The second law of thermodynamics is, as we have seen, an irreversible physical law, and seems to be the one distinguishing characteristic between the real universe and the reverse universe. At the same time, that law is of such a nature, that, for the ultimate particles of matter, it does not exist; it is essentially a law concerning transformations of energy of large masses. And yet all large bodies are made up of countless numbers of the ultimate particles of matter, the laws of whose motion are all perfectly reversible. All phenomena of the reverse universe, however strange they may look, are perfectly explicable in terms of the ordinary physical laws as applied to the smallest material particles. It would seem, then, as though there must be some reason in terms of the reversible physical laws why the second law of thermodynamics must be true; that is, the second law of thermodynamics, if true, should be a consequence of the reversible physical laws applicable to ultimate particles. We are, then, confronted with the paradox of having to deduce an irreversible law from perfectly reversible ones.
And yet, since the reverse universe consists of a perfectly consistent series of positions, obeying all reversible physical laws, it follows that any logical deduction from premises which are reversible laws must inevitably apply to the reverse universe, and that therefore the conclusion must be true in the reverse universe as well as in the real physical universe. That is to say, any deductive conclusion from reversible laws must itself be reversible. And yet, in the case of the second law of thermodynamics, the reversible laws which govern the motions of ultimate particles of matter seem to compound themselves somehow into the best possible example of an irreversible law governing the motions of large masses.
We are, therefore, inevitably led to the conclusion that the second law of thermodynamics cannot be deduced from the reversible laws by strict deductive reasoning. The reversible laws must of necessity leave some room for the possibility of the truth of the reverse of the second law of thermodynamics. But, since the second law of thermodynamics simply represents a general tendency, we come to the conclusion that the only possibility that the second law of thermodynamics represents a correct physical law, is, that it is to be deduced from the reversible laws not as a strict logical consequence, but as a great, or even an overwhelming probability. Such a solution of this paradox of the second law was propounded by Clerk-Maxwell and other physicists of the middle of the nineteenth century.
Let us, then, examine the reasoning by which Clerk-Maxwell was enabled to reconcile reversible premises with an irreversible conclusion. According to his reasoning, both processes are physically possible, concentration and diffusion of energy. The one process obeys the second law of thermodynamics, the other reverses it. Under the second law of thermodynamics, a collision of large masses will generate heat (conversion of molar energy into heat-energy); under its reversal, the heat generates molar motion in and of itself. Now, says Clerk-Maxwell, if particles move in a group, or rather in two approaching groups, the particles are likely to strike one another at all sorts of angles, so that, after the impact, the resulting velocities will become scattered, which means that some of the energy will be converted into heat. On the contrary, a reversal of the process means a concentration of the motions of the particles at the very point and time of the impact, which is a very much more improbable combination, and, requiring as it does that this concentration should happen in a particular direction, at a particular point, at a particular time, in order to have the desired effect, it follows that such a reversal of the second law of thermodynamics is so overwhelmingly improbable as to be almost impossible. The second law of thermodynamics is thus based not on necessity but on extreme probability. A reversal of the second law is possible under the reversible physical laws, as we have seen, but this reasoning tends to prove that it is overwhelmingly improbable, and therefore would almost never happen.
But, again, if the premises of the reasoning are, as we suppose, reversible physical laws, it must be possible to apply the same reasoning to the reverse universe. Consequently, a similar line of reasoning, which must be exactly as correct logically, can be followed by tracing events backwards from effect to cause instead of tracing from cause to effect, as Clerk-Maxwell has done.
Any momentary condition, of the universe may be regarded either as the cause of all future conditions of the universe or as the effect of all past conditions. And not only can a given momentary condition of all particles in the universe determine one and only one possible effect, one and only one possible future; that same given momentary condition (position and velocity of every particle) could only have been caused by one possible past series of conditions. Hence it is just as possible to trace our causal relations step by step backwards, as it is to trace them similarly forwards.
Now, tracing causation thus backwards, we find that molar motions, when traced backwards into the past, will, in all probability, bring us to a time when two masses which are now in motion have been together, in contact. Following Clerk-Maxwell's reasoning, we must say that, when two particles move away from contact with each other, an impact must have been the cause, at least some form of impact of particles, but it is a form of impact which produced molar motion. In all probability, those two particular masses will not have motions which trace back to a rebound of all particles at the same angle; which necessitates, according to the rules of elastic collision, that before the impact the motions of the particles must have been scattered. Thus, tracing the reasoning backwards, we arrive at the probability that the molar motions must have been partially at least caused by heat, that is, to the probability of a reversal of the second law of thermodynamics. On the contrary, in order to have a case in accordance with the second law of thermodynamics, on this analogous reasoning, it would be necessary to suppose two bodies being traced back to contact at some particular time, and that the heat-motions of those bodies, when thus traced back, should suddenly, at the particular moment and point of contact, trace back to a concentration of motion of the particles of each body away from the other, for only such concentration could be the effect of a molar motion bringing the bodies into collision. Now, the probability of such a combination is extremely small, so that, by merely shifting our reasoning gear into reverse, the very same reasoning tells us that the second law of thermodynamics is most extremely improbable, but that, on the contrary, its reversal is an overwhelming probability.
Tracing thus from a given momentary condition of the universe, our forward and backward reasoning combined might be interpreted, if such reasoning could be trusted, to mean that the second law of thermodynamics holds good as a probability as to the future, but that its reversal holds true as to the past. Aside from this result being untrue in point of fact, it is self-contradictory, for any given moment of time is always future as to moments that precede it, and past as to moments that follow it. It follows, then, that there must be some fallacy in Clerk-Maxwell's reasoning, which, when extended, gives us the second law of thermodynamics in the general form.
