### The Entropy of the Early Universe

There is something deeply puzzling about the time-asymmetry of the second law. As discussed earlier in this chapter, the fact that systems tend to evolve toward equilibrium can be explained probabilistically: high-entropy conditions are more likely to occur by chance than low-entropy ones. But it is not so easy to see why entropy was ever low in the first place! You can create a closed system in a low-entropy state, for instance by dropping ice cubes into a thermos of hot coffee and sealing the lid. But in order to do that, you have to use parts of the universe (e.g. ice cubes and coffee) that already have different temperatures or pressures. Similarly, the entropy of an open system can be lowered; but in order for that to happen, the entropy of its environment must have been low already (since the entropy of the environment is raised in the process). How did all of these low-entropy states come about in the first place?

The universe as a whole is very far from equilibrium. Moreover, the entropy of the universe must have been even lower in the past than it is now, because many physical processes increase the total entropy and none decrease it. Given that the universe began in a condition of extremely low entropy, it is unsurprising that its entropy has been increasing, because—as we saw earlier in this chapter—the random motion of microscopic particles tends to increase the entropy of a system. Thus, the time-asymmetry of the second law of thermodynamics can be explained in terms of the fact that the universe began in a condition of extremely low entropy. The details of this explanation are rather complicated, and numerous technical problems with the explanation remain unsolved. Some of these problems were the subject of my doctoral research at Princeton, so I am painfully aware of them; but I am optimistic that the explanation is mostly correct and that the problems will eventually be solved. See appendix C [link coming soon] for more details.

But why did the universe begin in such a low-entropy state? This question is especially pressing, because entropy is supposed to correspond to chance: the lower the entropy of a state, the lower the chance of that state occurring. Thus, it is absurdly unlikely that the universe just happened to start out in such a condition by chance. To get an appreciation for how amazingly improbable it is that the universe began in a low-entropy state merely by chance, remember the piggy-bank analogy discussed earlier. As you may recall, the probability of the coins being distributed unevenly depends on how many coins there are. If there are only a few coins, there’s a good chance that one piggy will get significantly more than the other. But if there are a lot of coins, the probability that one piggy will get a significantly larger proportion is almost zero. Analogously, the probability of a low-entropy state is much lower if a system contains very many particles. Moreover, the probability is also much lower if a system is very far from equilibrium. In other words, the probability of a state occurring by chance (or at random) depends on two factors:

1. The lower the entropy of a thermodynamic condition, the smaller the probability of that condition occurring by chance.
2. The bigger the system, the smaller the probability of a low-entropy condition occurring by chance.

The low-entropy condition of the early universe is extreme in both respects: the universe is a very big system, and it was once in a very low entropy state. The odds of that happening by chance are staggeringly small. Roger Penrose, a mathematical physicist at Oxford University, estimates the probability to be roughly 1/1010123.Roger Penrose, The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford: Oxford University Press, 1989) 445-445. That number is so small that if it were written out in ordinary decimal form, the decimal would be followed by more zeros than there are particles in the universe! It is even smaller than the ratio of the volume of a proton (a subatomic particle) to the entire volume of the visible universe. Imagine filling the whole universe with lottery tickets the size of protons, then choosing one ticket at random. Your chance of winning that lottery is much higher than the probability of the universe beginning in a state with such low entropy! Huw Price, a philosopher of science at Cambridge, has called the low-entropy condition of the early universe “the most underrated discovery in the history of physics.”Huw Price, “On the Origins of the Arrow of Time: Why There is Still a Puzzle about the Low Entropy Past,” in Contemporary Debates in the Philosophy of Science, ed. Christopher Hitchcock (Malden, MA: Blackwell, 2004)