Men with the skill and courage to do so led the scientific revolution of the seventeenth century. Since then, the frontiers of physics have moved far from the world of direct sense perception, and even the study of our immediate environment more often than not makes use of sophisticated tools and controlled experiments. Nevertheless, the ability to take a fresh look at the familiar and to contrast it with what would be the familiar in a different universe with different laws of nature remains a skill worth cultivating. For the student, and often for the scientist as well, useful insights come from looking at the familiar as if it were unfamiliar.
Consider the second law of thermodynamics. We need not go to the laboratory or to a machine or even to the kitchen to witness its impact on events. It is unlikely that you get through any five minutes of your waking life without seeing the second law at work. The way to appreciate this fact is by thinking backward. Imagine a motion picture of any scene of ordinary life run backward. You might watch a friend unwriting a letter by hand, each flourish of the pen erasing another word as the paper becomes fresher and the ink retreats into the pen. Or think of bits of hair clippings on a beauty-shop floor rising to join the hair on a customer’s head as the stylist unclips. Or a pair of mangled automobiles undergoing instantaneous repair as they back apart. Or a dead rabbit rising to scamper backward into the woods as a crushed bullet re-forms and flies backward into a rifle while some gunpowder is miraculously manufactured out of hot gas. Or something as simple as a cup of coffee on a table gradually becoming warmer as it draws heat from its cooler surroundings. All of these backward-in-time views and myriad others that you can quickly think of are ludicrous and impossible for one reason only—they violate the second law of thermodynamics. In the actual sequence of events, entropy is increasing. In the time reversed view, entropy is decreasing. We recognize at once the obvious impossibility of the hypothetical process in which entropy decreases, even though we may never have thought about entropy increase in the everyday world. In a certain sense everyone “knows” the second law of thermodynamics. It distinguishes the possible from the impossible in ordinary affairs.
In some of the examples cited above, the action of the second law is obvious, as in the increasing disorder produced by an automobile collision, or the increasing entropy associated with heat flow from a cup of coffee. In others, it is less obvious. But whether you can clearly identify the increasing entropy or not, you can be very confident that whenever a sequence of events occurs in our world in one order and not in the other, it is because entropy increase is associated with the possible order, entropy decrease with the impossible order. The reason for this confidence is quite simple. We know of no law other than the second law of thermodynamics that assigns to processes of change in the large-scale world a preferred direction in time.1 Here we have an apparent paradox. In order to understand the paradox and its resolution, one must first understand exactly what is meant by time-reversal invariance.
The principle of time-reversal invariance, valid in Newtonian physics, can be simply stated in terms of hypothetical moving pictures. If the filmed version of any physical process, or sequence of events, is shown backward, the viewer sees a picture of something that could have happened. In slightly more technical language, any sequence of events, if executed in the opposite order, is a physically possible sequence of events. This leads to the rather startling conclusion that it is, in fact, impossible to tell by watching a moving picture of events in nature whether the film is running backward or forward. How can this principle be reconciled with the gross violations of common sense contained in the backward view of a stylist cutting hair, a hunter firing a gun, a child breaking a plate, or the President signing his name? Does it mean that time-reversal invariance is not a valid law in the macroscopic world? No. As far as we know, time-reversal invariance governs every interaction that underlies processes of change in the large-scale world. The key to resolving the paradox is to recognize that possibility does not mean probability. Although the spontaneous reassembly of the fragments of an exploded bomb into a whole, unexploded bomb is wildly, ridiculously improbable, it is not, from the most fundamental point of view, impossible.
At every important point where the macroscopic and submicroscopic descriptions of matter touch, the concept of probability is crucial. The second law of thermodynamics is basically a probabilistic law whose approach to absolute validity increases as the complexity of the system it describes increases. For a system of half a dozen molecules, entropy decrease is not only possible, it is quite likely, at least some of the time. All six molecules might cluster in one corner of their container, or the three less energetic molecules might lose energy via collisions to the three more energetic molecules (“uphill” heat flow). For a system of 1020 molecules, on the other hand, entropy decrease becomes so improbable that it deserves to be called impossible. We could wait a billion times the known lifetime of the universe and still never expect to see the time-reversed view of something as simple as a piece of paper being torn in half. Nevertheless, it is important to realize that the time-reversed process is possible in principle.
Even in the world of particles, a sequence of events may occur with much higher probability in one direction than in the opposite direction. In the world of human experience, the imbalance of probabilities is so enormous that it no longer makes sense to speak of the more probable direction and the less probable direction. Instead we speak of the possible and the impossible. The action of molecular probabilities gives to the flow of events in the large-scale world a unique direction. The (almost complete) violation of time-reversal invariance by the second law of thermodynamics attaches an arrow to time, a one-way sign for the unfolding of events. Through this idea, thermodynamics impinges on philosophy.
