Do Dice Play God : Nature's Numbers Chapter 8

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The intellectual legacy of Isaac Newton was a vision of the clockwork universe, set in motion at the instant of creation but thereafter running in prescribed grooves, like a well-oiled machine. It was an image of a totally deterministic worldone leaving no room for the operation of chance, one whose future was completely determined by its present. As the great mathematical astronomer Pierre-Simon de Laplace eloquently put it in 1812 in his Analytic Theory of Probabilities:



An intellect which at any given moment knew all the forces that animate Nature and the mutual positions of the beings that comprise it, if this intellect were vast enough to submit its data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom: for such an intellect nothing could be uncertain, and the future just like the past would be present before its eyes. This same vision of a world whose future is totally predictable lies behind one of the most memorable incidents in Douglas Adams's 1979 science-fiction novel The Hitchhiker's Guide to the Galaxy, in which the philosophers Majikthise and Vroomfondel instruct the supercomputer "Deep Thought" to calculate the answer to the Great Question of Life, the Uni- verse, and Everything. Aficionados will recall that after five million years the computer answered, "Forty-two," at which point the philosophers realized that while the answer was clear and precise, the question had not been. Similarly, the fault in Laplace's vision lies. not in his answer-that the universe is in principle predictable, which is an accurate statement of a particular mathematical feature of Newton's law of motion-but in his interpretation of that fact, which is a serious misunderstanding based on asking the wrong question. By asking a more appropriate question, mathematicians and physicists have now come to understand that determinism and predictability are not synonymous.

In our daily lives, we encounter innumerable cases where Laplacian determinism seems to be a highly inappropriate model. We walk safely down steps a thousand times, until one day we turn our ankle and break it. We go to a tennis match, and it is rained off by an unexpected thunderstorm. We place a bet on the favorite in a horse race, and it falls at the last fence when it is six lengths ahead of the field. It's not so much a universe in which-as Albert Einstein memorably refused to believe-God plays dice: it seems more a universe in which dice play God.



Is our world deterministic, as Laplace claimed, or is it governed by chance, as it so often seems to be? And if Laplace is really right, why does so much of our experience indicate that he is wrong? One of the most exciting new areas of mathematics, nonlinear dynamics-popularly known as chaos theoryclaims to have many of the answers. Whether or not it does, it is certainly creating a revolution in the way we think about order and disorder, law and chance, predictability and randomness.

According to modern physics, nature is ruled by chance on its smallest scales of space and time. For instance, whether a radioactive atom-of uranium, say-does or does not decay at any given instant is purely a matter of chance. There is no physical difference whatsoever between a uranium atom that is about to decay and one that is not about to decay. None. Absolutely none.

There are at least two contexts in which to discuss these issues: quantum mechanics and classical mechanics. Most of this chapter is about classical mechanics, but for a moment let us consider the quantum-mechanical context. It was this view of quantum indeterminacy that prompted Einstein's famous statement (in a letter to his colleague Max Born) that "you believe in a God who plays dice, and 1 in complete law and order." To my mind, there is something distinctly fishy about the orthodox physical view of quantum indeterminacy, and 1 appear not to be alone, because, increasingly, many physicists are beginning to wonder whether Einstein was right all along and something is missing from conventional quantum mechanics-perhaps "hidden variables," whose values tell an atom when to decay. (I hasten to add that this is not the conventional view.) One of the best known of them, the Princeton physicist David Bohm, devised a modification of quantum mechanics that is fully deterministic but entirely consistent with all the puzzling phenomena that have been used to support the conventional view of quantum indeterminacy. Bohm's ideas have problems of their own, in particular a kind of "action at a distance" that is no less disturbing than quantum indeterminacy.

