How Nature Works — Per Bak on Sandpiles and Self-Organized Criticality

AuthorAthenarium / Posted on

How Nature Works — Per Bak on starquakes, traffic jams and self-organized criticality

Skip ahead

Sandpiles and self-organized criticality

Imagine, for example, that a gentle stream of sand is trickling from up high onto the floor. A small hill begins to form. The grains slide and shuffle mildly as the mound grows. Sooner or later, however, avalanches occur, both big and small. Grain upon grain, in a domino-like way, the system shifts from micro interactions to a global system-wide behavior. And this process repeats itself so long as the trickling stream continues. In How Nature Works, physicist Per Bak says that sandpiles like these are simple models of the tendency for complex systems to self-organize into a critical state.

As Bak explains:

“Most of the time things are completely calm around him, and it might appear to him that he is actually living in a stable equilibrium world, where nature is in balance. However, every now and then his quiet life is interrupted by a punctuation—a burst of activity where grains of sand keep tumbling around him. There will be bursts of all sizes… In the critical state, the sandpile is the functional unit, not the single grains of sand. No reductionist approach makes sense.”

Per Bak. (1996). How Nature Works.

Earthquakes and starquakes

To Bak, “complexity is a Chinese box phenomenon.” Within each box, from physics to biology to economics, are “all kinds of structure and surprises.” Sandpile-like phenomena are everywhere we look. Nuclear chain reactions are one such example. “If the concentration of fissionable atoms is low, the chain reaction will die out very soon.” But “if the concentration is high, there will be a nuclear explosion similar to that in an atomic bomb.”

Earthquakes are another case. “Bursts of intermittent, sometimes violent, activity” punctuate what is an otherwise peaceful event. And there are many more small earthquakes than there are large ones. The distribution is characterized by an empirical power law known as the Gutenberg-Richter law. Sandpile avalanches, by the way, appear to follow power law patterns too. All of this hints at a complicated organizational process that structures the patterns and order of our world.

In astrophysics, pulsars (rotating neutron stars) may exhibit a sort of self-organized criticality as well. The rotational velocity of these stars have been known to change or “glitch” suddenly. While the mechanics are not well understood, researchers attribute the size of pulsar glitches to starquakes on the pulsar’s surface—a product of tremendous gravitational pressures. What’s interesting, Bak writes, is that “pulsar glitches also follow the Gutenberg-Richter law.” 

Punctuated equilibria and mass extinctions

One statistical consequence of theories on sandpiles, earthquakes, and pulsar glitches “is that catastrophes can occur for no reason whatsoever.” We should wonder then if such dynamics are fundamental to nature and the economy as well.

Biologists, for example, have long debated the sources of evolution—between gradualism and punctuated equilibria. Charles Darwin himself, and the many gradualists after him, suggest that the theory on the origin of species by means of natural selection results in “steady, slow, and continuous” evolution over geologic time.

Darwin’s mechanism, however, may lead to “coevolutionary avalanches, or punctuations”, Bak explains. The Cambrian explosion, in particular, which took place around 500 million years ago, was characterized by “a punctuated change in the diversity of life.” Many fossil records too, like that of coral-like bryozoans, exhibit bursts of evolutionary activity.

Biologists Niles Eldridge and Stephen Jay Gould argue that “Darwin’s theory is incomplete because… it cannot explain the catastrophic extinctions”, Bak adds. Indeed, not much imagination is necessary to picture how external factors—like volcanic eruptions, climate change, and angry meteors—might reset and reshape things anew.

Perhaps it’s not surprising then that extinctions might conform to statistical patterns as well, given the empirical laws that govern earthquakes and other geophysical and celestial disasters. Mass extinction catastrophes, like the End Cretaceous that killed the dinosaurs, “occur with fairly well-defined probability and regularity,” Bak explains.

So while Darwin’s theory provides a powerful description of microevolution, our picture of macroevolution remains incomplete. That’s not to say that there is some new hidden evolutionary force awaiting discovery. Rather, it is a reminder, Bak says, that “simple extrapolation [may] not necessarily take us from the smallest to the largest scale.” As the physicist Philip Anderson writes, “more is different.”

Phantom traffic jams

If earthquakes and starquakes are self-organizing phenomena, maybe stock crashes and economic recessions are too. Economies, after all, consist of countless interactions. People influence and are influenced by everything around them.

