Welcome to this week’s installment of the From Eternity to Here book club. Finally we dig into the guts of the matter, as we embark on Chapter Eight, “Entropy and Disorder.”
Excerpt:
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.
Here’s something that’s been puzzling me. You write,
“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!”
And then you write,
“A very low spike could be the Big Bang, but the probability would be enormously greater that we would live in a much smaller spike.”
So it seems that the same logic — preferring higher probability to lower probability cases — that leads you to reject the idea of the Big Bang as a brute fact, should also lead you to believe that we are in fact living in a smaller spike: if not a Boltzmann brain, then a Boltzmann solar system or galaxy or Hubble volume, and some future observation will reveal thermal equilibrium outside. Obviously you don’t believe this; no one does. But that implies the preference for high-probability cases is not absolute, there’s some other principle that trumps it.
So my question is: Why is the principle of preferring high-probability cases strong enough to make you confident that the Big Bang is not a brute fact, but not strong enough to make you believe that you are living in a Boltzmann bubble?
That would be true, if the correct scenario of the universe were that we were fluctuating around an equilibrium state. My strategy is to reject that whole scenario, and look for one where environments like ours arise with high probability (compared to other anthropically allowed environments).
“that’s just a restatement of the reality of the Second Law.”
Well, if it’s a choice between denying unitarity or restating the Second Law, I’ll opt for the latter. I disagree that I’m “just” restating the Law though. There are plenty of low entropy initial conditions that would not allow for the evolution of dynamical processes like human memory — say a perfectly symmetric arrangement of millions of Windows installation CDs whose velocities are so aligned to collide and form a black hole at some point in the future. You’d get the second law out of this, but not much else.
But it is the quantum world that gets us in hot water, so to speak. With the canonical broken tea cup, in classical mechanics we can know all the positions and momentum of the electrons, and then run time backwards, or if the system is properly closed, allow time to run forwards long enough, and viola the cup reforms. But in the quantum world not only can we not know momentum and position precisely, we can only ever know scattering cross section probabilities which are time symmetric, so that if we allow time to run backwards the cup just continues to crumble.
I used to think I didn’t know enough mathematics, or physics, or wasn’t as bright as the best minds, because I didn’t understand how time evolution and classical reality emerged from the limit of quantum mechanics. But as the recent phaphing about in the theoretical community has demonstrated I can quite confidently say that no one understands where classical reality and time evolution comes from and how it emerges from quantum mechanics.
Take for example the oft repeated Schroedinger cat. There are three cheats, or slights of hand involved in it: First it is phenomenally difficulty to get a system that big to be that isolated, just getting photons to that degree of thermal isolation requires extraordinary lengths. Second for a coherent state to emerge can take quite a long time, so for the cat to be both dead and alive might require waiting longer than the lifespan of the cat. Finally and most important is that even when you open the box you are not observing dead or alive, but rather, alive, dead one minute ago, two minutes ago, three…There are many other observables available that can allow for an autopsy of the cat.
All this points to important gaps in our understanding quantum theory in the limit of both large numbers of observables, and the large eigenvalue limit.
I have enjoyed reading your book. Thanks for writing it.
I wonder if there any tension between the theories of relativity and the theories of entropy? Some ways of measuring entropy involve counting states and I wouldn’t think counting is affected by relativities. But classical definitions of entropy involve energy and temperature, and both of these involve kinetic energy. And since kinetic energy involve mass and velocity, I would think such measurements would depend on the reference frames of the measurers. People traveling at different speeds might measure different entropies for the same situation. Is this a problem?
I’m also curious about why you don’t mention or explain the units (joules per degree Kelvin, according to my college chemistry textbook) that entropy is measured in. All the mentions of entropy in your book were unitless, if I’m remembering correctly.
Entropy has the same units as Boltzmann’s constant — energy per Kelvin, as you say. And I do talk about it a little bit. But it’s easy (and very common) to simply use units where Boltzmann’s constant equals unity, and the units go away. It’s just a conversion between energy and temperature.
Some remarks are needed here.
(i)
S = k lnW is not “the definition” of entropy. The proper definition is
S = -k Tr ρ ln ρ
For an isolated system at equilibrium ρ= ρ_eq is given by ρ_eq= 1/W and then the above definition reduces to the equilibrium form S_eq = k lnW.
The universe as a whole is not in a state of equilibrium, and the expression S_eq = k lnW does not apply to it.
(ii)
One would do a strong distinction between thermodynamic entropy S and informational entropy I. Thermodynamic entropy is a physical quantity that has little to see with subjective informational entropies.
The irreversibility that we observe in Nature around us is independent of the level of coarse graining. Paper ages with independence if we observe it or not. It ages exactly the same if we describe the process macro, meso, or even nanoscopically. In fact, this is a known paradox of the informational approaches to entropy as a measure of ignorance.
(iii)
The principle of indifference is a principle in equilibrium statistical mechanics because this result cannot be obtained from mechanics. As a consequence you only can postulate it in equilibrium statistical mechanics).
There is not such principle in non-equilibrium statistical mechanics (NESM). And in fact, we cannot assign equal probability to every microstate within any given macrostate for nonequilibrium states.
Within the framework of NESM, the principle of indifference ρ_i = ρ_j for all i and j is a theorem which can be proved for equilibrium. I.e. NESM provides the foundation for the
principle of indifference postulated in the equilibrium theory.
(iv)
The “Past Hypothesis”, which states that the universe began in a state of very low entropy plays absolutely no role for any rigorous explanation of the second law of thermodynamics.
E.g. when deriving Boltzmann original H-theorem or any other modern generalized H-theorem we do absolutely no hypothesis about the initial state being a very low entropy state. In fact all the H-theorems apply to initial states with very high entropy as well. It is the irreversibility contained in the H-theorem which prevent that any system in an initial very high entropy will evolve to a final state with low entropy.