He said that throughout the sciences of physics and chemistry, symmetrical time everywhere suffices. This is of course not correct, because unidirectional time emerges in thermodynamics and statistical mechanics. It is even more important in kinetic theory, if we examine the quantum-mechanical interactions of particles.

Lewis hoped that the four-dimensional theories of spacetime of Hermann Minkowski and Albert Einstein could restore symmetric time, so we could not distinguish cause from effect.

This of course is the idea of a deterministic universe, in which information is conserved, that all times are visible in the eyes of a Laplacian super-intelligence. He found the second law of thermodynamics to be "in direct defiance to the law of symmetry of time."

Lewis hypothesized that to a believer in symmetric time he called "Dr.X"it would be...

a great satisfaction to read in a paper of Willard Gibbs that “the impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability”; and later to follow the development of this thesis by Boltzmann until near the end of the famous lectures on “Gastheorie” he found Boltzmann saying, “Hence, for the universe, both directions of time are indistinguishable, as in space there is no up or down.”

Boltzmann’s qualifications of this statement seemed unnecessary to Dr. X, who now definitely included thermodynamics among those branches of physics which require symmetrical time only. In his note book we read, “The statistical interpretation of thermodynamics offered by Gibbs and Boltzmann affords for the first time an understanding of entropy. The process irreversible in time does not exist. This corollary of the law of symmetry in time itself leads to further important consequences. Thence we may prove to those who are still skeptical the atomic structure of matter, as follows: if we imagine two continuous media to diffuse into one another, such a diffusion would in principle be a phenomenon which by no physical means could be reversed, but if two streams composed of discrete particles should diffuse, then, although it might be a matter of great difficulty to recapture the particles and restore each to its own kind, yet in principle the process is reversible and indeed, according to Boltzmann, the separation will occur spontaneously if the system be left to itself for a sufficiently long period.”

Dr. X adds a remark of much subtlety. “While we recognize the particulate nature of matter, we allow each particle to have a position and a velocity chosen from a whole continuum of possible values. Thus while we claim that an isolated system repeatedly returns nearly to its initial condition, we can not say that it returns exactly to that condition. If we start with a number of molecules all moving in precisely the same direction, we can not claim that after some disturbance they ever again move quite parallel to one another. This implies a sort of irreversibility, and while I am not sure that it is a contradiction to symmetrical time, I confess that I should be better satisfied if we could claim the exact recurrence of an initial state.” It is a pity that Dr. X did not live to see the universal acceptance of quantum theory, which assigns to an isolated system not an infinite continuum of states, but a finite number of discrete states. Thus every particular state exactly recurs within finite time. This modern picture is far simpler than that of Boltzmann, especially as we are going to see that each particular state occurs as often as every other. Hence molecular statistics furnishes quite elementary problems in the theory of probability, like the tossing of coins or the shuffling of cards.

In the main, however, the problems of thermodynamics to-day are not far different from those discussed by Boltzmann and Dr. X. In the note book of the latter we read, “The earth is constantly receiving energy from the sun, and in consequence water is continuously flowing over Niagara Falls, but these descriptive statements can not be called laws of physics. When we turn to the processes studied in the laboratory we find that when a hot and cold body are brought together, it is almost certain that the twro temperatures will become equalized until no discernible difference remains. If we mix two mutually soluble liquids, we may expect the concentration to become nearly uniform. I have learned that it is possible to perform an operation upon the brains of mice so that they respond to no external stimuli, but can still run aimlessly about. If a large number of these mice are placed in one end of a box, that end is now heavier than the other; but this distinction rapidly disappears as the mice, in their random movements, cover with greater uniformity the bottom of the box, so that we may no longer discern any tendency of the box in one direction or the other. I claim that in all these cases there is no phenomenon irreversible in time, and indeed nothing more formidable occurs than in the proverbial case of a needle dropped into a haystack.”

