“Many things in nature are much more ordered than you might expect from a description of the laws of physics,” observes Lionel Levine, Mathematics. Levine has a knack for finding order where someone else might see chaos. Levine’s expertise, probability, measures the likelihood of an event or outcome. Any single roll of the dice defies certainty, but with enough perspective—over the course of many crapshoots—even chance has a pattern.
“What always fascinated me,” Levine says, “is how many different types of order there are in our universe, all coexisting in the same space.” This fascination drew Levine to one focus of his research: abelian sandpiles. Levine describes abelian sandpiles as a toy universe—a mathematical model that captures aspects of the real world but with simpler rules.
Abelian Sandpiles, Visualizing Order
To visualize an abelian sandpile, start with a square grid. (This is for simplicity's sake. Mathematically, nothing prevents a sandpile from being built on polygonal lattices or having three or more dimensions.) Each square, or vertex, can hold a number of stackable chips, up to a predefined limit. Above that limit, the vertex becomes unstable and topples—sending one chip to each of its four adjacent vertices. One toppling vertex may in turn cause one or more of its neighboring vertices to topple. Every vertex continues to topple—losing four chips at a time to its neighbors—until its stack of chips arrives at or below the limit.
Say, for instance, each vertex can hold no more than three chips without toppling. Now add a million chips to a single vertex.
The result? In computer renderings, sandpiles collapse into otherworldly, polychromatic blossoms, each color indicating the number of chips remaining on a vertex. “When it stabilizes, it has this heterogeneous order,” Levine says. “Certain parts of the picture, if you zoom in on them, have a periodic pattern. And if you zoom on a different part, you find a different periodic pattern. A big strand of my work has been exploring how and why this sandpile is able to do this.”
For What Is Nature Optimizing?
The peculiarities don’t stop there. For any particular sandpile, the final outcome is the same regardless of the order of topplings. Most importantly, sandpiles are what Levine calls lazy. He means it as a compliment: “The sandpile is trying to optimize something. It’s trying to become stable in as few topplings as possible.”
“To me, the idea that nature is optimizing something is a recurring theme,” Levine says. “It’s a shift in thinking. Instead of thinking of systems as just doing something, if you think of them as optimizing something—or you ask what they are optimizing—that’s often a useful framework.”
He adds, “Sometimes I wonder what is human global society trying to optimize? I’m not sure we know the answer to that. Many would argue we are not trying to optimize for what we should be trying to optimize. That maybe we should be trying to optimize for sustainability or even for existence of humanity a couple hundred years from now. As soon as you ask what a system is trying to optimize, if it’s a system you have some control over, you can talk about whether you should try to influence what it’s optimizing.”
The Natural Long-Range Order of Sand Dunes, and Criticality
In 1987 physicists Per Bak, Chao Tang, and Kurt Wiesenfeld proposed abelian sandpiles to account for how often we observe long-range order in nature—like sand dunes. “If you go out to the Sahara somewhere and you look at sand dunes, you find that they have a characteristic slope,” Levine says. “It’s a certain angle that they like to sit at.”
Long-range order, in the Sahara and throughout the universe, exhibits a property that physicists call criticality. Prior to the sandpile, mathematical models suggested that criticality should be unusual, occurring only when a key variable stands at a precise value. Explains Levine, “What Per Bak and coauthors said is, ‘If there’s typically only one critical value, why should we expect systems in nature to be tuned to that exact value? It should be rare to observe long-range order. Yet we observe it everywhere.’”
“If your sandpile is too flat, then natural processes like wind blowing sand will tend to build it up.”
Abelian sandpiles, like actual sand dunes, do not require precise tuning to generate long-range order. Instead, they tend toward long-range order—what Bak, Tang, and Wiesenfeld call self-organized criticality. As Levine explains, “If your sandpile is too flat, then natural processes like wind blowing sand will tend to build it up, because whenever you get more sand, it might trickle down a little bit, but you’re not likely to get big avalanches that would flatten it out. On the other hand, if the sandpile somehow grew to be too steep, greater than the critical slope, then even little perturbations like a little gust of wind or something are very likely to set off a massive avalanche that would flatten it back out.”
Making Predictions about Natural Systems
Using sandpiles, Levine is building proofs to explain the behavior of critical systems. “What we’d like to be able to do is make predictions about them,” says Levine. “Instead of a pile of sand, think of natural disasters like forest fires or earthquakes. It would be great if we could predict when is the next giant earthquake going to hit, right? It’s something humanity should strive for.”
Levine also approaches sandpiles as a special kind of abelian network, a collection of independent processors (the vertices)—each performing one simple function, to topple or not to topple—that together generate an optimized solution even when the processors perform their updates in an unpredictable order. This makes sandpiles potentially useful models for many complex networks—the Internet, the economy, traffic patterns, and even the human brain.
Levine himself likes working collaboratively, and he gets energy from his students. Several years ago, a math major asked Levine for a senior thesis problem. When the student returned with the answer just a week later, Levine was amazed. For one subset of sandpiles, the student found a surprisingly simple solution. “It went against all my experience with sandpiles,” says Levine. Together they published a short paper building on the student’s thesis project. “In retrospect, it was staring us in the face,” says Levine. “It was just a matter of asking the right question and having the courage to think the answer could be so simple.”