The inexorable increase of entropy into the future

Let us try to get some understanding of why it is to be expected that the entropy should increase when a system evolves into the future, as the Second Law demands. Suppose we imagine that our system starts off in a state of reasonably low entropy—so that the point p, which is to move through phase space T thereby describing the time-evolution of the system, starts off at a point p0 in a fairly small coarse-graining region #0 (see Fig.1.15). We must bear in mind that, as noted above, the various coarse-graining regions tend to differ in size by absolutely enormous factors. Also, the huge dimensionality of phase space T will tend to imply that there are likely to be vast numbers of coarse-graining volumes neighbouring any one particular region. (Our 2- or 3-dimen-sional images are rather misleading in this particular respect, but we see that the number of neighbours is going up with increasing dimension—typically six in the 2-dimension case and fourteen in 3; see Fig.1.16.) Thus, it will be exceedingly probable that the evolution curve described by p, as it leaves the coarse-graining region #0 of the starting point p0 and enters the next coarse-graining region #1, will find that #1 has a hugely greater volume than #0—for to find, instead, an enormously smaller volume would seem a grossly unlikely action for the point p to take, as though p were to succeed, just by chance, in the proverbial search for a needle in a haystack, but here with an enormously more formidable task!

Fig. 1.15 The system starts off at a point of p0 in a fairly small coarse-graining region ^0.
Fig. 1.16 As the dimension n increases, the typical number of neighbouring coarse-graining regions increases rapidly. (a) n=2 with typically 6 neighbours. (b) n = 3 with typically 14 neighbours.

The logarithm of %'s volume will also, consequently, be somewhat greater than the logarithm of of volume, though only a good deal more modestly greater than is provided by the increase in actual volume (see §1.2 above), so the entropy will have been increased just somewhat. Then, as p enters the next coarse-graining region, say #2, we are again highly likely to find that the volume of #2 is hugely greater than that of #1, so the entropy value will again grow somewhat. We next expect to find thatp subsequently enters another region, say #3, even hugely larger than those it had been in before, and the entropy goes up again somewhat, and so on. Moreover, because of the vast increases in the volumes of these coarse-graining regions, once p has entered a larger region, we may regard it as a practical impossibility—i.e. as 'overwhelmingly unlikely'—that it will find a smaller one again, of the kind of far tinier sizes that provided the somewhat smaller entropy values that were encountered previously. Thus, as time marches forward, into the future, the entropy value must be expected to increase relentlessly, though far more modestly than do the actual volumes.

Of course, it is not strictly impossible that a smaller entropy value may be obtained in this way; it is merely that such occurrences of entropy reduction must be regarded as overwhelmingly unlikely. The entropy increase that we have obtained is simply the kind of trend that we must take to be the normal state of affairs where the evolution proceeds in a way that has no particular bias with regard to the coarse-graining of the phase space, and might as well be treated as though the track of p through phase space were essentially random, despite the fact that the evolution is actually governed by the well-defined and completely deterministic procedures of (say) Newtonian mechanics.

One might legitimately wonder why p does not simply directly enter #max, the (vastly) greatest coarse-graining region of all, rather than sequentially entering a succession of larger and larger coarse-graining regions as described above. Here, #max refers to what is commonly called thermal equilibrium, where the volume of #max would be likely to exceed the total of all the other coarse-graining regions put together. Indeed, it may be expected that p will eventually reach #max, and when it does so it will, for the most part, remain in this region, with merely the very occasional excursion into a smaller region (a thermal fluctuation). But the evolution curve must be regarded as describing a continuous evolution, where the state at one moment is not likely to differ greatly from the state a moment before. Accordingly, the coarse-graining volumes would be likely not to differ from their neighbours by such an enormous amount as would be represented by a direct leap to Rmax, despite the vast changes in coarse-graining volumes that the evolution curve would encounter. We would not expect that the entropy is likely to jump that discontinuously, but merely to pass fairly gradually to larger and larger values of the entropy.

This appears to be all pretty satisfactory, and might well lead us to believe that a gradual entropy increase into the future is a completely natural expectation which seems to be hardly in need of further deep deliberation—except perhaps for details of rigour that might be needed to satisfy the mathematical purist. The egg, referred to in the previous section, which starts, at the moment now, by being perched on the edge of the table, indeed has a likely entropy-increasing future evolution that would be consistent with its falling off the table and smashing on the ground. This is completely in accordance with the simple considerations of greatly increasing phase-space volumes, as indicated above.

However, let us pose another question, somewhat different from that of the expected future behaviour of the egg. Let us ask for the likely past behaviour of the egg. We want to know, instead: 'what is the most likely way for the egg to have found itself to be perched on the edge of the table in the first place?'

We can attempt to address this issue in just the same way as before, where we asked for the most likely future evolution of our system starting from now, but this time we are asking for the most probable past evolution of our system leading up to now. Our Newtonian laws work just as well in the past time-direction, and again give us a deterministic pastevolution. Thus, there is some evolution curve leading up to the point po, in the phase space T, which describes this past-evolution, and represents the way that the egg happened to become poised at the edge of the table. To find this 'most probable' past history of our egg, we again examine the coarse-graining regions adjoining Ro, and we again observe that there are vast differences in their sizes. Accordingly, there will be enormously more evolution curves ending at po which enter Ro from a huge region like #1, whose volume greatly exceeds that of #0, than there will be that enter #0 from much smaller regions. Let us say that the evolution curve enters #0 from the region #1', very much larger than #0. Prior to this, there would again be neighbouring regions of vastly differing sizes, and we again note that the enormous majority of evolutions entering #1' would come from coarse-graining regions far larger than #1'. Accordingly, it appears that we may again suppose that our past-evolution curve entering #1' comes from some region #2', of vastly greater volume than #1', and that likewise it entered #2', from a region #3', of even larger volume than #2', and so on. See Fig. 1.15.

This is the conclusion that our reasoning seems to have led us to, but does it make sense? Such evolution curves would be hugely more numerous than the evolution curves leading up to p0 from the succession of much smaller volumes, say . . . , #-3, #-2, #-1, #0, which would be likely to have actually occurred, whose volumes would be greatly increasing from smaller volumes, in the direction of increasing time, as would be consistent with the Second Law. Rather than providing us with support for the Second Law, our line of reasoning now seems to have led us to a completely wrong answer, namely to expect continual gross violations of the Second Law in the past!

Our reasoning seems to have led us to expect, for example, that an exceedingly probable way that our egg originally found itself to be perched at the edge of the table was that it started as a mess of broken eggshell at the bottom of the table, mixed with yolk and albumen all churned up together and partly absorbed between the floorboards. This mess then spontaneously collected itself together, removing itself cleanly from the floor, with yolk and albumen thoroughly separating out and becoming completely enclosed by the miraculously self-assembling eggshell to provide a perfectly constructed egg, which propels itself from the ground at exactly the right speed so that it can become delicately perched on the edge of the table. Such behaviour would be of the kind that our above reasoning led to, with a 'probable' evolution curve successively passing through regions with volumes of greatly reducing size, like . . . , #3', #2', #1', #0. But this would be grossly in conflict with what presumably actually happened, namely that some careless person placed the egg on the table, not realizing that it was in danger of rolling too close to the edge. That evolution would have been consistent with the Second Law, being represented in the phase space T, by an evolution curve passing through the succession of greatly increasing volumes . . . , #-3, #-2, #-1, #0. When applied in the past time-direction, our argument has indeed given us an answer that is about as completely wrong as one could imagine.

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