However, even if quantum mechanics is correct about indeterminacy on the smallest scales, on macroscopic scales of space and time the universe obeys deterministic laws, This results from an effect called decoherence, which causes sufficiently large quantum systems to lose nearly all of their indeterminacy and behave much more like Newtonian systems. In effect, this reinstates classical mechanics for most humanscale purposes. Horses, the weather, and Einstein's celebrated dice are not unpredictable because of quantum mechanics.

On the contrary, they are unpredictable within a Newtonian model, too. This is perhaps not so surprising when it come to horses-living creatures have their own hidden variables, such as what kind of hay they had for breakfast. But it was definitely a surprise to those meteorologists who had been developing massive computer simulations of weather in the hope of predicting it for months ahead. And it is really rather startling when it comes to dice, even though humanity perversely uses dice as one of its favorite symbols for chance. Dice are just cubes, and a tumbling cube should be no less predictable than an orbiting planet: after all, both objects obey the same laws of mechanical motion. They're different shapes, but equally regular and mathematical ones.

To see how unpredictability can be reconciled with determinism, think about a much less ambitious system than the entire universe-namely, drops of water dripping from a tap. * This is a deterministic system: in principle, the flow of water into the apparatus is steady and uniform, and what happens to it when it emerges is totally prescribed by the laws of fluid motion. Yet a simple but effective experiment demonstrates that this evidently deterministic system can be made to behave unpredictably; and this leads us to some mathematical "lateral thinking," which explains why such a paradox is possible.

If you turn on a tap very gently and wait a few seconds for the flow to settle down, you can usually produce a regular series of drops of water, falling at equally spaced times in a regular rhythm. It would be hard to find anything more predictable than this. But if you slowly turn the tap to increase the flow, you can set it so that the sequence of drops falls in a very irregular manner, one that sounds random. It may take a little experimentation to succeed, and it helps if the tap turns smoothly. Don't turn it so far that the water falls in an unbroken stream; what you want is a medium-fast trickle. If you get it set just right, you can listen for many minutes without any obvious pattern becoming apparent.

In 1978, a bunch of iconoclastic young graduate students at the University of California at Santa Cruz formed the Dynamical Systems Collective. When they began thinking about this water-drop system, they realized that it's not as random as it appears to be. They recorded the dripping noises with a microphone and analyzed the sequence of intervals between each drop and the next. What they found was shortterm predictability. If I tell you the timing of three successive drops, then you can predict when the next drop will fall. For example, if the last three intervals between drops have been 0.63 seconds, 1.17 seconds, and 0.44 seconds, then you can be sure that the next drop will fall after a further 0.82 seconds. (These numbers are for illustrative purposes only.) In fact, if you know the timing of the first three drops exactly, then you can predict the entire future of the system.

So why is Laplace wrong? The point is that we can never measure the initial state of a system exactly. The most precise measurements yet made in any physical system are correct to about ten or twelve decimal places. But Laplace's statement is correct only if we can make measurements to infinite precision, infinitely many decimal places-and of course there's no way to do that. People knew about this problem of measurement error in Laplace's day, but they generally assumed that provided you made the initial measurements to, say, ten decimal places, then all subsequent prediction would also be accurate to ten decimal places. The error would not disappear, but neither would it grow.

Unfortunately, it does grow, and this prevents us from stringing together a series of short-term predictions to get one that is valid in the long term. For example, suppose I know the timing of the first three water drops to an accuracy of ten decimal places. Then I can predict the timing of the next drop to nine decimal places, the drop after that to eight decimal places, and so on. At each step, the error grows by a factor of about ten, so I lose confidence in one further decimal place. Therefore, ten steps into the future, I really have no idea at all what the timing of the next drop will be. (Again, the precise figures will probably be different: it may take half a dozen drops to lose one decimal place in accuracy, but even then it takes only sixty drops until the same problem arises.)

