## Does God Play Dice?

What is Randomness?
First, we'd better take a closer look at what we mean by 'random'. Chaos has taught us that we must be careful to distinguish between what happens in mathematical systems where we assume perfect and infinitely precise knowledge, and what happens in practice when our knowledge is imperfect and imprecise. The meaning of the word 'random' depends heavily upon this distinction.

A dynamical system is one whose state changes over time, according to some rule or procedure which I'll call a dynamic. A dynamic is a rule for getting to the 'next' state from the current one. The key to thinking about randomness is to imagine such a system to be in some particular state, and to let it do whatever that particular system does. Then imagine putting it back into exactly that initial state and running the whole experiment again. If you always get exactly the same result, then the system is deterministic. If not, then it's random. Notice that in order to show that a system is deterministic, we don't actually have to predict what it will do: we just have to be assured that on both occasions it will do the same thing.

For example, suppose the system is a cannonball, being dropped off the edge of a cliff under controlled, repeatable conditions. Suppose that the dynamic is the action of gravity according to Newton's laws. You drop the cannonball and it falls, accelerating as it does so. Obviously if you repeat the same experiment under identical conditions, the ball will do exactly the same thing as before, because Newton's laws prescribe the future motion uniquely. So this system is deterministic.

In contrast, the system may be a pack of cards, and the dynamic may be to shuffle the pack and then take the top card. Imagine that the current top card is the ace of spades, and that after shuffling the pack the top card becomes the seven of diamonds. Does that imply that whenever the top card is the ace of spades then the next top card will always be the seven of diamonds? Of course not. So this system is random.

Even this distinction is not so clear cut when we think about the real world. In fact, it's difficult to imagine circumstances in which you can be absolutely sure that the real world 'is' random rather than deterministic, or vice versa. The distinction is about appearances, not deep realities - an apparently random universe could be obeying every whim of a deterministic deity who chooses how the dice roll; a universe that has obeyed perfect mathematical laws for the last ten billion years could suddenly start to play truly random dice. So the distinction is about how we model the system, and what point of view seems most useful, rather than about any inherent feature of the system itself.

In modelling terms, the difference between randomness and determinacy is clear enough. The randomness in the pack of cards arises from our failure to prescribe unique roles for getting from the current state to the next one. There are lots of different ways to shuffle a pack. The determinism of the cannonball is a combination of two things: fully prescribed rules of behaviour, and fully defined initial conditions. Notice that in both systems we are thinking on a very short timescale: it is the next state that matters - or, if time is flowing continuously, it is the state a tiny instant into the future. We don't need to consider long-term behaviour to distinguish randomness from determinacy. Scientists have devised many models of the real world: some deterministic some not. In a clockwork Newtonian model nothing is truly random. If you run a deterministic model twice from the same initial state, it will do the same thing both times. However, we currently have a different model of the universe a quantum mechanical one. In quantum mechanics - at least, as currently formulated there is genuine randomness.Just as the rule 'shuffle the cards' permits many different outcomes, so the rules of quantum mechanics permit a particle to be in many different states. When we observe its state, we pin it down to a particular value in the same way that turning over the top card of the pack reveals a particular card. But while a quantum system is following its own nose, unobserved, it has a random choice of possible futures. So whether we think that our universe as a whole is random depends on what brand of physics we currently espouse, and since we can't actually run the entire universe twice from the same initial conditions the whole discussion becomes a trifle moot. [Note that "moot" here suggests the meaning "void" -LB]

However, instead of asking 'is the entire universe really random?' we can ask a less ambitious question - but a more useful one. Given some particular subsystem of the real world, is it best modelled by a deterministic mathematical system or a random one? And now we can make a genuine distinction. It is clear from the start that any real world system might be suddenly influenced by factors outside our knowledge or control. If a bird smashes into the falling cannonball then its path will deviate from what we expect. We could build the bird into the mathematics as well, but then what of the cat that may or may not capture the bird before it can crash into the cannonball? The best we can do is choose a subsystem that we think we understand, and agree that unexpected outside influences don't count. Because our knowledge of the system is necessarily limited by errors of measurement, we can't guarantee to return it to exactly the same initial state. The best we can do is return it to a state that is experimentally indistinguishable from the previous initial state. We can repeat the cannonball experiment with something that looks like the same cannonball in the same place moving at the same speed; but we can't control every individual atom inside it to produce the identical initial state with infinite precision. In fact, whenever we touch the cannonball a few atoms rub off and a few others transfer themselves to its surface, so it is definitely different every time.

So now - provided we remember that only short timescales are important - we can formulate a practical version of the distinction between deterministic chaos and true randomness. A real subsystem of the universe looks deterministic if ignoring unexpected outside effects, whenever you return it to what looks like the same initial state it then does much the same thing for some non-zero period of time. It is random if indistinguishable initial states can immediately lead to very different outcomes.