Before analyzing further these problems, we may consider a very interesting discussion of one-way time by Professor Eddington, in “The Nature of the Physical World.” He arrives at a compromise, first by stating that one-way time does not occur in physics outside of thermodynamics, and then by reducing the principle of the increase of entropy from a “primary” to a “secondary” law, which does not prevent him, however, from deducing therefrom a “running down of the universe.” To this compromise I can not agree.* The first statement will be answered by the cases which will be discussed in the following sections, and the second can not be regarded as consistent with the new conception of thermodynamics.

We must be cautious about extending to the whole cosmos the rules which we have obtained from limited experiments in our small laboratories. The chance of obtaining valid results from such an extrapolation is very small, but it can be made in a purely formal way. If the universe is finite, as is now frequently supposed, then the formal application of our existing ideas of thermodynamics and statistics leads directly to the following statement: The precise present state of the universe has occurred in the past and will recur in the future, and in. each case within finite time. Whether the universe actually is running down is, of course, another matter. All we can say is that such an assumption obtains no support *from thermodynamics. Let us, however, turn from the behavior of the universe, about which we know almost nothing, to the phenomena of the laboratory, about which we know a little more. Even in this limited domain it is going to be difficult enough to persuade ourselves that such a phenomenon as an explosion is wholly compatible with the thesis of symmetrical time. If a statement runs counter to a fixed habit of thought which has become nearly instinctive, it may be accepted by many, but believed by few. The use of one-way time

("The Symmetry of Time in Physics.", Science, Vol. 71, No. 1849 (Jun. 6, 1930), p.570)

Turning now to the irreversible thermodynamic process, we shall choose an illustration which is not quite so complicated as an explosion, but involves all essentials. A chemist has spent days in preparing a flask of nearly pure alcohol. This he places in a water bath, and then by accident the flask overturns and the alcohol diffuses through the water. His disappointment is in no way allayed by the fact that none of his material is really lost, nor by the belief that ultimately the molecules of alcohol will accidentally come together to give once more a nearly pure sample. That the chemist would be obliged to wait an unconscionable time for this chance restoration must be given no weight. If it occurred every ten minutes, the principle would be the same. It would still be necessary for him to devise rapid analytical methods to ascertain just when the fortunate event occurred. There is no question but that the accident has involved an element of loss which typifies the irreversible process (which is also spoken of as a process of dissipation, or degradation), but we shall see that this loss in no way implies a dissymmetry of time, nor indeed that it has any temporal implications whatever. Without losing any of the characteristics of the reversible process, we may next examine one of the simplest of systems. Suppose that we have a cylinder closed at each end, and with a middle wall provided with a shutter. In this cylinder are one molecule each of three different gases, A, B and C, and the cylinder is in a thermostat at temperature T. In dealing with the individual molecules we are perhaps arrogating to ourselves the privileges of Maxwell's demon; but in recent years, if I may say so without offense, physicists have become demons.

Begarding each molecule, we shall ask only whether it is in the right or the left half of the cylinder. Obviously eight distributions are possible, such as A and B on the left and C on the right; or B on the left and A and C on the right. According to our ordinary assumptions, each of these distributions is equally probable, or, in other words, the system averages to be in each distribution one eighth of the time. Moreover, each of the eight distributions can be easily described and remembered, so that we are not troubled by a large number of nondescript states. Each distribution occurs over and over, but in no particular order, and in these occurrences there is no trace of dissymmetry with respect to time—there is no “running-down” process here. Yet we may have a typical irreversible process. Suppose that the shutter is closed so as to trap a particular distribution, say all three molecules on the left. We become familiar with this one distribution and wish to study it further, but accidentally the shutter is opened, and instead of the one distribution, we have all eight succeeding one another in a random way. This is a complete analogy to the overturn of the flask of alcohol. If we desire once more to obtain and keep the one distribution in which all the molecules are on the left-hand side of the cylinder, we may . exercise our prerogatives as Maxwell demons by closing the shutter from time to time and determining by spectroscopic means or otherwise which distribution is trapped. In about eight trials we shall obtain the desired result. Unless, however, there is in sentient beings the power to defy the second law of thermodynamics, we shall find that this method of obtaining the desired distribution requires at least as much work as the old-fashioned thermodynamical method of forcing the system into the particular distribution without the aid of demoniacal devices. This classical method consists in slowly pushing a piston from the extreme right of the cylinder as far as the middle wall. In this typical reversible process the work required to overcome the pressure of the three molecules is 3 k T In 2 = k T In 8. At the same time the entropy of the gas is diminished by 3 k In 2.