This amplification of error is the logical crack through which Laplace's perfect determinism disappears. Nothing short of total perfection of measurement will do. If we could measure the timing to a hundred decimal places, our predictions would fail a mere hundred drops into the future (or six hundred, using the more optimistic estimate). This phenomenon is called "sensitivity to initial conditions," or more informally "the butterfly effect." (When a butterfly in Tokyo flaps its wings, the result may be a hurricane in Florida a month later.) It is intimately associated with a high degree of irregularity of behavior. Anything truly regular is by definition fairly predictable. But sensitivity to initial conditions renders behavior unpredictable-hence irregular. For this reason, a system that displays sensitivity to initial conditions is said to be chaotic. Chaotic behavior obeys deterministic laws, but it is so irregular that to the untrained eye it looks pretty much random. Chaos is not just complicated, patternless behavior; it is far more subtle. Chaos is apparently complicated, apparently patternless behavior that actually has a simple, deterministic explanation.

The discovery of chaos was made by many people, too numerous to list here. It came about because of the conjunction of three separate developments. One was a change of scientific focus, away from simple patterns such as repetitive cycles, toward more complex kinds of behavior. The second was the computer, which made it possible to find approximate solutions to dynamical equations easily and rapidly. The third was a new mathematical viewpoint on dynamics-a geometric rather than a numerical viewpoint. The first provided motivation, the second provided technique, and the third provided understanding.

The geometrization of dynamics began about a hundred years ago, when the French mathematician Henri Poincare-a maverick if ever there was one, but one so brilliant that his views became orthodoxies almost overnight-invented the concept of a phase space. This is an imaginary mathematical space that represents all possible motions of a given dynamical system. To pick a nonmechanical example, consider the population dynamics of a predator-prey ecological system.

The predators are pigs and the prey are those exotically pungent fungi, truffles. The variables upon which we focus attention are the sizes of the two populations-the number of pigs (relative to some reference value such as one million) and the number of truffles (ditto). This choice effectively makes the variables continuous-that is, they take real-number values with decimal places, not just whole-number values. For example, if the reference number of pigs is one million, then a population of 17,439 pigs corresponds to the value 0.017439. Now, the natural growth of truffles depends on how many truffles there are and the rate at which pigs eat them: the growth of the pig population depends on how many pigs there are and how many truffles they eat. So the rate of change of each variable depends on both variables, an observation that can be turned into a system of differential equations for the population dynamics. I won't write them down, because it's not the equations that matter here: it's what you do with them.

These equations determine-in principle-how any initial population values will change over time. For example, if we start with 17,439 pigs and 788,444 truffles, then you plug in the initial values 0.017439 for the pig variable and 0.788444 for the truffle variable, and the equations implicitly tell you how those numbers will change. The difficulty is to make the implicit become explicit: to solve the equations. But in what sense? The natural reflex of a classical mathematician would be to look for a formula telling us exactly what the pig population and the truffle population will be at any instant. Unfortunately, such "explicit solutions" are so rare that it is scarcely worth the effort of looking for them unless the equations have a very special and limited form. An alternative is to find approximate solutions on a computer; but that tells us only what will happen for those particular initial values, and most often we want to know what will happen for a lot of different initial values.

Poincare's idea is to draw a picture that shows what happens for all initial values. The state of the system-the sizes of the two populations at some instant of time-can be represented as a point in the plane, using the old trick of coordinates. For example, we might represent the pig population by the horizontal coordinate and the truffle population by the vertical one. The initial state described above corresponds to the point with horizontal coordinate 0.017439 and vertical coordinate 0.788444. Now let time flow. The two coordinates change from one instant to the next, according to the rule expressed by the differential equation, so the corresponding point moves. A moving point traces out a curve; and that curve is a visual representation of the future behavior of the entire system. In fact, by looking at the curve, you can "see" important features of the dynamics without worrying about the actual numerical values ofthe coordinates.

