The relentless march of randomness

The Second Law of thermodynamics—what law is this? What is its central role in physical behaviour? And in what way does it present us with a genuinely deep mystery? In the later sections of this book, we shall try to understand the puzzling nature of this mystery and why we may be driven to extraordinary lengths in order to resolve it. This will lead us into unexplored areas of cosmology, and to issues which I believe may be resolved only by a very radical new perspective on the history of our universe. But these are matters that will be our concern later. For the moment let us restrict our attention to the task of coming to terms with what is involved in this ubiquitous law.

Usually when we think of a 'law of physics' we think of some assertion of equality between two different things. Newton's second law of motion, for example, equates the rate of change of momentum of a particle (momentum being mass times velocity) with the total force acting upon it. As another example, the law of conservation of energy asserts that the total energy of an isolated system at one time is equal to its total energy at any other time. Likewise, the law of conservation of electric charge, of momentum, and of angular momentum, each asserts a corresponding equality for the total electric charge, for the total momentum, and for total angular momentum. Einstein's famous law E=mc2 asserts that the energy of a system is always equal to its mass multiplied by the square of the speed of light.

As yet another example, Newton's third law asserts that the force exerted by a body A on a body B, at any one time, is always equal and opposite to the force acting on A due to B. And so it is for many of the other laws of physics.

These are all equalities—and this applies also to what is called the First Law of thermodynamics, which is really just the law of conservation of energy again, but now in a thermodynamic context. We say 'thermodynamic' because the energy of the thermal motions is now being taken into account, i.e. of the random motions of individual constituent particles. This energy is the heat energy of a system, and we define the system's temperature to be this energy per degree of freedom (as we shall be considering again later). For example, when the friction of air resistance slows down a projectile, this does not violate the full conservation law of energy (i.e. the First Law of thermodynamics)—despite the loss of kinetic energy, due to the projectile's slowing—because the air molecules, and those in the projectile, become slightly more energetic in their random motions, from heating due to the friction.

However, the Second Law of thermodynamics is not an equality, but an inequality, asserting merely that a certain quantity referred to as the entropy of an isolated system—which is a measure of the system's disorder, or 'randomness'—is greater (or at least not smaller) at later times than it was at earlier times. Going along with this apparent weakness of statement, we shall find that there is also certain vagueness or subjectivity about the very definition of the entropy of a general system. Moreover, in most formulations, we are led to conclude that there are occasional or exceptional moments at which the entropy must be regarded as actually (though temporarily) reducing with time (in a fluctuation) despite the general trend being that the entropy increases.

Yet, set against this seeming imprecision inherent in the Second Law (as I shall henceforth abbreviate it), this law has a universality that goes far beyond any particular system of dynamical rules that one might be concerned with. It applies equally well, for example, to relativity theory as it does to Newtonian theory, and also to the continuous fields of Maxwell's theory of electromagnetism (that we shall be coming to briefly in §2.6, §3.1 and §3.2, and rather more explicitly in Appendix A1) just as well as it does to theories involving only discrete particles. It applies also to hypothetical dynamical theories that we have no good reason to believe have relevance to the actual universe that we inhabit, although it is most pertinent when applied to realistic dynamical schemes, such as Newtonian mechanics, which have a deterministic evolution and are reversible in time, so that for any allowed evolution into the future, reversing the time direction gives us another equally allowable evolution according to the dynamical scheme.

