by Brian Greene
Although it may not be immediately apparent, we have now come to an intriguing point. The second law of thermodynamics seems to have given us an arrow of time, one that emerges when physical systems have a large number of constituents. If you were to watch a film of a couple of carbon dioxide molecules that had been placed together in a small box (with a tracer showing the movements of each), you'd be hard pressed to say whether the film was running forward or in reverse. The two molecules would flit this way and that, sometimes coming together, sometimes moving apart, but they would not exhibit any gross, overall behavior distinguishing one direction in time from the reverse. However, if you were to watch a film of 10 24 carbon dioxide molecules that had been placed together in the box (as a small, dense cloud of molecules, say), you could easily determine whether the film was being shown forward or in reverse: it is overwhelmingly likely that the forward time direction is the one in which the gas molecules become more and more uniformly spread out, achieving higher and higher entropy. If, instead, the film showed uniformly dispersed gas molecules swooshing together into a tight group, you'd immediately recognize that you were watching it in reverse.
The same reasoning holds for essentially all the things we encounter in daily life—things, that is, which have a large number of constituents: the forward-in-time arrow points in the direction of increasing entropy. If you watch a film of a glass of ice water placed on a bar, you can determine which direction is forward in time by checking that the ice melts—its H 2 O molecules disperse throughout the glass, thereby achieving higher entropy. If you watch a film of a splattering egg, you can determine which direction is forward in time by checking that the egg's constituents become more and more disordered—that the egg splatters rather than unsplatters, thereby also achieving higher entropy.
As you can see, the concept of entropy provides a precise version of the "easy versus difficult" conclusion we found earlier. It's easy for the pages of War and Peace to fall out of order because there are so many out-of-order arrangements. It's difficult for the pages to fall in perfect order because hundreds of pages would need to move in just the right way to land in the unique sequence Tolstoy intended. It's easy for an egg to splatter because there are so many ways to splatter. It's difficult for an egg to unsplatter, because an enormous number of splattered constituents must move in perfect coordination to produce the single, unique result of a pristine egg resting on the counter. For things with many constituents, going from lower to higher entropy—from order to disorder—is easy, so it happens all the time. Going from higher to lower entropy—from disorder to order—is harder, so it happens rarely, at best.
Notice, too, that this entropic arrow is not completely rigid; there is no claim that this definition of time's direction is 100 percent foolproof. Instead, the approach has enough flexibility to allow these and other processes to happen in reverse as well. Since the second law proclaims that entropy increase is only a statistical likelihood, not an inviolable fact of nature, it allows for the rare possibility that pages can fall into perfect numerical order, that gas molecules can coalesce and reenter a bottle, and that eggs can unsplatter. By using the mathematics of entropy, the second law expresses precisely how statistically unlikely these events are (remember, the huge number on pages 152-53 reflects how much more likely it is that pages will land out of order), but it recognizes that they can happen.
This seems like a convincing story. Statistical and probabilistic reasoning has given us the second law of thermodynamics. In turn, the second law has provided us with an intuitive distinction between what we call past and what we call future. It has given us a practical explanation for why things in daily life, things that are typically composed of huge numbers of constituents, start like this and end like that, while we never see them start like that and end like this. But over the course of many years— and thanks to important contributions by physicists like Lord Kelvin, Josef Loschmidt, Henri Poincaré, S. H. Burbury, Ernst Zermelo, and Willard Gibbs—Ludwig Boltzmann came to appreciate that the full story of time's arrow is more surprising. Boltzmann realized that although entropy had illuminated important aspects of the puzzle, it had not answered the question of why the past and the future seem so different. Instead, entropy had redefined the question in an important way, one that leads to an unexpected conclusion.