In these terms the cannonball system, using a real cannonball, a real cliff, and real gravity, still looks pretty deterministic. The experiment is 'repeatable' - which is what makes Newton's laws of motion so effective in their proper sphere of application. In contrast, a real card-shuffling experiment looks random. So does the decay of a radioactive atom. The randomness of the card-shuffle is of course caused by our lack of knowledge of the precise procedure used to shuffle the cards. But that is outside the chosen system, so in our practical sense it is not admissible. If we were to change the system to include information about the shuffling rule - for example, that it is given by some particular computer code for pseudo-random numbers, starting with a given 'seed value' - then the system would look deterministic. Two computers of the same make running the same 'random shuffle' program would actually produce the identical sequence of top cards.

We can also look at the card system in a different way. Suppose the choice of card is determined by just the first few digits of the pseudo-random number, which is fairly typical of how people write that kind of program. Then we don't know the 'complete' state of the system at any time - only the few digits that tell us the current top card. Now, even with a fixed pseudo-random number generator, the next card after an ace of spades will be unpredictable, so our model has become random again. The randomness results from lack of information about some wider system that includes the one we think we are looking at. If we knew what those 'hidden variables' were doing, then we would stop imagining that the system was random.

Suppose we are observing a real system, and we think it looks random. There are two distinct reasons why this might happen: either we are not observing enough about it, or it truly is irreducibly random. It's very hard to decide between these possibilities. Would the decay of a radioactive atom become deterministic if only we knew the external rules for making it decay (the shuffling rule) or some extra 'internal' dynamic on 'the entire pack of cards'? Fine but right now we don't, and maybe we never will because maybe there is no such internal dynamic. (See Chapter 16 for some speculations on this topic.)

I repeat, we are in the business of comparing observations of the real world with some particular model, and it is only the model that can safely be said to be random or deterministic. And if it is one or the other, then so is the real world, as far as the aspects of it that our model captures are concerned.

Chance and Chaos
Having sorted out what we mean or at any rate, what I mean by 'random' and 'deterministic', we can turn to the relation between chance and chaos. This is not a simple story with a single punchline. The main source of potential confusion is the multifaceted nature of chaos: it takes on different guises when viewed in different lights.

On the surface, a chaotic system behaves much like a random one. Think about computer models of the Earth's weather system, which are chaotic and so suffer from the butterfly effect. Run the computer model starting from some chosen state, and you get a pleasant, sunny day a month later. Run the same computer model starting from some chosen state plus one flap of a butterfly's wing surely an indistinguishable state in any conceivable practical experiment - and now you get a blizzard. Isn't that what a random system does? Yes but the timescale is wrong. The 'randomness' arises on large timescales - here months. The distinction between determinacy or randomness takes place on short timescales; indeed it should be immediate. After a day that flapping wing may just alter the local pressure by a tenth of a millibar. After a second, it may just alter the local pressure by a ten billionth of a millibar. And indeed in the computer models that's just what happens. It takes time for the errors to grow and we can quantify that time using the Liapunov exponent. So we can safely say that on short timescales the computer model of the weather is not random: it is deterministic (but chaotic).

To add to the scope for confusion, in certain respects a chaotic system may behave exactly like a random one. Remember the 'wrapping mapping' of Chapter 6, which pulls out successive decimal places of its initial condition using the dynamical rule 'multiply by ten and drop anything in front of the decimal point'? There is nothing random about the rule - when presented with any particular number, it always leads to the same result. But even though the rule is deterministic, the behaviour that it produces need not be.

The reason is that the behaviour does not depend solely on the rule: it depends on the initial condition as well. If the initial condition has a regular pattern to its digits, such as 0.3333333..., then the behaviour (as measured by the first digit after the decimal point) is regular too: 3, 3, 3, 3, 3, 3, 3. However, if the initial condition was determined by randomly throwing a die, say 0.1162541 ... then the behaviour will appear equally random: 1,1, 6, 2, 5, 4, 1.

In the sense described, the 'multiply by ten' dynamical system displays absolutely genuine random behaviour exactly as random as the die that produced 1, 1, 6, 2, 5, 4,1 in the first place. However, it would be a gross distortion to say that the system 'is' random, for at least two reasons. The first is that the measurement we are considering, the first digit after the decimal point, is not a complete description of the state of the system. A more accurate representation is 0.3333333,0.333333, 0.33333,0.3333,0.333, 0.33, 0.3; or in the random case 0.1162541, 0.162541, 0.62541, 0.2541, 0.541, 0.41, 0.1. That second sequence doesn't look totally random if you see the whole thing. The second reason is that it is the initial condition that provides the source of randomness; the system merely makes this randomness visible. You might say that chaos is a mechanism for extracting and displaying the randomness inherent in initial conditions, an idea that the physicist Joseph Ford has advocated for many years as part of a general theory of the information-processing capabilities of chaos.

