MIDNIGHT in Berlin, one night in the summer of 1753. A weary man in his fifties, well-dressed but unshaven, sits alone in his dark, quiet study. With trembling hands, he leafs through a thin booklet entitled Essay on Cosmology. It is a manuscript that has shattered his life. Three years earlier, Pierre-Louis Moreau de Maupertuis had published his pamphlet in a bid to prove the existence of God through a peculiar mathematical and theological reworking of Newton’s laws of mechanics. He had anticipated acclaim. But instead, his arguments were ridiculed by intellectuals across Europe. Tonight, Maupertuis tries once more to reassure himself. But his faith is undermined. He decides to resign his post as president of the Berlin Academy of Sciences and flee the city.
Maupertuis’s proof of the existence of God has by now been long forgotten. But in the 245 years since his hasty flight, Maupertuis has been vindicated. The mathematical portion of his repudiated text contained a curious and subtle twist on Newton’s laws. The “principle of least action”, as Maupertuis called it, has turned out to be one of the most influential ideas in theoretical physics.
To get a feel for Maupertuis’s idea, imagine a comet in orbit around the Sun. In Newton’s view, the comet moves along its particular path because of the way the Sun’s gravity pulls on it. Maupertuis saw the same thing quite differently. For him, there were no forces. Or, at least, he argued that you don’t need to think that way. Maupertuis took Newton’s equations and remoulded them into an entirely new form: a comet, he said, moves as it does because it tries to minimise something called its “action”. This action, he claimed, is an abstract quantity, like energy but different, that depends on the path of the comet between a starting point and an ending point. In principle, a comet might follow any one of an infinite number of different paths-it might follow a straight line, for example, or a complicated path that wiggles back and forth. Every conceivable path has a different action, and of all these, the path actually taken has the least (see Diagram).
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Maupertuis showed that comets or planets (or, in fact, anything moving according to Newton’s laws) generally try to move so as to spend as little action as possible, as if action were some kind of precious fuel. So, in a sense, the world is lazy, and tries to do as little as it possibly can.
That was only the beginning. In the 1940s, the American physicist Richard Feynman showed that all the weird laws of quantum theory also follow from a version of Maupertuis’s principle. Since then, physicists have found that least action also underlies the fundamental gauge theories of particle physics, such as the theories of the weak and strong interactions.
But the really surprising thing about Maupertuis’s idea is that it applies even outside physics. From linguistics to finance, you don’t have to look far to find one of these “least action” principles at work. You can even stumble on Maupertuis’s ideas during an innocent walk through a public park.
In most parks, the geometrically perfect concrete walks laid down by landscape architects bear little resemblance to the paths people actually use. It’s easy enough to understand why walkers might cut corners and wear in dirt paths between two concrete paths that come close to one another. After all, people are lazy, and like to take the shortest path possible. But what about the “islands” and other oddities that often emerge in trails? It takes many walkers acting over a long time to form such patterns (see Diagram). What drives them to do it?
Last year, German researchers Dirk Helbin, Joachim Keltsch, and Peter Molnar of the University of Stuttgart showed that these features of trail systems arise from a simple variation on Maupertuis’s idea (Nature, vol 388, p 47). People, it seems, don’t merely follow the shortest path, they do something a bit more sophisticated-they try to minimise their discomfort in moving from one place to another.
Suppose you take a pristine grassy field and let people loose to walk across it. At first, of course, they’ll take the most direct paths to wherever they want to go. But after some time, trails begin to form, and things get more complicated. New travellers face a choice: they can strike out on their own, or they can use one of the existing trails, which is easier than clearing a new path. The researchers made a computer model to simulate the evolution of such paths on the assumption that people follow a principle of “least discomfort” and sometimes follow longer paths if they are easier to walk on.
Alongside this assumption, Helbin and his colleagues took two other factors into account. Frequently used trails are more attractive to new arrivals than paths less travelled and the reinforcing effect of these walkers makes them even more distinct and attractive. On the other hand, the effects of weather tend to wear trails away, as nature reclaims its territory. In a sense, then, trails compete with each other and with the weather. This leads to a kind of selection between trails. A finite number of walkers can only keep a finite length of pathway trodden down. As a result, in the simulations, the trails begin after a time to bundle together in groups, especially wherever a number of trails meet or pass close together. What happens is that walkers with different destinations use and reinforce common parts of the trail system and neglect others.
Evolving pathways
The simulations also suggest that a single number (K) determines the patterns into which the trails ultimately develop. To work out K you multiply the number of trail users by the time it would take the weather to wipe out a trail, and divide by the walkers’ visibility-how far they can see. A small value for K means that there are few walkers, that the paths aren’t very durable, and that the walkers can’t see well enough to choose the well-worn paths anyway. Not surprisingly, in the simulations, when K is small the trail system evolves so that every entry point and destination is connected by a minimal-distance path. For larger and more realistic values of K the evolution of the trail system stops before reaching this shortest-path structure.
The quantity K seems to be related to the typical size of the detour that walkers make in crossing the field, and the researchers believe that the trail system that emerges has the shortest total path length compatible with an acceptable level of detour. The striking similarity between how these trails form in simulations and in the real world strongly suggests that this “least discomfort” approach to the way people follow paths is correct. Walkers make the best compromise between convenience and distance to walk.
