He asked how probable a state should be. Imagine a grid of 100 holes into which you drop 1000 marbles at random. It is hugely improbable that they will all finish up in one hole. I am not going to give numbers, but it is simple to work out the probability that all will land in one hole or, say, in four adjacent holes. In fact, one can list every possible distribution of the marbles in the grid, and then see in how many of these distributions all the marbles fall in one hole, in four adjacent holes, eight adjacent holes, and so on. If each distribution is assumed to be equally probable, the number of ways a particular outcome can happen becomes the relative probability of that outcome, or state. Boltzmann had the inspired idea that, applied to atoms, this probability (which must also take into account the velocities of the atoms) is a measure of the entropy that had been found through study of the thermodynamics of steam engines.
There is no need to worry about the technical details. The important thing is that states with low entropy are inherently improbable. Boltzmann’s idea was brilliantly successful, and much of modern chemistry, for example, would be unthinkable without it. However, his attempt to explain the more fundamental issues associated with the unidirectionality of physical processes was only partly successful.
He wanted to show that, matching the behaviour of macroscopic entropy, his microscopic entropy would necessarily increase solely by virtue of Newton’s laws. This seems plausible. If a large number of atoms are in some unlikely state, say all in a small region, so that they have a low entropy, it seems clear that they will pass to a more probable state with higher entropy. However, it was soon noted that there are exactly as many dynamically possible motions of the atoms that go from states of low probability to states of high probability as vice versa. This is a straight consequence of the fact that Newton’s laws have the same form for the two directions of time. Newton’s laws alone cannot explain the arrow of time.
Only two ways have ever been found to explain the arrow: either the universe was created in a highly unlikely special state, and its initial order has been ‘degrading’ ever since, or it has existed for ever, and at some time in the recent past it entered by chance an exceedingly improbable state of very low entropy, from which it is now emerging. The second possibility is entirely compatible with the laws of physics. For example, if a collection of atoms (which obey Newton’s laws) is confined in a box and completely isolated, it will, over a sufficiently long period of time, visit (or rather come arbitrarily near) all the states that it can in principle ever reach, even those that are highly ordered and statistically very unlikely. However, the intervals of time between returns to states of very low entropy are stupendously long (vastly longer than the presently assumed age of the universe), and neither explanation is attractive.
The fact is that mechanical laws of motion allow an almost incomprehensibly large number of different possible situations. Interesting structure and order arise only in the tiniest fraction of them. Scientists feel they should not invoke miracles to explain the order we see, but that leaves only statistical arguments, which give bleak answers (only dull situations can be expected), or the so-called anthropic principle that if the world were not in a highly structured but extremely unlikely state, we should not exist and be here to observe it.
One of my reasons for writing this book is that timeless physics opens up new ways of thinking about structure and entropy. It may be easier to explain the arrow of time if there is no time!
CHAPTER 2
Time Capsules
THE PHYSICAL WORLD AND CONSCIOUSNESS
The discussion in Chapter 1 prompts the question of how our sense of the passage of time arises. Before we can begin to answer this, we have to think about another mystery – consciousness itself. How does brute inanimate matter become conscious, or rather self-conscious?
No one has any idea. Consciousness and matter are as different as chalk and cheese. Nothing in the material world gives a clue as to how parts of it (our brains) become conscious. However, there is increasing evidence that certain mental states and activities are correlated with certain physical states in different specific regions of the brain. This makes it natural to assume, as was done long ago, that there is psychophysical parallelism: conscious states somehow reflect physical states in the brain.
Put in its crudest form, a brain scientist who knew the state of our brain would know our conscious state at that instant. The brain state allows us to reconstruct the conscious state, just as musical notes on paper can be transformed by an orchestra into music we can hear. By the ‘state’ of a system, say a collection of atoms, scientists usually mean the positions of all its parts and the motions of those parts at some particular instant. It is widely assumed that conscious states, in which, after all, we are aware of motion directly, are at the least correlated with (correspond to) brain states that involve not only instantaneous positions but also motions and, more generally, change (associated with flow of electric currents or chemicals, for example). This is a natural assumption. Our awareness of motion and change is vivid and often exciting: think of watching gymnastics, or the 100-metre sprint final in the Olympic Games. We suppose that the impression of motion must be created by some motion or change in the brain.
However, if the physical processes in the brain are controlled by laws like Newton’s, such an assumption runs up against the problem that they distinguish no direction of time. Figure 1, with its impossibility of saying in which direction time flows, makes this clear. It is no help to go from its three particles to billions of them. Observed effects should have a real cause. The chain from cause to effects may be quite long and take surprising forms, but a cause there must be. It is unsatisfactory to suppose that we have a direct awareness of an invisible flow of time. Our sense of the passage of time and, even more basically, of seeing motion and knowing its direction, ought to have a cause we can get our hands on.
The lack of time direction in the bare laws of motion led Boltzmann to a remarkable suggestion (quoted in the Notes). As we have seen, Newtonian systems can enter highly ordered phases. These are exceptionally rare periods separated by ‘deserts’ of monotony. Nevertheless, every now and then a system will enter one. Its entropy will go down, reach a minimum, and then start to increase.
We should not think of this happening in a definite direction of time. Instead, we should picture the states of the system strung out in a line, as in Figure 1, which we could ‘walk along’ in either direction. Every now and then, with immense stretches between them, we will come upon regions in which the entropy decreases and the order increases. Then the entropy will start to increase again. Someone ‘walking’ in the opposite direction would have the same experience. Now, such a line of states can represent the entire universe, including human beings. Since we are very complicated and exhibit much order, we can be present only in the exceptional regions of low entropy.