However, a dynamical system is not just a response to a single initial condition: it is a response to all initial conditions. We just tend to observe that response one initial condition at a time. When we start thinking like that, we can soon distinguish regular patterns lurking among the chaos. The most basic is that for a time systems whose initial conditions differ by a small amount follow approximately similar paths. Thanks to the butterfly effect this similarity eventually breaks down, but not straight away. If the initial condition had been 0.3333334 then the behaviour would have been 3, 3, 3, 3, 3, 3 - so far so good - and then 4, well, it couldn't last for ever. In exactly the same way, if the initial condition had been 0.1162542 instead of 0.1162541, the two behaviours would also have looked very similar for the first six steps, with the difference becoming apparent only on the seventh. In fact, if we compare the exact values (rather than just our 'observations' of the first decimal place) then we can see how the divergence goes. The first initial condition goes
0.1162541,0.162541, 0.62541,0.2541,0.541, 0.41,0.1
and the second goes
0.1162542,0.162542, 0.62542,0.2542,0.542, 0.42,0.2.

The differences between corresponding values go
0.0000001, 0.000001, 0.00001, 0.0001, 0.001, 0.01, 0.1,
and each is ten times bigger than the previous difference. So we can actually see how the error is growing; we can watch how the butterfly's flapping wing cascades into ever - bigger discrepancies.
This kind of regular growth of tiny errors - I'll use the word 'error' for any small difference in initial conditions, whether or not it's a mistake - is one of the simplest tests for chaos. The technicians call it the system's Liapunov exponent, named after A. M. Liapunov, a famous Russian mathematician who invented many of the basic concepts of dynamical systems theory in the early 1 900s. Here the Liapunov exponent is 10, meaning that the error grows by a factor of ten at each step. (Well, strictly speaking the Liapunov exponent is log 10, which is about 2.3026, because the rate of growth is e raised to the power of the Liapunov exponent, not the exponent itself .But that's a technical nicety. To avoid confusion I'll talk about the 'growth rate', which here really is 10.)

Of course the growth rate of a tiny error is not always constant: the only reason that the growth here is exactly tenfold at every step is that the dynamics multiplies everything by ten. If the dynamics were more variable, multiplying some numbers by 9 and others by 11, say, then you'd get a more complicated pattern of error growth; but on average and for very small initial errors it would still grow by some rate between 9 and 11. In fact Liapunov proved that every deterministic dynamical system has a well-defined rate of growth of errors, provided the errors are taken to be sufficiently small.

The Liapunov growth rate provides a quantitative test for chaos. If the Liapunov growth rate is bigger than 1, then initial errors, however small increase exponentially. This is the butterfly effect in action, so such a system is chaotic. However, if the Liapunov growth rate is less than 1, the errors die away, and the system is not chaotic. That's wonderful if you know you have a deterministic system to begin with, and if you can make the extremely accurate observations required to calculate the growth rate from experiments. It's much less useful if you don't, or can't. Nonetheless, we see that deterministic systems behave differently from random ones, and that certain features of that difference lead to quantitative measures of the degree of chaos that is present. The Liapunov exponent is just one diagnostic of chaos. Another is the fractal dimension of the attractor (see Chapter 11). A steady state attractor has fractal dimension 0, a periodic cycle has fractal dimension 1, a torus attractor formed by superposing n independent periodic motions has fractal dimension n. These are all whole numbers. So if you can measure the fractal dimension of a system's attractor, and you get numbers like 1.356 or 2.952, then that's an extra piece of evidence for chaos. How can we measure such a fractal dimension? There are two main steps. One is to reconstruct the qualitative form of the attractor using the Ruelle-Takens method of Chapter 9 or one of the many variants that have appeared since. The other is to perform a computer analysis on the reconstructed attractor to calculate its fractal dimension. There are many methods for doing this, the simplest being a 'box-counting' technique that works out what proportion of different-sized boxes is occupied by the attractor. As the size of the box decreases, this proportion varies in a manner that is determined by the fractal dimension. The mathematics of phase space reconstruction then guarantees that the fractal dimension of the reconstructed attractor is the same as that of the original one - provided there is an original one, which means that your system must be describable by a deterministic dynamical system to begin with.

It is absolutely crucial not to be naive about this process. You can take any series of measurements whatsoever - say the prices in your last twelve months' shopping lists - push them through the Ruelle-Takens procedure, and count boxes. You will get some number maybe 5.277, say. This does not entitle you to assume that your shopping list is chaotic and lives on a 5.277- dimensional strange attractor. Why not? Firstly, because there is no good reason to assume that your shopping list comes from a deterministic dynamical system. Secondly, because even if it did, your shopping list data contains too little information to give any confidence in a dimension that big. The bigger the dimension of an attractor, the more data points you need to pin the structure of the attractor down. In fact any fractal dimension over about 4 should be viewed with great suspicion.