For example, if the curve closes up into a loop, then the two populations are following a periodic cycle, repeating the same values over and over again-just as a car on a racetrack keeps going past the same spectator every lap. If the curve homes in toward some particular point and stops, then the populations settle down to a steady state, in which neither changes-like a car that runs out of fuel. By a fortunate coincidence, cycles and steady states are of considerable ecological significance-in particular, they set both upper and lower limits to populations sizes. So the features that the eye detects most easily are precisely the ones that really matter. Moreover, a lot of irrelevant detail can be ignored: for example, we can see that there is a closed loop without having to work out its precise shape (which represents the combined "waveforms" of the two population cycles).

What happens if we try a different pair of initial values? We get a second curve. Each pair of initial values defines a new curve; and we can capture all possible behaviors of the system, for all initial values, by drawing a complete set of such curves. This set of curves resembles the flow lines of an imaginary mathematical fluid, swirling around in the plane.

We call the plane the phase space of the system, and the set of swirling curves is the system's phase portrait. Instead of the symbol-based idea of a differential equation with various initial conditions, we have a geometric, visual scheme of points flowing through pig/truffle space. This differs from an ordinary plane only in that many of its points are potential rather than actual: their coordinates correspond to numbers of pigs and truffles that could occur under appropriate initial conditions, but may not occur in a particular case. So as well as the mental shift from symbols to geometry, there is a philosophical shift from the actual to the potential.

The same kind of geometric picture can be imagined for any dynamical system. There is a phase space, whose coordinates are the values of all the variables; and there is a phase portrait, a system of swirling curves that represents all possible behaviors starting from all possible initial conditions, and that are prescribed by the differential equations. This idea constitutes a major advance, because instead of worrying about the precise numerical details of solutions to the equations, we can focus upon the broad sweep of the phase portrait, and bring humanity's greatest asset, its amazing image processing abilities, to bear. The image of a phase space as a way of organizing the total range of potential behaviors, from among which nature selects the behavior actually observed, has become very widespread in science.

The upshot of Poincare's gre'at innovation is that dynamics can be visualized in terms of geometric shapes called attractors. If you start a dynamical system from some initial point and watch what it does in the long run, you often find that it ends up wandering around on some well-defined shape in phase space. For example, the curve may spiral in toward a closed loop and then go around and around the loop forever. Moreover, different choices of initial conditions may lead to the same final shape. If so, that shape is known as an attractor. The long-term dynamics of a system is governed by its attractors, and the shape of the attractor determines what type of dynamics occurs.

For example, a system that settles down to a steady state has an attractor that is just a point. A system that settles down to repeating the same behavior periodically has an attractor that is a closed loop. That is, closed loop attractors correspond to oscillators. Recall the description of a vibrating violin string from chapter 5; the string undergoes a sequence of motions that eventually puts it back where it started, ready to repeat the sequence over and over forever. I'm not suggesting that the violin string moves in a physical loop. But my description of it is a closed loop in a metaphorical sense: the motion takes a round trip through the dynamic landscape of phase space.

Chaos has its own rather weird geometry: it is associated with curious fractal shapes called strange attractors. The butterfly effect implies that the detailed motion on a strange attractor can't be determined in advance. But this doesn't alter the fact that it is an attractor. Think of releasing a Ping-Pong ball into a stormy sea. Whether you drop it from the air or release it from underwater, it moves toward the surface. Once on the surface, it follows a very complicated path in the surging waves, but however complex that path is, the ball stays on-or at least very near-the surface. In this image, the surface of the sea is an attractor. So, chaos notwithstanding, no matter what the starting point may be, the system will end up very close to its attractor. Chaos is well established as a mathematical phenomenon, but how can we detect it in the real world? We must perform experiments-and there is a problem. The traditional role of experiments in science is to test theoretical predictions, but if the butterfly effect is in operation-as it is for any chaotic system- how can we hope to test a prediction? Isn't chaos inherently untestable, and therefore unscientific?