Entropy: Past and Future
Earlier, we introduced the dilemma of past versus future by comparing our everyday observations with properties of Newton's laws of classical physics. We emphasized that we continually experience an obvious directionality to the way things unfold in time but the laws themselves treat what we call forward and backward in time on an exactly equal footing. As there is no arrow within the laws of physics that assigns a direction to time, no pointer that declares, "Use these laws in this temporal orientation but not in the reverse," we were led to ask: If the laws underlying experience treat both temporal orientations symmetrically, why are the experiences themselves so temporally lopsided, always happening in one direction but not the other? Where does the observed and experienced directionality of time come from?
In the last section we seemed to have made progress, through the second law of thermodynamics, which apparently singles out the future as the direction in which entropy increases. But on further thought it's not that simple. Notice that in our discussion of entropy and the second law, we did not modify the laws of classical physics in any way. Instead, all we did was use the laws in a "big picture" statistical framework: we ignored fine details (the precise order of War and Peace 's unbound pages, the precise locations and velocities of an egg's constituents, the precise locations and velocities of a bottle of Coke's CO 2 molecules) and instead focused our attention on gross, overall features (pages ordered vs. unordered, egg splattered vs. not splattered, gas molecules spread out vs. not spread out). We found that when physical systems are sufficiently complicated (books with many pages, fragile objects that can splatter into many fragments, gas with many molecules), there is a huge difference in entropy between their ordered and disordered configurations. And this means that there is a huge likelihood that the systems will evolve from lower to higher entropy, which is a rough statement of the second law of thermodynamics. But the key fact to notice is that the second law is derivative: it is merely a consequence of probabilistic reasoning applied to Newton's laws of motion.
This leads us to a simple but astounding point: Since Newton's laws of physics have no built-in temporal orientation, all of the reasoning we have used to argue that systems will evolve from lower to higher entropy toward the future works equally well when applied toward the past. Again, since the underlying laws of physics are time-reversal symmetric, there is no way for them even to distinguish between what we call the past and what we call the future. Just as there are no signposts in the deep darkness of empty space that declare this direction up and that direction down, there is nothing in the laws of classical physics that says this direction is time future and that direction is time past. The laws offer no temporal orientation; it's a distinction to which they are completely insensitive. And since the laws of motion are responsible for how things change—both toward what we call the future and toward what we call the past—the statistical/probabilistic reasoning behind the second law of thermodynamics applies equally well in both temporal directions. Thus, not only is there an overwhelming probabilitythat the entropy of a physical system will be higher in what we call the future, but there is the same overwhelming probability that it was higher in what we call the past. We illustrate this in Figure 6.2.
This is the key point for all that follows, but it's also deceptively subtle. A common misconception is that if, according to the second law of thermodynamics, entropy increases toward the future, then entropy necessarily decreases toward the past. But that's where the subtlety comes in. The second law actually says that if at any given moment of interest, a physical system happens not to possess the maximum possible entropy, it is extraordinarily likel
y that the physical system will subsequently have and previously had more entropy. That's the content of Figure 6.2b. With laws that are blind to the past-versus-future distinction, such time symmetry is inevitable.
Figure 6.2 (a) As it's usually described, the second law of thermodynamics implies that entropy increases toward the future of any given moment. (b) Since the known laws of nature treat forward and backward in time identically, the second law actually implies that entropy increases both toward the future and toward the past from any given moment.
That's the essential lesson. It tells us that the entropic arrow of time is double-headed. From any specified moment, the arrow of entropy increase points toward the future and toward the past. And that makes it decidedly awkward to propose entropy as the explanation of the one-way arrow of experiential time.
Think about what the double-headed entropic arrow implies in concrete terms. If it's a warm day and you see partially melted ice cubes in a glass of water, you have full confidence that half an hour later the cubes will be more melted, since the more melted they are, the more entropy they have. 11 But you should have exactly the same confidence that half an hour earlier they were also more melted, since exactly the same statistical reasoning implies that entropy should increase toward the past. And the same conclusion applies to the countless other examples we encounter every day. Your assuredness that entropy increases toward the future— from partially dispersed gas molecules' further dispersing to partially jumbled page orders' getting more jumbled—should be matched by exactly the same assuredness that entropy was also higher in the past.