So least-action principles seem to be at work in more than just the motions of planets or comets. Still, they seem to be most apparent in the laws of physics. For example, a century before Maupertuis, his countryman the French mathematician Pierre de Fermat invented one that applies to the movement of light. Think of a ray of light streaming through your living-room window. Suppose you want to know what path the light ray will take in getting to your eye (see Diagram). Try to imagine all possible paths that the ray could take from its source to your eye. Some paths go straight through the window without bending. Others curve through space haphazardly. According to Fermat’s principle of least time, the path that the light will actually follow is the one that gets it there quickest.FIG-mg21335502.JPG
For light, then, time plays the role of Maupertuis’s action. But Fermat’s principle is limited. It only applies to light, whereas Maupertuis’s is far more general. There are other least action principles too. A common principle of this sort lies behind a kind of empirical relation that comes up often in complex systems studies-something known as a “power law”. This is a special mathematical relationship that sometimes holds between two quantities. An intriguing example was discovered by George Zipf of Harvard University in 1949.
Suppose we list the words of the English language according to how frequently they appear in a large block of text. The most common word is “the”, which we assign rank one. The next most common word is “of”, which is given rank two, followed by “and”, “to” and so on. Zipf found that if we plot word frequency in a body of English text versus the word rank, the result is a nearly perfect straight line. That is, the frequency of word appearance is related to the word rank in a very simple way (see Diagram).
Mathematically, Zipf found that the frequency of a word is inversely proportional to its rank. That is, for a word of rank r, the frequency fis given by the formula f = constant/r. Zipf found that for any language, the constant takes the value 1/log(1.78R), Rbeing the number of words in the language. Any relation in which something is proportional to r raised to some power is called a power law, and the power in Zipf’s law is -1. Using Zipf’s law, you can work out that for a good writer with an active vocabulary of, say, R = 100 000 words, the 10 highest ranking words will occupy about 24 per cent of a text, while for a popular novel or newspaper using a slimmed-down vocabulary of R =10 000 words, this percentage will increase marginally to nearly 30 per cent.
To illustrate an in-between case, amateur mathematician William Bowers of Thousand Oaks, California, once did a Zipfian analysis of Conan Doyle’s Sherlock Holmes classic story, The Hound of the Baskervilles. Using the vocabulary size of R = 10 000 words, Bowers’s analysis showed that the story conformed closely to Zipf’s theoretical prediction, except that the word “to”, which is ranked four in general English text, dropped down to fifth place in Conan Doyle’s usage.
Zipf titled the volume in which he presented this idea Human Behavior and the Principle of Least Effort, and he tried to derive the power-law form of his law by appealing to a principle of least effort. In this case the principle is: communicate as much information as possible in the least number of symbols. Zipf didn’t succeed in his demonstration, but he was on the right track. For it is possible to show, using arguments from information theory, that out of all conceivable languages, the natural languages we use are those that convey the maximum amount of information at a given cost, measured by the average time needed to produce the words of the language. Only languages that obey the inverse power law above have this property. Once again, a principle in Maupertuis’s style seems to reign.
But there’s a snag. It turns out that monkeys hitting typewriter keys at random will also produce a “language” obeying Zipf’s law. Suppose we consider a nine-letter alphabet with a space character, so that our mythical monkey strikes each character with a likelihood of 0.1. Many years ago, Benoit Mandelbrot, now at the IBM Thomas J Watson Research Center in Yorktown Heights, New York, showed that the exponent in Zipf’s law is -1.048 for this monkey language, hardly any different than the value -1 for English.
But there’s a striking difference between this monkey language and ordinary English. Fully half of an ordinary English text is made up of just 100 of the most popular-lowest ranking-words. Literary giants have somewhat larger vocabularies, and it may take 500 words to fill half the text of a great literary work. In contrast, a monkey would need 1 895 761 of the most frequently used simian words to fill half of a large block of text. So it’s fair to say that monkey language is a bit wordy. Apparently, we apply an additional constraint in shaping language-to minimise the burden on our memories, perhaps, we develop languages that don’t have too many words.
Somewhere between English and the monkey language is the language of DNA. Working with geneticists, physicist Eugene Stanley from the University of Boston has been applying the Zipf test to the part of yeast DNA that does not correspond to the coding region for any genes, the “junk” DNA. When the researchers arbitrarily divided up the junk into “words” between 3 and 8 base pairs long, what emerged was a graph that looked surprisingly similar to the Zipf curve with exponent -1. And when the researchers applied a second test to quantify the “redundancy”-the amount of repeated information-in the yeast language, they found a level significantly higher than what would be expected if the “junk” was random. These findings suggest that something is written in these mysterious regions, and that the junk may not be so much junk, after all. As Harvard biologist Walter Gilbert described it, “I think the junk is like the stuff in a junk shop. You can find lovely things in it.”
What all this adds up to is that if you look hard enough, almost every aspect of life seems to have a least-action principle underlying it. Poor old Maupertuis may have been humiliated and driven from Berlin, but that doesn’t mean that he thought his ideas were wrong. His booklet held a profound secret. Perhaps he just felt that convincing his contemporaries was more trouble than it was worth.