The answer is a resounding no, because the word "prediction" has two meanings. One is "foretelling the future," and the butterfly effect prevents this when chaos is present. But the other is "describing in advance what the outcome of an experiment will be." Think about tossing a coin a hundred times. In order to predict-in the fortune-teller's sense-what happens, you must list in advance the result of each of the tosses. But you can make scientific predictions, such as "roughly half the coins will show heads," without foretelling the future in detail-even when, as here, the system is random. Nobody suggests that statistics is unscientific because it deals with unpredictable events, and therefore chaos should be treated in the same manner. You can make all sorts of predictions about a chaotic system; in fact, you can make enough predictions to distinguish deterministic chaos from true randomness. One thing that you can often predict is the shape of the attractor, which is not altered by the butterfly effect. All the butterfly effect does is to make the system follow different paths on the same attractor. In consequence, the general shape of the attractor can often be inferred from experimental observations.

The discovery of chaos has revealed a fundamental misunderstanding in our views of the relation between rules and the behavior they produce-between cause and effect. We used to think that deterministic causes must produce regular effects, but now we see that they can produce highly irregular effects that can easily be mistaken for randomness. We used to think that simple causes must produce simple effects (implying that complex effects must have complex causes), but now we know that simple causes can produce complex effects. We realize that knowing the rules is not the same as being able to predict future behavior.

How does this discrepancy between cause and effect arise? Why do the same rules sometimes produce obvious patterns and sometimes produce chaos? The answer is to be found in every kitchen, in the employment of that simple mechanical device, an eggbeater. The motion of the two beaters is simple and predictable, just as Laplace would have expected: each beater rotates steadily. The motion of the sugar and the egg white in the bowl, however, is far more complex. The two ingredients get mixed up-that's what eggbeaters are for. But the two rotary beaters don't get mixed up-you don't have to disentangle them from each other when you've finished. Why is the motion of the incipient meringue so different from that of the beaters? Mixing is a far more complicated, dynamic process than we tend to think. Imagine trying to predict where a particular grain of sugar will end up! As the mixture passes between the pair of beaters, it is pulled apart, to left and right, and two sugar grains that start very close together soon get a long way apart and follow independent paths. This is, in fact, the butterfly effect in action-tiny changes in initial conditions have big effects. So mixing is a chaotic process. Conversely, every chaotic process involves a kind of mathematical mixing in Poincare's imaginary phase space. This is why tides are predictable but weather is not. Both involve the same kind of mathematics, but the dynamics of tides does not get phase space mixed up, whereas that of the weather does. It's not what you do, it's the way that you do it.

Chaos is overturning our comfortable assumptions about how the world works. It tells us that the universe is far stranger than we think. It casts doubt on many traditional methods of science: merely knowing the laws of nature is no longer enough. On the other hand, it tells us that some things that we thought were just random may actually be consequences of simple laws. Nature's chaos is bound by rules. In the past, science tended to ignore events or phenomena that seemed random, on the grounds that since they had no obvious patterns they could not be governed by simple laws. Not so. There are simple laws right under our noses-laws governing disease epidemics, or heart attacks, or plagues of locusts. If we learn those laws, we may be able to prevent the disasters that follow in their wake.

Already chaos has shown us new laws, even new types of laws. Chaos contains its own brand of new universal patterns. One of the first to be discovered occurs in the dripping tap. Remember that a tap can drip rhythmically or chaotically, depending on the speed of the flow. Actually, both the regularly dripping tap and the "random" one are following slightly different variants of the same mathematical prescription. But as the rate at which water passes through the tap increases, the type of dynamics changes. The attractor in phase space that represents the dynamics keeps changing-and it changes in a predictable but highly complex manner.

Start with a regularly dripping tap: a repetitive drip-dripdrip- drip rhythm, each drop just like the previous one. Then tum the tap slightly, so that the drips come slightly faster. Now the rhythm goes drip-DRIP-drip-DRIP, and repeats every two drops. Not only the size of the drop, which governs how loud the drip sounds, but also the timing changes slightly from one drop to the next.

