Last time I started talking about entropy and a puzzle it presents in cosmology. To understand the puzzle we have to understand entropy, which is a crucial part of our view of physics. In fact, we consider entropy to be a (statistical) law about the behavior of reality. That law says: Entropy always increases.
There are some nuances to this, though. For example we can decrease entropy in a system by expending energy. But expending that energy increases the entropy in some other system. Overall, entropy does always increase.
Previously I talked about air molecules in a room. For another example, now I’ll look at the CD Collection example in light of the more formal definition presented last time.
[The CD Collection example is from my Barrel of Wine; Barrel of Sewage post, which you should probably read if you haven’t.]
For reference, here’s the entropy formula again:
(See the previous post for details.)
The macro-state of the CD collection is how sorted it is. The micro-states are the locations of each CD (first, second, third, etc).
If every CD is in the correct place, the sort is perfect, and the entropy (relative to being sorted) is zero.
Per the entropy formula, we have: K log 1. Omega (Ω) is 1 because there is just one arrangement of CDs (of micro-states) that is perfectly sorted. The log of 1 is 0.0, and K×0.0 is still 0.0, so the entropy of a perfectly sorted CD system is zero. (Relative to being sorted!)
Suppose we take one CD and misfile it. We’ve actually introduced two errors:  there is a CD missing from where it should be;  there is a CD inserted where it shouldn’t be.
If there are N CDs in the collection, then there are N configurations with a single CD missing. Given a CD in hand, there are N locations it could be placed (including the correct slot).
That makes N2 arrangements of the CD collection with a single CD out of place. If we have 1,000 CDs, there are 1,000,000 such arrangements.
Per the formula, we have K log 1000000, which is K×6.0.
So in moving from the perfect sort to a single misplaced CD, the entropy of the collection jumps from K×0.0 to K×6.0
(Experts: I’m going to leave it at that, because I’m being a little shameless with logs. In fact the formula uses the natural log, but we can accommodate that by modifying Boltzmann’s K. For now just focus on the relative difference between the entropy numbers calculated here.)
(Also: Since K is constant and always present, from now on I’ll just use the numbers. So in this case we say entropy goes from 0.0 to 6.0 when there is just one misplaced CD.)
The worst case is the earthquake that knocks the CDs off the shelves, leaving a random pile on the floor. Zero sort order!
For 1000 CDs, there are 1000! (one-thousand factorial) possible arrangements. The resulting number has over 2,500 digits.
Per the formula: K log 4.02×102567 = K×2567.6
So the maximum entropy of the 1000-CD collection is roughly 2568 — much larger than the entropy of a perfect sort or one CD out of place. (This is the same level of entropy as the 125 “gas” molecules in the two-inch cube with 5-per-inch fixed spacing.)
The maximum entropy of the sort order of just 1000 CDs isn’t very large. (As I showed you last time, when dealing with real world particle systems the numbers are huge beyond comprehension.)
Even so, there is still considerable difference between the one-million micro-states of the “one CD misplaced” macro-state versus the 2568-digit number that numbers the micro-states of the “random pile of CDs” macro-state.
It’s easy to see how shaking the pile is not at all likely to result in accidental sorting (although it’s not completely impossible).
One might think black holes should have low entropy because they’re so simple. They have no hair; they only have mass, charge, and spin.
But that’s exactly why they have such high entropy.
Consider the vast number of atoms in a large star that ends its life by collapsing into a black hole. Prior to the collapse, the star’s macro state includes all its properties: the light it emits, its magnetic field, the stellar wind, flares, etc.
While there are a vast number of micro-states for the star’s particles — and hence a fair amount of entropy — the star’s macro-states are complex. There are comparatively fewer arrangements that produce the macro-state.
After collapse, all those micro-states no longer affect the properties of the black hole, except for its mass, charge, and spin. All those micro-states become irrelevant resulting in a massive increase of entropy.
The supermassive black holes in the hearts of galaxies contain the micro-states of millions of stars reduced down to three properties: mass, charge, and spin.
Most of the entropy of the universe is in these supermassive black holes.
Entropy, to me, seems the result of the behavior of particles plus time. Some believe time arises from entropy, that entropy is the fundamental property. It’s a variation of the notion that change is fundamental and that time arises from change.
I do not find these notions persuasive. They are in some part motivated by another odd notion: that there’s no time in physics. (Which, in turn, seems to come from the idea that physics works forwards and backwards, rather than a genuine perceived lack.)
The thing is, one is hard-pressed to find any physics equation that does not, in some way, contain the ubiquitous t (for time) variable. Time is very much a fundamental notion in the mathematics of physics.
Kant saw time as one of our most fundamental intuitions, even more so than our other fundamental intuition, space. (And how interesting that many years later Einstein would knit them into fundamental spacetime.)
FWIW, I believe time is fundamental and axiomatic to reality. Time just is. It may be more fundamental than space in that time probably existed before the big bang (which is where our space began).
The point of which is that big crunch scenarios, which posit the eventual end of cosmic expansion and the resulting collapse back to singularity, sometimes suggest time runs backwards during collapse. This is (supposedly) because time arises from entropy, and such scenarios predict a reverse of entropy.
Penrose suggests, and I agree, this is very unlikely. It is far more likely entropy will, as always, and as the Second Law requires, continue to increase. Time will certainly not rewind.
One thing Penrose points out is that, by the time the universe stopped expanding and started collapsing, there would be lots of black holes.
As the universe collapses, those black holes merge to form ever larger black holes. As the collapse approaches the singularity, the universe consists of a churning mass of black holes.
This is nothing like the astonishingly smooth cosmic expansion resulting from the big bang. The big crunch will not be its mirror image, but something else entirely.
On this count alone, we see the collapse of the universe — should it happen — must involve a massive increase of entropy as black holes merge.
The last of Penrose’s points for now the above analysis of black holes is a clue to how the early universe was low entropy.
We see that black holes increase entropy when they swallow matter. Since black holes are a phenomenon of gravity, perhaps perhaps that’s the clue we need.
[As an aside, every form of energy ultimately is due to gravity. The reason the universe gets away with big banging from nothing is that expansion against gravity acts like stretching a spring — it puts energy into the spring. Or, in our case, matter and energy into the universe.]
What characterizes the early universe from the later one is the clumping of matter due to gravity.
Penrose’s argument suggests that the even distribution of matter in the beginning implies gravity was “switched off” (as Penrose puts it) at first. Gravity kicked in later and the diffuse cloud of hydrogen, helium, and a dash of lithium, began to collapse into the first stars and galaxies.
Given the canonical air-molecules-in-a-room example of entropy, it’s hard to see the diffuse early universe as the low-entropy state and modern clumpy galaxies as the higher. Structure doesn’t seem “disordered.”
The difference is that gravity plays no role with gas molecules. The situations aren’t analogous. In the air, the diffuse state is the macro-state with the most arrangements. Squeezing the air into a cube is an unlikely state, so naturally the evolution of the system is towards diffusion.
But with gravity, the diffuse state can’t survive. Any disturbance or imbalance starts collapse towards clumping. And the number of micro-states that promote those changes is much larger than what amounts to the sorted perfect state of diffusion.
I may never understand Penrose’s comformal cyclic cosmology (CCC) idea — let alone be able to judge it — but the book is a success for me just on the entropy discussion.
There’s a lot I didn’t go into, but Penrose’s treatment was one of the most thorough and nuanced that I’ve read. And, being it’s Penrose, most mathematical. Say what you will about his metaphysics, but it is grounded in hard math.
Stay entropic, my friends!