In the latter part of the nineteenth century, long before time-reversal invariance was appreciated as a fundamental law of submicroscopic nature, physicists realized that the second law had something quite general to say about our passage through time. There are two aspects of the idea of the arrow of time: first, that the universe, like a wound-up clock, is running down, its supply of available energy ever dwindling; second, that the spontaneous tendency of nature toward greater entropy is what gives us a conception of the unique one-way direction of time.
The second law of thermodynamics had not long been formulated in a general way before scientists reflected on its implications for the universe at large. In 1865, Clausius wrote, without fanfare, as grand a pair of statements about the world as any produced by science: “We can express the fundamental laws of the universe which correspond to the two fundamental laws of the mechanical theory of heat in the following simple form.
“1. The energy of the universe is constant.
“2. The entropy of the universe tends toward a maximum.”
These are the first and second laws of thermodynamics extended to encompass all of nature. Are the extensions justifiable? If so, what are their implications? We know in fact no more than Clausius about the constancy of energy and the steady increase of entropy in the universe at large. We do know that energy conservation has withstood every test since he wrote, and that entropy increase is founded on the very solid principle of change from arrangements of lesser to those of greater probability. Nevertheless, all that we have learned of nature in the century since Clausius leaped boldly to the edge of existence should make us cautious about so great a step. In 1865, the single theory of Newtonian mechanics seemed to be valid in every extremity of nature, from the molecular to the planetary. A century and a half later we know instead that it fails in every extremity—in the domain of small sizes, where quantum mechanics rules; in the domain of high speed, where special relativity changes the rules; and in the domain of the very large, where general relativity warps space and time.
The logical terminus of the universe, assuming it to be a system obeying the same laws as the macroscopic systems accessible to experiment, is known as the “heat death,” a universal soup of uniform density and uniform temperature, devoid of available energy, incapable of further change, a perfect and featureless final disorder. If this is where the universe is headed, we have had no hints of it as yet. Over a time span of some thirteen billion years, the universe has been a vigorously active place, with new stars still being born as old ones die. Presently it looks like the universe will expand “forever,” but much remains to be learned. We have no evidence at all to confirm or contradict the applicability of thermodynamics to the universe as a whole. Even if we choose to postulate its applicability, we need not be led inevitably to the idea of the ultimate heat death. The existence of a law of time-reversal invariance in the world of the small and the essential probabilistic nature of the second law leave open the possibility that one grand improbable reversal could occur in which disorder is restored to order.
Finally, we can link this line of thought to the second aspect of the arrow of time, the uniqueness of the direction of our course through time, with this challenging thought. If it is the second law that gives us a sense of time’s direction, the very construction of the human machine forces us to see the universe running down. In a world that we might look in upon from the outside to see building order out of disorder, the less probable from the more probable, we would see creatures who remembered their future and not their past. For them the trend of events would seem to be toward disorder and greater probability and it is we who would seem to be turned around.
In the more than three centuries since Newton, time has evolved from the obvious to the mysterious. In the Principia, Newton wrote, “Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration.” This view of time as something flowing constantly and inexorably forward, carrying humankind with it, persisted largely intact until the revolution of relativity at the beginning of the twentieth century. The nineteenth century brought only hints of a deeper insight, when it was appreciated that the second law of thermodynamics differentiated between forward and backward in time, as the laws of mechanics had failed to do. If time were run backward, the reversed planetary orbits would be reasonable and possible, obeying the same laws as the actual forward-in-time orbits. But the reversal of any entropy-changing transformation would be neither reasonable nor possible. The second law of thermodynamics points the way for Newton’s equable flow.
Relativity had the most profound effect on our conception of time. The merger of space and time made unreasonable a temporal arrow when there was no spatial arrow. More recently, time-reversal invariance has confirmed the equal status of both directions in time (for most of the laws of nature). Relativity also brought time to a stop. It is more consistent with the viewpoint of modern physics to think of people and things moving through time (as they move through space) than to think of time itself as flowing.
All of the new insights about time make clear that we must think about it in very human terms: Its definition, its measurement, its apparently unique direction stem not from “absolute, true and mathematical time” but from psychological time. These insights also reinforce the idea that the second law of thermodynamics must ultimately account for our sense of time.
It is a stimulating idea that the only reason we are aware of the past and not the future is that we are complicated and highly organized structures. Unfortunately, simpler creatures are no better off. They equalize future and past by remembering neither. An electron, being precisely identical with every other electron, is totally unmarked by its past or by its future. We humans are intelligent enough to be scarred by our past. But the same complexity that gives us a memory at all is what keeps our future a mystery.
1 In the submicroscopic world, time-reversal invariance governs most, but not all, interactions.