If you allow the water to flow slightly faster still, you get a four-drop rhythm: drip-DRIP-drip-DRIP. A little faster still, and you produce an eight-drop rhythm: drip-DRIP-drip-DRIPdrip- DRIP-drip-DRIP. The length of the repetitive sequence of drops keeps on doubling. In a mathematical model, this process continues indefinitely, with rhythmic groups of 16, 32, 64 drops, and so on. But it takes tinier and tinier changes to the flow rate to produce each successive doubling of the period; and there is a flow rate by which the size of the group has doubled infinitely often. At this point, no sequence of drops repeats exactly the same pattern. This is chaos. We can express what is happening in Poincare's geometric language. The attractor for the tap begins as a closed loop, representing a periodic cycle. Think of the loop as an elastic band wrapped around your finger. As the flow rate increases, this loop splits into two nearby loops, like an elastic band wound twice around your finger. This band is twice as long as the original, which is why the period is twice as long. Then in exactly the same way, this already-doubled loop doubles again, all the way along its length, to create the period-four cycle, and so on. After infinitely many doublings, your finger is decorated with elastic spaghetti, a chaotic attractor.

This scenario for the creation of chaos is called a perioddoubling cascade. In 1975, the physicist Mitchell Feigenbaum discovered that a particular number, which can be measured in experiments, is associated with every period-doubling cascade. The number is roughly 4.669, and it ranks alongside 1t (pi) as one of those curious numbers that seem to have extraordinary significance in both mathematics and its relation to the natural world. Feigenbaum's number has a symbol, too: the Greek letter () (delta). The number 1t tells us how the circumference of a circle relates to its diameter. Analogously, Feigenbaum's number () tells us how the period of the drips relates to the rate of flow of the water. To be precise, the extra amount by which you need to turn on the tap decreases by a factor of 4.669 at each doubling of the period.

The number 1t is a quantitative signature for anything involving circles. In the same way, the Feigenbaum number () is a quantitative signature for any period-doubling cascade, no matter how it is produced or how it is realized experimentally. That very same number shows up in experiments on liquid helium, water, electronic circuits, pendulums, magnets, and vibrating train wheels. It is a new universal pattern in nature, one that we can see only through the eyes of chaos; a quantitative pattern, a number, emerges from a qualitative phenomenon. One of nature's numbers, indeed. The Feigenbaum number has opened the door to a new mathematical world, one we have only just begun to explore.

The precise pattern found by Feigenbaum, and other patterns like it, is a matter of fine detail. The basic point is that even when the consequences of natural laws seem to be patternless, the laws are still there and so are the patterns. Chaos is not random: it is apparently random behavior resulting from precise rules. Chaos is a cryptic form of order. Science has traditionally valued order, but we are beginning to appreciate the fact that chaos can offer science distinct advantages. Chaos makes it much easier to respond quickly to an outside stimulus. Think of tennis players waiting to receive a serve. Do they stand still? Do they move regularly from side to side? Of course not. They dance erratically from one foot to the other. In part, they are trying to confuse their opponents, but they are also getting ready to respond to any serve sent their way. In order to be able to move quickly in any particular direction, they make rapid movements in many different directions. A chaotic system can react to outside events much more quickly, and with much less effort, than a non chaotic one. This is important for engineering control problems. For example, we now know that some kinds of turbulence result from chaos-that's what makes turbulence look random. It may prove possible to make the airflow past an aircraft's skin much less turbulent, and hence less resistant to motion, by setting up control mechanisms that respond extremely rapidly to cancel out any small regions of incipient turbulence. Living creatures, too, must behave chaotically in order to respond rapidly to a changing environment.