To put things in familiar terms, if we have a moving-picture film depicting some action that is in accordance with dynamical laws—such as Newton's—that are reversible in time, then the situation depicted when the film is run in reverse will also be in accordance with these dynamical laws. The reader might well be puzzled by this, for whereas a film depicting an egg rolling off a table, falling to the ground, and smashing would represent an allowable dynamical process, the time-reversed film— depicting the smashed egg, originally as a mess on the floor, miraculously assembling itself from the broken pieces of shell, with the yolk and albumen separately joining up to become surrounded by the self-assembling shell, and then jumping up on to the table—is not an occurrence that we expect ever to see in an actual physical process (Fig. 1.1). Yet the full Newtonian dynamics of each individual particle, with its accelerated response (in accordance with Newton's second law) to all forces acting upon it, and the elastic reactions involved in any collision between constituent particles, is completely reversible in time. This also would be the case for the refined behaviour of relativistic and quantum-mechanical particles, according to the standard procedures of modern physics—although there are some subtleties arising from the black-hole physics of general relativity, and also with regard to quantum mechanics, that I do not wish to get embroiled in just yet. Some of these subtleties will actually be crucially important for us later, and will be considered particularly in §3.4. But for the moment, an entirely Newtonian picture of things will suffice.

time progresses

Fig. 1.1 An egg rolling off a table, falling to the ground and smashing according to time-reversible dynamical laws.

We have to accustom ourselves to the fact that the situations that are depicted by both directions of film-running are consistent with Newtonian dynamics, but the one showing the self-assembling egg depicts an occurrence that is inconsistent with the Second Law, and would be such an enormously improbable sequence of events that we can simply reject it as a realistic possibility. What the Second Law indeed states, roughly speaking, is that things are getting more 'random' all the time. So if we set up a particular situation, and then let the dynamics evolve it into the future, the system will evolve into a more random-looking state as time progresses. Strictly, we should not say that it will evolve into a more random-looking state but that, in accordance with what has been said above, it is (something like) overwhelmingly likely to evolve into such a more random state. In practice, we must expect that, according to the Second Law, things are indeed getting progressively more and more random with time, but that this represents merely an overwhelming probability, not quite an absolute certainty.

Nevertheless we can assert, with a considerable amount of confidence, that what we shall experience will be an entropy increase—in other words an increase in randomness. Stated that way, the Second Law sounds perhaps like a council of despair, for it tells us that things are just getting more and more disorganized as time progresses. This does not sound like any kind of a mystery, however, as the title of Part 1 seems to be suggesting that it should. It's just an obvious feature of the way things would behave if left entirely to themselves. The Second Law appears to be just expressing an inevitable and perhaps depressing feature of everyday existence. Indeed, from this point of view, the Second Law of thermodynamics is one of the most natural things imaginable, and certainly something that reflects a completely commonplace experience.

Some might worry that the emergence of life on this Earth, with its seemingly unbelievable sophistication, represents a contradiction with this increase of disorder that the Second Law demands. I shall be explaining later (see §2.2) why there is in fact no contradiction. Biology is, as far as we know, entirely consistent with the overall entropy increase that the Second Law demands. The mystery referred to in the title of Part 1 is a mystery of physics of an entirely different order of scale. Although it has some definite relation to that mysterious and puzzling organization that we are continually being presented with through biology, we have good reason to expect that the latter presents no paradox with regard to the Second Law.

One thing should be made clear, however, with regard to the Second Law's physical status: it represents a separate principle that must be adjoined to the dynamical laws (e.g. to Newton's laws), and is not to be regarded as a deduction from them. The actual definition of the entropy of a system at any one moment is, however, symmetrical with regard to the direction of time (so we get the same entropy definition, for our filmed falling egg, at any one moment, irrespective of the direction in which the film is shown), and if the dynamical laws are also symmetrical in time (as is indeed the case with Newtonian dynamics), the entropy of a system being not always constant in time (as is clearly so with the smashing egg), then the Second Law cannot be a deduction from these dynamical laws. For if the entropy is increasing in a particular situation (e.g. egg smashing), this being in accordance with the Second Law, then the entropy must be decreasing in the reversed situation (egg miraculously assembling), which is in gross violation of the Second Law. Since both processes are nevertheless consistent with the (Newtonian) dynamics, we conclude that the Second Law cannot simply be a consequence of the dynamical laws.

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