The Fabric of the Cosmos: Space, Time, and the Texture of Reality (21 page)

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.

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 thermodynamics 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

Figure 6.3 A comparison of two proposals for how the ice cubes got to their partially melted state, right now, at 10:30 p.m. Proposal 1 aligns with your memories of melting ice, but requires a comparatively low-entropy starting point at 10:00 p.m. Proposal 2 challenges your memories by describing the partially melted ice you see at 10:30 p.m. as having coalesced out of a glass of water, but starts off in a high-entropy, highly probable configuration of disorder at 10:00 p.m. Every step of the way toward 10:30 p.m., Proposal 2 involves states that are more likely than those in Proposal 1—because, as you can see in the graph, they have higher entropy—and so Proposal 2 is statistically favored.

It was a small step for Boltzmann to realize that the whole of the universe is subject to this same analysis. When you look around the universe right now, what you see reflects a great deal of biological organization, chemical structure, and physical order. Although the universe could be a totally disorganized mess, it's not. Why is this? Where did the order come from? Well, just as with the ice cubes, from the standpoint of probability it is extremely unlikely that the universe we see evolved from an even more ordered—an even less likely—state in the distant past that has slowly unwound to its current form. Rather, because the cosmos has so many constituents, the scales of ordered versus disordered are magnified intensely. And so what's true at the bar is true with a vengeance for the whole universe: it is
far
more likely—breathtakingly more likely—that the whole universe we now see arose as a statistically rare fluctuation from a normal, unsurprising, high-entropy, completely disordered configuration.

Think of it this way: if you toss a handful of pennies over and over again, sooner or later they will all land heads. If you have nearly the infinite patience needed to throw the jumbled pages of
War and Peace
in the air over and over again, sooner or later they will land in correct numerical order. If you wait with your open bottle of flat Coke, sooner or later the random jostling of the carbon dioxide molecules will cause them to reenter the bottle. And, for Boltzmann's kicker, if the universe waits long enough—for nearly an eternity, perhaps—its usual, high-entropy, highly probable, totally disordered state will, through its own bumping, jostling, and random streaming of particles and radiation, sooner or later just happen to coalesce into the configuration that we all see right now. Our bodies and brains would emerge fully formed from the chaos—stocked with memories, knowledge, and skills—even though the past they seem to reflect would never really have happened. Everything we know about, everything we value, would amount to nothing more than a rare but every-so-often-expectable statistical fluctuation momentarily interrupting a near eternity of disorder. This is schematically illustrated in Figure 6.4.

Figure 6.4 A schematic graph of the universe's total entropy through time. The graph shows the universe spending most of its time in a state of total disorder—a state of high entropy—and every so often experiencing fluctuations to states of varying degrees of order, varying states of lower entropy. The greater the entropy dip, the less likely the fluctuation. Significant dips in entropy, to the kind of order in the universe today, are extremely unlikely and would happen very rarely.