Traffic jams illustrate this well. Indeed, each driver’s speed on a busy highway depends on the speed of everybody else. Road quality, weather conditions, braking and accelerating, response times, and whatnot, add friction to traffic happenings. 

What’s curious, Bak notes, is that “phantom traffic jams… [can] emerge for no reason whatsoever.” Random slowdowns—perhaps due to some daredevil pigeon—will reverberate across the highway. Because of friction and stop-start dynamics, minor shocks can lead to small buildups or large scale jams (even without a major accident).

What’s more, “the jams are fractal”. We find “small subjams inside big jams.” Computer simulations suggest that jam distributions may follow a power law as well. Such models are a snapshot in time, of course. One has to wonder what self-driving networks might do to jams someday. Efficiency, complexity and surprises are abound, I’m sure.

Out-of-equilibrium thinking

Traffic jams suggest that power-law like fluctuations, as we’ve seen in geophysics, astrophysics and biology, might occur in social systems too. And economic theories that treat out-of-equilibrium swings as abnormal outliers might be ignoring something fundamental about the economy. Avalanches might be built in.

Similarly, Benoit Mandelbrot and others observe that fluctuations in stock, cotton, and other commodity prices conform to a scale-free Lévy distribution. They have fatter tails than what we’d assume with Gaussian distributions. Given the latter remains the standard assumption, we might very well be underestimating systemic risks in global finance. 

Bak is even more dismal in his outlook. Even if you could identify and remove the grains that would otherwise lead to some foreseeable avalanche, the reshuffling of grains would set the conditions for another somewhere down the track. Control is difficult in the face of complexity. Meteorologists and environmental scientists understand this well.

Bak further suggests that attempts to stabilize the economy is likely to create unanticipated consequences elsewhere in the economic system. The long range history of economic development so far, I believe, would disagree with him somewhat. But the sandpile-like features of economic systems are helpful to keep in mind, nonetheless. 

Natural and social sandpiles

Is it naive then to assume that economies are like sandpiles? Or is it hubris to assume that it is not? I’ll leave the details to you. But the answer, I suspect, is probably somewhere in between. For starters, the nature of beliefs, expectations, fallibility, and reflexivity, makes social systems a very complicated sandpile. 

Either way, Bak warns that we must be careful not to conflate empirical probabilities with periodicity. Just because a large earthquake, market crash, or extinction has or hasn’t happened recently may not tell us much about what’s around the corner. (Although humans appear well on track to trigger the sixth mass extinction.) 

Bak parallels this to gambling in roulette. A history of repeating or alternating colors tells us little about the very next outcome. That is, we can say more about the distributions we study than any of its individual instances. From geology to sociology, protracted calms can turn suddenly and unexpectedly into tumult. Such are the vicissitudes of life.

“Historical events depend on freak accidents, so if the tape of history is replayed many times with slightly different initial conditions, the outcome will differ vastly each time. … An unlikely event is likely to happen because there are so many unlikely events that could happen.”

Per Bak. (1996). How Nature Works.

Forecasters and the principle of universality

Given all of this, are there any principles that we can take home? For one, there is the chicken-and-egg nature of complex systems. “Earthquakes and the fault structures”, Bak writes, are “two sides of the same coin, different manifestations of one unique underlying critical dynamical process.” The same is similarly true of many processes in nature and society.

Secondly, theories are likely to “be statistical and therefore cannot produce specific details.” Scientists, of course, are no strangers to this. Statistical mechanics, quantum mechanics, and chaos theory are good examples of computational or fundamental uncertainties in nature. Ecology and economics might progress further along similar lines too.

True, we can describe many events with hindsight, narratives and models. But even individual avalanches on simple sandpiles are impossible to predict and fully describe. They depend on fine structures and details over a long course of time. As the seismologist Charles Francis Richter—inventor of the Ricther magnitude scale—reminds:

“Only fools, charlatans, and liars predict earthquakes.”

Charles Francis Richter in Per Bak. (1996). How Nature Works.  

There remains, of course, a humbler task of measuring distributions. But this does rely on some belief in the principle of universality—that recurring structures and patterns emerge from the hierarchical layers before it. “Without the concept of universality”, Bak writes, “we would be in bad shape… There would be no fundamental emergent laws of nature to discover, only a big mess.” But the fact that we can observe physical and empirical laws in nature should give us some assurance.

Sources and further reading

  • Bak, Per. (1996). How Nature Works: The Science of Self-Organized Criticality.