Why is mixing easy and unmixing hard? When we mix two liquids, we see them swirl together and gradually blend into a uniform texture. By itself, that process doesn’t offer much clue into what is really going on. So instead let’s visualize what happens when we mix together two different kinds of colored sand. The important thing about sand is that it’s clearly made of discrete units, the individual grains. When we mix together, for example, blue sand and red sand, the mixture as a whole begins to look purple. But it’s not that the individual grains turn purple; they maintain their identities, while the blue grains and the red grains become jumbled together. It’s only when we look from afar (“macroscopically”) that it makes sense to think of the mixture as being purple; when we peer closely at the sand (“microscopically”) we see individual blue and red grains.
Okay cats and kittens, now we’re really cooking. We haven’t exactly been reluctant throughout the book to talk about entropy and the arrow of time, but now we get to be precise. Not only do we explain Boltzmann’s definition of entropy, but we give an example with numbers, and even use an equation. Scary, I know. (In fact I’d love to hear opinions about how worthwhile it was to get just a bit quantitative in this chapter. Does the book gain more by being more precise, or lose by intimidating people away just when it was getting good?)
In case you’re interested, here is a great simulation of the box-of-gas example discussed in the book. See entropy increase before your very eyes!
Explaining Boltzmann’s definition of entropy is actually pretty quick work; the substantial majority of the chapter is devoting to digging into some of the conceptual issues raised by this definition. Who chooses the coarse graining? (It’s up to us, but Nature does provide a guide.) Is entropy objective, or does it depend on our subjective knowledge? (Depends, but it’s as objective as we want it to be.) Could entropy ever systematically decrease? (Not in a subsystem that interacts haphazardly with its environment.)
We also get into the philosophical issues that are absolutely inevitable in sensible discussions of this subject. No matter what anyone tells you, we cannot prove the Second Law of Thermodynamics using only Boltzmann’s definition of entropy and the underlying dynamics of atoms. We need additional hypotheses from outside the formalism. In particular, the Principle of Indifference, which states that we assign equal probability to every microstate within any given macrostate; and the Past Hypothesis, which states that the universe began in a state of very low entropy. There’s just no getting around the need for these extra ingredients. While the Principle of Indifference seems fairly natural, the Past Hypothesis cries out for some sort of explanation.
Not everyone agrees. Craig Callender, a philosopher who has thought a lot about these issues, reviewed my book for New Scientist and expresses skepticism that there is anything to be explained. (A minority view in the philosophy community, for what it’s worth.) He certainly understands the need to assume that the early universe had a low entropy — as he says in a longer article, “By positing the Past State the puzzle of the time asymmetry of thermodynamics is solved, for all intents and purposes,” with which I agree. Callender is simply drawing a distinction between positing the past state, which he’s for, and trying to explain the past state, which he thinks is a waste of time. We should just take it as a brute fact, rather than seeking some underlying explanation — “Sometimes it is best not to scratch explanatory itches,” as he puts it.
While it is absolutely possible that the low entropy of the early universe is simply a brute fact, never to be explained by any dynamics or underlying principles, it seems crazy to me not to try. If we picked a state of the universe randomly out of a hat, the chances we would end up with something like our early universe are unimaginably small. To most of us, that’s a crucial clue to something deep about the universe: it’s early state was not picked randomly out of a hat! Something should explain it. We can’t be completely certain that such an explanation exists, but cosmology is hard enough without choosing to ignore the most blatant clues that nature is sticking under our noses.
This chapter and the next two are the heart and soul of the book. I hope that the first part of the book is interesting enough that people are drawn in this far, because this is really the payoff. It’s all interesting and fun, but these three chapters are crucial. Putting it into the context of cosmology, as we’ll do later in the book, is indispensable to the program we’re outlining, but the truth is that we don’t yet know the final answers. We do know the questions, however, and here is where they are being asked.