Why has our reasoning gone so sadly astray—this being apparently just the same reasoning that seemed convincingly to lead us to expect that the Second Law, with overwhelming probability, must hold for the future evolution of an ordinary physical system? The trouble with the reasoning, as I have provided it, lies in the assumption that the evolution can be regarded as effectively 'random' in relation to the coarse-graining regions. Of course it is not really random, as noted above, since it is precisely determined by the dynamical (e.g. Newton's) laws. But we have taken it that there is no particular bias in this dynamical behaviour, in relation to these coarse-graining regions, and this supposition seemed to be fine for the future evolution. When we consider the past evolution, however, we find that this is manifestly not the case. There is a great deal of bias, for example, in the past-evolved behaviour of the egg, where it would appear to be guided inexorably—if viewed from a time-reversed perspec-tive—from an original messy broken state, through exceptionally improbable actions albeit all in accordance with the dynamical laws, to the exceedingly improbable state of being balanced, complete and unbroken, at the edge of the table. If such behaviour were to be observed in future-directed behaviour it would be regarded as an impossible form of teleology or magic. Why do we regard such clearly focused behaviour as being perfectly acceptable if it is directed towards the past, whereas it would be rejected as scientifically unacceptable if directed into the future?
The answer—though hardly a 'physical explanation'—is simply that such 'past-teleology' is common experience, whereas 'future-teleology' is just something that we never seem to encounter. For it is just a fact of the observed universe that we do not encounter such 'future-teleology'; it is just observational fact that the Second Law holds good. In the universe we know, the dynamical laws appear not to be guided in any way to a future goal and can be regarded as being completely unconcerned with coarse-graining regions; whereas such 'guidance' of the evolution curve in past directions is utterly commonplace. If we examine the evolution curve in its past behaviour, it seems to be 'deliberately' seeking ever smaller and smaller coarse-graining regions. That we do not regard this as strange is simply a matter of it being such a familiar part of our everyday experience. The experience of an egg rolling off the edge of a table and smashing on the floor below is not regarded as strange, whereas a movie film of such an occurrence which is run in the reverse time-direction does indeed look extremely odd, and it represents something that in the ordinary time-direction is simply not part of our experience of the physical world. Such 'teleology' is perfectly acceptable if we are looking towards the past, but it is not a feature of our experience that it apply towards the future.
In fact we can understand this seeming past-teleology of behaviour if we simply suppose that the very origin of our universe was represented in phase space by a coarse-graining region of quite exceptional tininess, so that the initial state of the universe was one of particularly small entropy. So long as we may take it that the dynamical laws are such that there is an appropriate degree of continuity in the way that the entropy of the universe behaves, as noted above, then we need merely suppose that the universe's initial state—what we call the Big Bang—had, for some reason, an extraordinarily tiny entropy (a tininess which, as we shall be seeing in the next part, has a rather subtle character). The appropriate continuity of entropy would then imply a relatively gradual increase of the universe's entropy from then on (in the normal time-direction), giving us some kind of theoretical justification of the Second Law. So the key issue is indeed the specialness of the Big Bang, and the extraordinary minuteness of the initial coarse-graining region B that represents the nature of this special initial state.
The issue of the Big-Bang specialness is central to the arguments of this book. In §2.6 we shall be seeing how extraordinarily special the Big Bang must actually have been and we shall have to confront the very particular nature of this initial state. The underlying deep puzzles that this raises will later lead us into the strange line of thought that provides the distinctive underlying theme of this book. But just for the moment, we may simply take note of the fact that once we accept that such an extraordinarily special state did indeed originate the universe that we know, then the Second Law, in the form that we observe it, is a natural consequence. Provided that there is no corresponding low-entropy ultim ate state of the universe, or some such, providing us with a teleological demand that the universe's evolution curve has to terminate in some other extraordinarily tiny 'future' region T in T, then our reasoning for the increase of entropy in the future time-direction seems to be perfectly acceptable. It is the initial low-entropy constraint, demanding that the evolution curve originate within the extraordinarily tiny region B that gives us a theoretical basis for the Second Law that we actually experience in our universe.
A few points of clarification should however be addressed before we venture (in Part 2) into a more detailed examination of the Big-Bang state. In the first place, it has occasionally been argued that the existence of a Second Law holds no mystery, for our experience of the passage of time is dependent upon an increasing entropy as part of what constitutes our conscious feeling of the passage of time; so whatever time-direction we believe to be the 'future' must be that in which entropy increases. According to this argument, had the entropy been decreasing with respect to some time-parameter t, then our conscious feelings of temporal flow would project in the reverse direction, so that we would regard the small values of entropy to lie in what we think of as our 'future' and the large values in our 'past'. We would therefore take the parameter t to be the reverse of a normal time parameter, so that the entropy would still be increasing into what we experience as being the future. Thus, so the argument goes, our psychological experiences of the passage of time would always be such that the Second Law holds true, irrespective of the physical direction of the progression of entropy.