Take the special case that we have been using as an illustration. Molar motion without heat, it is true, is likely, as a matter of pure theory, to produce, after impact, less molar motion and some heat (the total amount of energy remaining invariable). But such an initial condition is, in itself, extremely improbable. If initial velocities of particles may be selected initially as in any direction: and in any amount, it is extremely improbable that all the velocities will have the same direction and amount, or even approximately so. The smaller the number of particles, the greater the probability of a concentrated motion resulting. Also, the smaller the mass, the greater the probable average velocity of the mass at a given time, when the particles are moving at random. Hence, when there is impact of bodies in which particles move at random, the probabilities are that, at that moment, at the point of contact, the small mass of particles in the immediate vicinity will have a greater speed in all probability than the entire mass. Thus, when the collision occurs, the force available for producing molar motion will consist, in the immediate vicinity of the point of contact, of two average speeds greater than those of the respective masses. If those greater speeds tend to be more towards one another than the masses as a whole, then it would be most probable that some of the heat-energy of the two bodies will be converted into molar motion. On the other hand, if the respective speeds in the vicinity of the point of contact are more away from each other than the velocities of the masses themselves, the reverse will happen. Besides, while we have this possibility of heat turning into molar energy or into some other form of energy, and of differences of energy concentration building themselves up in this manner, we have the contrary tendency supplied by Clerk-Maxwell's reasoning. The result is, that we as yet can form no conclusions as to which tendency is more likely.
If, furthermore, we consider that we must regard for a given moment of time, all positions and velocities as equally likely, and that for all such initial positions and velocities which will give a universe obeying the second law of thermodynamics, there is a reverse universe, equally probable, reversing that law, we come to the conclusion that the second law and its reverse are equally probable. If this is true for any given event, then the probability of the observed facts, that is to say, that all events obey the second law, must be infinitesimally small. So that, again, we are forced to the conclusion that the second law of thermodynamics, being an observed fact which can only be explained as an extremely probable result of the reversible physical laws is, on the contrary, most extremely improbable.
Not merely that, but the second law of thermodynamics, when pushed to its logical conclusion, produces rather absurd results. In the first place, we have, seen that it involves a sort of death of the universe in the remote future, a time when all will be one dead level of heat; though all this will, in all probability, come about slowly. But the rate of decrease of the available energy under this second law is approximately proportional to the amount of available energy in the universe; therefore the rate of the running down of energy into the unavailable form must be constantly decreasing. Tracing backwards, we find that, in the past, the farther back we go, the more we get a larger percentage of available energy in the universe, increasing at an ever greater rate. Therefore it follows that we must arrive at some definite time in the past—and that not at an infinite time back—when the available energy was 100% of the total energy of the universe. At a time probably not much farther back, all the motion in the universe must have consisted of molar motion of masses which, as we go back, must have increased in size till we arrive at a time when all the energy must have consisted of the energy of two halves of the universe moving together, each half of the universe being at a temperature of absolute zero and all its parts moving side by side at exactly the same velocity. This possibility, it is true, is somewhat corroborated by the fact that at present the stars are moving in two opposite directions, in two opposite currents, as it were, which may be supposed to be the remnants of the two original large groups of stars whose collision formed the present universe according to this hypothesis.
At the same time the two original halves of the universe cannot have been altogether mutually impenetrable, for in that case the result of the collision would but have made them rebound, though producing a great amount of internal heat-energy in each, and possibly breaking some small pieces off each. It would seem, then, as though the original halves of the universe must have consisted of separate dark stars, with a structure somewhat similar to the present universe. At the time of the collision, all the stars, even all the particles, in each semi-universe must all be moving together at the same speed and in the same direction.
The second law of thermodynamics, then, must date from some sort of Great Collision out of which the present universe evolved. But what happened before this Great Collision? The answer would have to be, everything was at a temperature of absolute zero, there were two semi-universes which were moving towards each other, in each of which there was not even a trace of relative motion. Although each of the two semi-universes was in motion, yet within each there was no motion, no internal energy.
But if such was the situation at the time of the Great Collision, it cannot have been so for an eternity past, unless we conceive of the law of gravitational attraction not to have been true in those times. Taking each semi-universe by itself, its reverse universe will also show the same conditions as we have already described, except that the semi-universes are moving away from each other, so that we can proceed in peace without danger from the impending Great Collision. Each semi-universe may, for the purpose of internal occurrences, be regarded as at rest. Gravitation will then draw all the stars of each semi-universe towards its center of gravity, till all of them fall in there. Reversing once more, so as to obtain the process as it must have been supposed to happen, we get the following result: Each semi-universe originally consisted of one great body; suddenly, somehow, that body exploded into pieces, which formed stars, each piece, though, remaining at a temperature of absolute zero. Finally, in each semi-universe, mutual gravitation of the stars slowed them down to relative rest. Just when this relative rest was reached, the two semi-universes collided, and out of this collision came our present universe. Thus we trace a little farther back to the Great Explosions; but these explosions cannot possibly be traced back any farther according to the known physical laws without violating the second law of thermodynamics. In consequence, if we wish to preserve the second law of thermodynamics, we must either dispense with some of the other physical laws, or as some physicists have done, intersperse a creation. In other words, the second law of thermodynamics cannot have been true for an eternity past, though it may be true for on eternity in the future. And even the assumption of a creation would be assuming a process different from the processes coming under the ordinary physical laws.
In other words, we come to the inevitable conclusion that the subsistence of the irreversible second law of thermodynamics in the same universe as the reversible laws concerning the motion of particles is a paradox, both from that point of view and from the fact that this second law, pushed to its logical conclusion, leads back to a mysterious creation which denies all physical laws whatever.