The troubling thing is that half of these conclusions seem to be flatout wrong. Entropic reasoning yields accurate and sensible conclusions when applied in one time direction, toward what we call the future, but gives apparently inaccurate and seemingly ridiculous conclusions when applied toward what we call the past. Glasses of water with partially melted ice cubes do not usually start out as glasses of water with no ice cubes in which molecules of water coalesce and cool into chunks of ice, only to start melting once again. Unbound pages of War and Peace do not usually start thoroughly out of numerical order and through subsequent tosses get less jumbled, only to start getting more jumbled again. And going back to the kitchen, eggs do not generally start out splattered, and then coalesce into a pristine whole egg, only to splatter some time later.
Or do they?
Following the Math
Centuries of scientific investigations have shown that mathematics provides a powerful and incisive language for analyzing the universe. Indeed, the history of modern science is replete with examples in which the math made predictions that seemed counter to both intuition and experience (that the universe contains black holes, that the universe has anti-matter, that distant particles can be entangled, and so on) but which experiments and observations were ultimately able to confirm. Such developments have impressed themselves profoundly on the culture of theoretical physics. Physicists have come to realize that mathematics, when used with sufficient care, is a proven pathway to truth.
So, when a mathematical analysis of nature's laws shows that entropy should be higher toward the future and toward the past of any given moment, physicists don't dismiss it out of hand. Instead, something akin to a physicists' Hippocratic oath impels researchers to maintain a deep and healthy skepticism of the apparent truths of human experience and, with the same skeptical attitude, diligently follow the math and see where it leads. Only then can we properly assess and interpret any remaining mismatch between physical law and common sense.
Toward this end, imagine it's 10:30 p.m. and for the past half hour you've been staring at a glass of ice water (it's a slow night at the bar), watching the cubes slowly melt into small, misshapen forms. You have absolutely no doubt that a half hour earlier the bartender put fully formed ice cubes into the glass; you have no doubt because you trust your memory. And if, by some chance, your confidence regarding what happened during the last half hour should be shaken, you can ask the guy across the way, who was also watching the ice cubes melt (it's a really slow night at the bar), or perhaps check the video taken by the bar's surveillance camera, both of which would confirm that your memory is accurate. If you were then to ask yourself what you expect to happen to the ice cubes during the next half hour, you'd probably conclude that they'd continue to melt. And, if you'd gained sufficient familiarity with the concept of entropy, you'd explain your prediction by appealing to the overwhelming likelihood that entropy will increase from what you see, right now at 10:30 p.m., toward the future. All that makes good sense and jibes with our intuition and experience.
But as we've seen, such entropic reasoning—reasoning that simply says things are more likely to be disordered since there are more ways to be disordered, reasoning which is demonstrably powerful at explaining how things unfold toward the future—proclaims that entropy is just as likely to also have been higher in the past. This would mean that the partially melted cubes you see at 10:30 p.m. would actually have been more melted at earlier times; it would mean that at 10:00 p.m. they did not begin as solid ice cubes, but, instead, slowly coalesced out of room-temperature water on the way to 10:30 p.m., just as surely as they will slowly melt into room-temperature water on their way to 11:00 p.m.
No doubt, that sounds weird—or perhaps you'd say nutty. To be true, not only would H 2 O molecules in a glass of room-temperature water have to coalesce spontaneously into partially formed cubes of ice, but the digital bits in the surveillance camera, as well as the neurons in your brain and those in the brain of the guy across the way, would all need to spontaneously arrange themselves by 10:30 p.m. to attest to there having been a collection of fully formed ice cubes that melted, even though there never was. Yet this bizarre-sounding conclusion is where a faithful application of entropic reasoning—the same reasoning that you embrace without hesitation to explain why the partially melted ice you see at 10:30 p.m. continues to melt toward 11:00 p.m.—leads when applied in the time-symmetric manner dictated by the laws of physics. This is the trouble with having fundamental laws of motion with no inbuilt distinction between past and future, laws whose mathematics treats the future and past of any given moment in exactly the same way. 12
Rest assured that we will shortly find a way out of the strange place to which an egalitarian use of entropic reasoning has taken us; I'm not going to try to convince you that your memories and records are of a past that never happened (apologies to fans of The Matrix ). But we will find it very useful to pinpoint precisely the disjuncture between intuition and the mathematical laws. So let's keep following the trail.