However, even leaving aside the very dubious nature of any such argument from our 'experience of time progression'—when we know almost nothing about what physical prerequisites might be required for 'conscious experience'—this argument misses the crucial point that the very usefulness of the notion of entropy depends upon our universe being enormously far from thermal equilibrium, so that coarse-graining regions that are far smaller than #max are involved in our common experience. In addition to this, the very fact that entropy is either uniformly increasing or uniformly decreasing depends upon the actuality of one or the other end (but not both ends) of the evolution curve in phase space being constrained to a very tiny coarse-graining region, and this is the case for only a very minute fraction of possible universe histories. It is the very tininess of the coarse-graining region B that our evolution curve appears to have encountered that needs explaining, and this issue is completely untouched by the aforementioned argument.
Sometimes the argument is made (perhaps in conjunction with the above) that the presence of a Second Law is an essential prerequisite for life, so that living beings like ourselves could only exist in a universe (or a universe epoch) in which the Second Law holds true, this law being a necessary ingredient of natural selection, etc. This is an example of 'anthropic reasoning' and I shall be returning briefly to this general issue in §3.2 (end) and §3.3. Whatever value this type of argument may have in other contexts, it is next to useless here. Again there is the very dubious aspect of such reasoning that we do not have a great deal more understanding of the physical requirements for life than we do for consciousness. But even apart from this, and even assuming that natural selection is indeed an essential prerequisite for life, and that it does require the Second Law, this still provides no explanation for the fact that the same Second Law operative here on Earth appears to hold everywhere in the observable universe to distances far beyond those of any relevance to local conditions, such as in galaxies thousands of millions of light years distant, and to times far earlier than the beginnings of life on Earth.
One further point to bear in mind is the following. If we do not assume the Second Law, or that the universe originated in some extraordinarily special initial state, or something else of this general nature, then we cannot use the 'improbability' of the existence of life as a premise for a derivation of a Second Law that is operative at times earlier than the present. No matter how curious and non-intuitive it may seem, the production of life would (if we do not make a prior assumption of the Second Law) be far less probable to come about by natural means—be it by natural selection or any by other seemingly 'natural' process—than by a 'miraculous' creation simply out of random collisions of the constituent particles! To see why this must be, we again examine our evolution curve in the phase space T. If we consider the coarse-graining region £ which represents our present-day Earth, teeming with life as it is, and we ask for the most probable way that this situation can have come about, then we again find that—as with our sequence of greatly decreasing coarse-graining regions . . . , #3', #2', #1', #0 considered in §1.5 above—the 'most probable' way in which £ would have been reached would have been via some corresponding sequence of coarse-graining regions . . ., £3', £2', £1', £ of greatly decreasing volume, representing some completely random-looking teleological assembly of life, completely at odds with what actually happened, this being violently in disagreement with the Second Law, rather than providing a demonstration of it. Accordingly, the mere existence of life provides, in itself, no argument whatever for the full validity of the Second Law.
There is a final point that should be addressed here, having to do with the future. I have argued that it is just a matter of observational fact that the Second Law, with its consequence of an enormous constraint on the initial state, actually holds true in our universe. It is again just a matter of observation that there appears not to be a corresponding constraint in the very remote future. But do we really know that the latter is actually the case? We do not really have much direct evidence of what, in detail, the very remote future will be like. (The evidence that we do have will be discussed in §3.1, §3.2, and §3.4.) We can certainly say that there is no indication available to us now, that hints that the entropy might eventually start to come down again, so that ultimately the very reverse of the Second Law might hold in the remote future. Yet, I do not see that we can absolutely rule out such a thing for the actual universe that we inhabit. Although the ~ 1.4 x 1010 years that have elapsed since the Big
Bang may seem to be a very long time (see §2.1), and no such reversed Second-Law effects have been observed, this time-span is as nothing when compared with what seems to be projected for the complete future time-span of the universe (as we shall be coming to address in §3.1)! In a universe constrained to have an evolution curve that terminates within some tiny region T, its very late evolution must ultimately begin to experience curious correlations between particles that would eventually lead to the kind of teleological behaviour that would seem to us now to be as strange as the self-assembling egg described in §1.5 above.
There is no inconsistency with (say Newtonian) dynamics for the evolution curve of the universe, in its phase space T, to be constrained so that it originates in one very tiny coarse-graining region B and also terminates in another one T. There would be far fewer such curves than there would be those which merely originate in B, but we must already be accustomed to the fact that those which merely originate in B, as appears to be the case with the actual universe we inhabit, represent but an extremely minute proportion of the totality of possibilities. The situations in which the evolution curve is indeed constrained to have both end-points in very tiny regions would represent an even far tinier fraction of all the possibilities, but their logical status is not very different, and we cannot just rule them out. For such evolutions there would be a Second Law operating in the early stages of the universe, just as seems to be the case for the universe we know, but in the very late stages we should find that a reverse Second Law holds true, with entropy ultimately decreasing with time.
I do not myself regard as at all plausible this possibility that the Second Law might eventually reverse—and it will play no significant role in the main proposal that I shall be making in this book. Yet it should be made clear that while our experience provides no hint of such an ultimate reversal of the Second Law, such an eventuality is not intrinsically absurd. We must keep an open mind that exotic possibilities of this kind cannot necessarily be ruled out. In Part 3 of this book, I shall be making a different proposal, and an open mind will be helpful, also, for appreciating what I have to say. Yet the ideas are based on some remarkable facts about the universe, about which we can be reasonably certain. So let us start, in Part 2, with what we actually know about the Big Bang.
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