If we wish to obtain any other one of the particular distributions, from the general distribution, the same amount of work is required. Suppose the particular distribution desired is B on the left, A and C on the right. At the extreme left we have a piston permeable only to B, and at the extreme right a piston permeable only to A and C, and these pistons are moved slowly to the middle wall. We thus obtain the given distribution, and the sum of the work done upon the two pistons is 3 k T In 2. In every case, in passing from the general distribution to a particular known distribution, the gas loses entropy in the amount 3 k In 2. All these processes are completely reversible. If we start with any known distribution and let the proper pistons move outward from the center to the ends of the cylinder, we obtain the general distribution, the system does work in the amount 3 k T In 2, and the entropy of the gas increases by 3 k In 2.

The entropy of the general unknown distribution is greater than the entropy of any one known distribution by 3 k In 2. This, therefore, is the increase in entropy in the irreversible process which occurs when, after trapping any one known distribution, we open the shutter. It is evident, however, that the mere trapping of one distribution makes no change in the entropy, for the shutter may be made as frictionless as we please, and the mere act of opening or closing it will not change the entropy of the system. If we start with the shutter open, with all the eight distributions occurring one after another, and then close the shutter, the system is trapped in one distribution, but there is no change of entropy. Whence we have now reached our most important conclusion. The increase in entropy comes when a known distribution goes over into an unknown distribution. The loss, which is characteristic of an irreversible process, is loss of information. In the simplest case, if we have one molecule which must be in one of two flasks, the entropy becomes less by k In 2, if we know which is the flask in which the molecule is trapped.

Gain in entropy always means loss of information, and nothing more. It is a subjective concept, but we can express it in its least subjective form, as follows. If, on a page, we read the description of a physicochemical system, together with certain data which help to specify the system, the entropy of the system is determined by these specifications. If any of the essential data are erased, the entropy becomes greater; if any essential data are added, the entropy becomes less. Nothing further is needed to show that the irreversible process neither implies one-way time, nor has any other temporal implications. Time is not one of the variables of pure thermodynamics.

("The Symmetry of Time in Physics.", Science, Vol. 71, No. 1849 (Jun. 6, 1930), p.570)

Lewis argued for what he called

the law of entire equilibrium, [which] may be stated as follows. Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process... Moreover if there are various paths by which the first process occurs, there is an equal number of paths by which the second process occurs, and the rate is the same in both directions along every path. This will be true no matter how detailed are the specifications which define the several groups and the several paths..

("A New Principle of Equilibrium", Proceedings of the National Academy of Sciences, 11.3 (1925): 179.

Lewis was familiar with Albert Einstein's 1916 work on the absorption and emission coefficients for radiation interactions with matter, and Einstein's independent derivation of the Planck radiation law.. Unlike Tolman, who did not think quantum processes might make some microscopic interactions irreversible, Lewis considered Einstein's work and suggested it might include errors. He wrote

I believe that some of the ideas contained in this paper have been suggested by the work of Einstein, but he has not proposed this law of equilibrium. Indeed one of the first applications which I shall make, in a subsequent paper, will be to the interaction between matter and light, where I shall attempt to demonstrate the invalidity of Einstein's derivation of Planck's radiation formula.

(ibid., p. 183.