A Quagmire
Your intuition balks at a past with higher entropy because, when viewed in the usual forward-time unfolding of events, it would require a spontaneous rise in order: water molecules spontaneously cooling to 0 degrees Celsius and turning into ice, brains spontaneously acquiring memories of things that didn't happen, video cameras spontaneously producing images of things that never were, and so on, all of which seem extraordinarily unlikely—a proposed explanation of the past at which even Oliver Stone would scoff. On this point, the physical laws and the mathematics of entropy agree with your intuition completely. Such a sequence of events, when viewed in the forward time direction from 10 p.m. to 10:30 p.m., goes against the grain of the second law of thermodynamics—it results in a decrease in entropy—and so, although not impossible, it is very unlikely.
By contrast, your intuition and experience tell you that a far more likely sequence of events is that ice cubes that were fully formed at 10 p.m. partially melted into what you see in your glass, right now, at 10:30 p.m. But on this point, the physical laws and mathematics of entropy only partly agree with your expectation. Math and intuition concur that if there really were fully formed ice cubes at 10 p.m., then the most likely sequence of events would be for them to melt into the partial cubes you see at 10:30 p.m.: the resulting increase in entropy is in line both with the second law of ther
modynamics and with experience. But where math and intuition deviate is that our intuition, unlike the math, fails to take account of the likelihood, or lack thereof, of actually having fully formed ice cubes at 10 p.m., given the one observation we are taking as unassailable, as fully trustworthy, that right now, at 10:30 p.m., you see partially melted cubes.
This is the pivotal point, so let me explain. The main lesson of the second law of thermodynamics is that physical systems have an overwhelming tendency to be in high-entropy configurations because there are so many ways such states can be realized. And once in such high-entropy states, physical systems have an overwhelming tendency to stay in them. High entropy is the natural state of being. You should never be surprised by or feel the need to explain why any physical system is in a high-entropy state. Such states are the norm. On the contrary, what does need explaining is why any given physical system is in a state of order, a state of low entropy. These states are not the norm. They can certainly happen. But from the viewpoint of entropy, such ordered states are rare aberrations that cry out for explanation. So the one fact in the episode we are taking as unquestionably true—your observation at 10:30 p.m. of low-entropy partially formed ice cubes—is a fact in need of an explanation.
And from the point of view of probability, it is absurd to explain this low-entropy state by invoking the even lower- entropy state, the even less likely state, that at 10 p.m. there were even more ordered, more fully formed ice cubes being observed in a more pristine, more ordered environment. Instead, it is enormously more likely that things began in an unsurprising, totally normal, high-entropy state: a glass of uniform liquid water with absolutely no ice. Then, through an unlikely but every-so-often-expectable statistical fluctuation, the glass of water went against the grain of the second law and evolved to a state of lower entropy in which partially formed ice cubes appeared. This evolution, although requiring rare and unfamiliar processes, completely avoids the even lower-entropy, the even less likely, the even more rare state of having fully formed ice cubes. At every moment between 10 p.m. and 10:30 p.m., this strange-sounding evolution has higher entropy than the normal ice-melting scenario, as you can see in Figure 6.3, and so it realizes the accepted observation at 10:30 p.m. in a way that is more likely— hugely more likely—than the scenario in which fully formed ice cubes melt. 13 That is the crux of the matter. 13