This idea has been turned into an extremely useful practical technique by a group of mathematicians and physicists, among them William Ditto, Alan Garfinkel, and Jim Yorke: they call it chaotic control. Basically, the idea is to make the butterfly effect work for you. The fact that small changes in initial conditions create large changes in subsequent behavior can be an advantage; all you have to do is ensure that you get the large changes you want. Our understanding of how chaotic dynamics works makes it possible to devise control strategies that do precisely this. The method has had several successes. Space satellites use a fuel called hydrazine to make course corrections. One of the earliest successes of chaotic control was to divert a dead satellite from its orbit and send it out for an encounter with an asteroid, using only the tiny amount of hydrazine left on board. NASA arranged for the satellite to swing around the Moon five times, nudging it slightly each time with a tiny shot of hydrazine. Several such encounters were achieved, in an operation that successfully exploited the occurrence of chaos in the three-body problem (here, Earth/Moon/satellite) and the associated butterfly effect.

The same mathematical idea has been used to control a magnetic ribbon in a turbulent fluid-a prototype for controlling turbulent flow past a submarine or an aircraft. Chaotic control has been used to make erratically beating hearts return to a regular rhythm, presaging invention of the intelligent pacemaker. Very recently, it has been used both to set up and to prevent rhythmic waves of electrical activity in brain tissue, opening up the possibility of preventing epileptic attacks. Chaos is a growth industry. Every week sees new discoveries about the underlying mathematics of chaos, new applications of chaos to our understanding of the natural world, or new technological uses of chaos-including the chaotic dishwasher, a Japanese invention that uses two rotating arms, spinning chaotically, to get dishes cleaner using less energy; and a British machine that uses chaos-theoretic data analysis to improve quality control in spring manufacture.

Much, however, remains to be done. Perhaps the ultimate unsolved problem of chaos is the strange world of the quantum, where Lady Luck rules. Radioactive atoms decay "at random"; their only regularities are statistical. A large quantity of radioactive atoms has a well-defined half-life-a period of time during which half the atoms will decay. But we can't predict which half. Albert Einstein's protest, mentioned earlier, was aimed at just this question. Is there really no difference at all between a radioactive atom that is not going to decay, and one that's just about to? Then how does the atom know what to do?

Might the apparent randomness of quantum mechanics be fraudulent? Is it really deterministic chaos? Think of an atom as some kind of vibrating droplet of cosmic fluid. Radioactive atoms vibrate very energetically, and every so often a smaller drop can split off-decay. The vibrations are so rapid that we can't measure them in detail: we can only measure averaged quantities, such as energy levels. Now, classical mechanics tells us that a drop of real fluid can vibrate chaotically. When it does so, its motion is deterministic but unpredictable. Occasionally, "at random," the vibrations conspire to split off a tiny droplet. The butterfly effect makes it impossible to say in advance just when the drop will split; but that event has precise statistical features, including a well defined half-life. Could the apparently random decay of radioactive atoms be something similar, but on a microcosmic scale? After all, why are there any statistical regularities at all? Are they traces of an underlying determinism? Where else can statistical regularities come from? Unfortunately, nobody has yet made this seductive idea work-though it's similar in spirit to the fashionable theory of superstrings, in which a subatomic particle is a kind of hyped-up vibrating multidimensional loop. The main similar feature here is that both the vibrating loop and the vibrating drop introduce new "internal variables" into the physical picture. A significant difference is the way these two approaches handle quantum indeterminacy. Superstring theory, like conventional quantum mechanics, sees this indeterminacy as being genuinely random. In a system like the drop, however, the apparent indeterminacy is actually generated by a deterministic, but chaotic, dynamic. The trick-if only we knew how to do it-would be to invent some kind of structure that retains the successful features of superstring theory, while making some of the internal variables behave chaotically. It would be an appealing way to render the Deity's dice deterministic, and keep the shade of Einstein happy.

Chapter 6 : Broken Symmetry
Chapter 7 : The Rhythm of Life
Chapter 8 : Do dice Play God 
Chapter 9 : Drops Dynamics and Daisies