The search for a grain of truth in a pile of rice

September 29, 2000

Could a falling grain of rice help historians explain the outbreak of war or a stock-market crash? Some physicists believe it could. Mark Buchanan reports

In 1934, the historian H. A. L. Fisher came to a pessimistic conclusion about the character of history. "Men wiser and more learned than I have discerned in history a plot, a rhythm, a predetermined pattern," he wrote. "These harmonies are concealed from me. I can see only one emergency following upon another I and only one safe rule for the historian: that he should recognise in the development of human destinies the play of the contingent and the unforeseen."

It is easy to see why Fisher might have thought this way. He was writing just five years after the great stock-market crash of 1929 and only 20 years after a chauffeur in Sarajevo had sparked the first world war by taking a wrong turn, thereby allowing an assassin to gun down the Austro-Hungarian archduke. To Fisher and his contemporaries, history seemed a string of meaningless but dangerous accidents, with tumultuous upheavals always lurking around the corner.

Historians nowadays may see this attitude as unduly despairing. And yet it is safe to say that history as a science is little closer to answering a host of deep questions about the natural dynamics and rhythms of the past. Are there any "harmonies" in history? Can gradual trends be trusted to continue? Can anyone forecast the future with even a modicum of success, or was Fisher right?

A fresh perspective on these questions may emerge from an unlikely direction - theoretical physics. New mathematical ideas point to the possibility that Fisher was correct - history is unpredictable and may always resist understanding in terms of general causes and effects. Something akin to a law of nature appears to dictate that the course of history be punctuated, sporadically but frequently, by seemingly inexplicable upheavals.

Over the past decade, a growing number of physicists have become intrigued by a deceptively simple problem. Imagine sprinkling grains of rice onto a table, one by one, so that a pile begins to grow. As it grows, avalanches will occasionally carry some grains away. This is obvious, and yet a detailed understanding of what happens is difficult to come by. Where and when should we expect the avalanches? What is their typical size? What sets up conditions for the really big ones?

Physicists care about this problem only because it is a simple example of a process that is out of equilibrium - that is, a process in which the forces at work cannot find their immediate release. Sprinkle droplets of water over a full bathtub and each drop will quickly melt into the pool. Here the forces find their release and the result is monotonous sameness because nothing builds up. Not so with the pile of grains. In this case, stress accumulates slowly, only to be released in avalanches that follow some inscrutable rhythm.

In studying this problem, physicists have been hoping to gain clues about the typical rhythms of things that are out of equilibrium. But there is a connection to history: what makes the pile "out of equilibrium" also gives it a history. When a grain falls in one place rather than another, the event alters the course of the future. As in human history, the future emerges out of a continuous stream of events with irreversible consequences that mingle and fight with one another.

So a pile of rice is a kind of mathematical toy that distills the essence of a historical process. And if this pile is one of the simplest of all conceivable historical processes, we might expect to discover some of its basic features reflected in more complicated settings. Conversely, if we cannot come to terms with the rhythm of a rice pile, how can we hope to make headway with 6 billion human beings?

So how does the pile work?

Physicists have discovered that at first, as the pile grows, a falling grain can trigger only small avalanches and the resulting chain reaction will die out fast. But as the pile increases, many grains come to be balanced uneasily on steep slopes. Now a single grain can potentially trigger a domino-like reaction that can sweep through the entire pile. Physicists call this condition the "critical state".

At this point, we might ask a simple question - if we drop another grain, how big an avalanche will it trigger? The remarkable feature of the critical state is that we cannot answer the question. There is no "typical" or "expected" size of avalanche. The pile evolves naturally to a state in which the next grain may stick and cause no avalanche at all, or may trigger anything from the tumbling of a few grains to a system-wide cataclysm. Anything is always just about to happen.

What could this teach the historian? It is one of our deepest beliefs that great events have correspondingly great causes - that it is sensible to look for exceptional conditions that made the first world war take place, for example, conditions that a perceptive observer might have been able to recognise as the precursor to disaster. If possible, we would like to learn lessons from history and so be better prepared for the future.

But consider a historian of the rice pile who approaches his task with such a belief. In the rice world, every event begins in the same way - when one more grain falls on the pile. Yet because the pile is tuned to the critical state, the consequences of each grain are wildly unpredictable. If one day a great avalanche carries half the pile away, our historian might well narrate the grain-by-grain history of what took place. But this "explanation" would offer no insight into why one particular grain triggered such a great event, nor would it offer any lessons about when we might expect another.

The rice-pile historian would learn the disconcerting truth that there was nothing special about that particular grain and nothing unusual about the condition of the pile at that moment. The pile is always in the critical state. He would have to agree with Fisher.

The good news is that with this deeper insight, the rice-pile historian would not remain puzzled as to why great and catastrophic avalanches strike occasionally and unpredictably. Instead, he would see it as the consequence of an organisational law of nature that counsels against always searching for special causes behind great events.

Mathematically, the signature of the critical state is known as a "power law" - a special algebraic form for the relative frequencies of small, intermediate and large avalanches. What makes it interesting - and bolsters the plausibility of the link between rice piles and history - is that similar power laws have been discovered for events ranging from earthquakes and forest fires to mass extinctions. Two decades of research have convinced many scientists that the earth's crust, forests and ecosystems are organised, like the rice pile, into critical states of extreme sensitivity.

Putting similar numbers on human history is not easy, of course. But in the 1920s, the British physicist Lewis Richardson undertook a study of 82 wars that flared up between 1820 and 1929. In reckoning the size of a war, Richardson chose the simplest and grimmest statistic - the number of deaths - and found that wars, just like earthquakes and forest fires, conform to a power law. The message: there is no "typical"size for a war and no sensible expectation of what might happen in any specific instance.

There are other reasons to suspect that the mathematical link between the rice pile and human history may be instructive. The rhythm of the pile is the consequence of the natural build-up of stress and its release through avalanches. A few years ago, the American historian Paul Kennedy suggested that much the same thing could be found behind the rhythm of international politics over five centuries. In Kennedy's view, as nations naturally wax and wane in economic strength, some are left clinging to power their economic base can no longer support, while others find new economic strength and so seek greater influence. Tension grows until it passes some threshold and something gives. Usually, the stress finds release through armed conflict, after which each nation's influence falls back in line with its true economic strength. In this view, armed conflict is analogous to the avalanching of grains in the pile. Such conflicts have no typical or expected size and are utterly unforeseeable.

Whether or not historians can make useful inferences from the workings of a pile of rice is debatable. But then there is no reason to assume that the complex features of our world do not in fact have origins far simpler than we would expect.

Ubiquity by Mark Buchanan is published by Weidenfeld and Nicolson this week, Pounds 20.00.

The Tipping Point, books, page 26

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