(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.

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.

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.

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.

“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?

“the space of allowed states doesn’t expand, and that’s what matters.”

Suppose I squeeze a gas into a small volume in a cylinder with a piston then let it come to equilibrium. Then I pull the piston out rapidly (rapidly expanding the volume). The gas is now concentrated at the bottom and no longer in equilibrium because the space of states (maximum allowable entropy) has increased. The gas will expand into the volume until it comes to a new equilibrium at a much higher entropy. By pulling the piston out rapidly, I did not change the entropy of the gas, but I changed the maximum allowable entropy, so the actual entropy was now low wrt the new maximum.

The universe was initially squeezed up into a singularity (or something close) like the gas in the piston at maximum entropy for that configuration. The big bang acted like a sudden pulling out of the piston, rapidly increasing the maximum allowable entropy. The actual entropy of the universe didn’t change, but it was no longer at maximum; it was low wrt the new maximum. Obviously, your statement disagrees with this view. I haven’t read the final chapter of the book yet, so I don’t know what your answer is.

CF– You could very (very) easily have had a Big Bang with much higher entropy. It would have been extremely inhomogeneous, not at all smooth. More later on this, as well.

drm– It’s hard to think of the physical system describing the universe as being “at fixed energy” when we take gravity into account. See previous post!

Metre– The existence of gravity changes the way you would naively count states. When things are bunched together, there are actually more states of that form than if things were spread randomly. That’s not completely surprising; a similar thing happens in oil and water, where there are more states when the two liquids are separate than when they are fully mixed.

Ray– Yes, I’m accusing you of non-unitarity. Of course “there are more and more allowed microstates whose macrostate is consistent with the general story,” if by “the general story” you mean the kind of evolution we actually observe — that’s just a restatement of the reality of the Second Law. But it’s not right to exclude highly inhomogeneous states by fiat, or just because information would be hidden behind horizons. This is Part Four kind of stuff, but the underlying assumption is that the full evolution is completely unitary, even when gravity is taken into account. So for every possible microstate in the current macrostate of the universe, there is exactly one microstate of a much denser (higher Hubble parameter) universe from which it could have evolved — that’s the content of “unitarity.” Most of them would have white holes and wild inhomogeneities. But even without such exotica, there are still a lot of very lumpy states that are inconsistent with the extreme smoothness of the early universe as we find it.

So if you are not working with the entropy and energy of momentum and position, then what are you working with?

I’m confused by this sentence. Did you mean to say the opposite, or are you accusing me of assuming some kind of non-unitarity?

Anyway, I didn’t mean to imply that the entire space of states for the universe is expanding (this probably doesn’t even make sense, since the universe in the broadest sense is probably infinite.)

I meant that in a given Hubble volume there are more and more allowed microstates whose macrostate is consistent with the general story: “the universe has been expanding since the big bang, and the initial fireball didn’t contain any large black holes.” (Large black holes in the early universe would foil the Laplace Demon program, since there’s no way for the demon to know what’s inside of them until they evaporate — which takes a long time.) The point is that this simplifying assumption restricts the space of allowed states more, the closer you get to the big bang.

As far as the expansion of the universe being important, I suppose the Laplace demon argument would still work if the simplifying assumption was something else, but in our case it does seem that it ultimately derives from big-bang cosmology — so I don’t see why it isn’t relevant. Didn’t you have an entire chapter on it?

Even in a gas, a spread out configuration seem more probable than a concentrated one (all particles in one corner of the box) if for no other reason than the mean free path is larger, so the number of collisions is reduced. When you try to concentrate the particles into one corner, the mfp becomes small and the number of collisions goes up, making such a configurations less likely than a more spread out one?

This has always puzzled me, as I don’t see why, given the big bang, such a state is seen as so unlikely. Certainly, if we leave the big bang out of it, and just consider all possible states for the early universe, the chances of something like our early universe being selected is extraordinarily unlikely. But would it be possible for the big bang to not have had low entropy? To my mind, if it didn’t, the big bang would no longer resemble anything like a big bang. That is, it seems to me that given the big bang, laws of physics and constants, such a low entropy state is to be expected. The more pertinent question becomes how the big bang?

Aaron– I’m not using that definition, since there’s no need to for any of what I’m discussing. We proceed by assuming that the universe is in some particular microstate, even if we don’t know what it is, and the entropy is a property of the macrostate to which it belongs. Also, none of the relevant spaces are infinite dimensional. (Even if we’re doing quantum mechanics, you can describe what happens within a comoving patch of space with a finite-dimensional Hilbert space.) So these issues just don’t apply.

Graham– There’s not really any difference between antimatter and any particular species of matter, as far as entropy is concerned. Also, there’s very little antimatter compared to matter in the observable universe.