', or some other journal. They didn't realize that it was big deal."
Oddly enough, the fact that DLA-like growth is so ubiquitous in nature was also a hindrance to acknowledgment of the algorithm's importance. Sander points out that numerous researchers had come close to discovering DLA. "There were many people who just didn't know the implications of what they had," says Sander, "I attribute [our insight] to the fact that we were able to visualize the structure, and to Mandelbrot's work effectively saying 'You know, you really ought to think about shapes, and particularly about odd shapes.'"
By the mid 1980's, a flood of papers citing Witten and Sander started to appear in the literature as other researchers began to notice that DLA linked two emerging areas of research. "The reason this idea eventually struck lightning," Witten explains, "is that it connected two big things which people really didn't see how to connect before: dendritic growth on the one hand, and critical phenomena on the other hand. And they both need to be there for people to gloom onto it."
Although, the broad applicability and simplicity of DLA algorithms certainly help explain initial interest in the ideas of Witten and Sander, both researchers believe the challenge of finding an underlying theory is the likely reason that their paper continues to rack up citations. "I would have felt that it was amazing to hit on something that captured peoples' imagination even a tenth as much as this, and for it to keep going on is surprising," says Witten, "But I think what's truly unexpected is that it turned out to be one of these problems that just resists being solved. I would have thought would have been solved before 1990. That's really why I think it keeps going on."
Sander is confident that a workable theory will eventually emerge. "We should be able to reduce the problem to something less complex than the algorithm itself, "says Sander. "It should be possible to gain ideas about the overall behavior of DLA clusters without actually doing simulations. We think we've made a good deal of progress along those lines. Much remains to be done, but the sort of glimmerings of a real theory are now, I think, available."
Although Witten achieved a certain amount of renown, particularly in Europe, for his involvement with early DLA research, he has spent most of his time in the last two decades pursuing other subjects. As a professor of physics at the University of Chicago, he is currently concentrating on the study of granular materials and crumpling of polymer sheets.
Sander has been more or less involved in pursuing a theory of DLA for much of his career. In glancing at the collection of student theses on the bookshelf of his office in the University of Michigan physics department, Sander notes that few of the theses from the early 90's are related to DLA. The lull corresponds to a period that his own interest in the subject waned. At times, Sander says that he has felt about DLA the way that Arthur Conan Doyle felt about Sherlock Holmes-that is, Doyle sometimes wished he could kill off Holmes, but the popularity and success of the character forced him to continually revive the legendary detective.
How does Sander feel about DLA now? "Well, Sherlock Holmes is still alive," laughs Sander. "Seriously though, DLA is a nice picture, it sneaks up on you. When you look at it sort of cursorily, you can get a little bit bored. But when you start looking deep into it, it's pretty fascinating."
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