In a previous post on Entropy, I offered a garbled notion of "statistical symmetry." I'm currently reading Carlo Rovelli's The Order of Time, and chapter two laid out the idea that I was grasping at concisely, clearly and -- because Rovelli is an actual physicist -- correctly.
What follows is a fairly long and rambling discussion of the same toy system as the previous post, of five marbles in a square box with 25 compartments. It does eventually circle back to the idea of symmetry, but it's really more of a brain dump of me trying to make sure I've got the concepts right. If that sounds interesting, feel free to dive in. Otherwise, you may want to skip this one.
In the earlier post, I described a box split into 25 little compartments with marbles in five of the compartments. If you start with, say, all the marbles on one row (originally I said on one diagonal, but that just made things a bit messier) and give the box a good shake, the odds that the marbles all end up in the same row that they started in are low, about one in 50,000 for this small example. So far, so good.
But this is really true for any starting configuration -- if there are twenty-five compartments in a five-by-five grid, numbered from left to right then top to bottom, and the marbles start out in, say, compartments 2, 7, 8, 20 and 24, the odds that they'll still be in those compartments after you shake the box are exactly the same, about one in 50,000.
On the one hand, it seems like going from five marbles in a row to five marbles in whatever random positions they end up in is making the box more disordered. On the other hand, if you just look at the positions of the individual marbles, you've gone from a set of five numbers from 1 to 25 ... to a set of numbers from 1 to 25, possibly the one you started with. Nothing special has happened.
This is why the technical definition of entropy doesn't mention "disorder". The actual definition of entropy is in terms of microstates and macrostates. A microstate is a particular configuration of the individual components of a system, in this case, the positions of the marbles in the compartments. A macrostate is a collection of microstates that we consider to be equivalent in some sense.
Let's say there are two macrostates: Let's call any microstate with all five marbles in the same row lined-up, and any other microstate scattered. In all there are 53,130 microstates (25 choose 5). Of those, five have all the marbles in a row (one for each row), and the other 53,125 don't. That is, there are five microstates in the lined-up microstate and 53,125 in the scattered microstate.
The entropy of a macrostate is related to the number of microstates consistent with that macrostate (for more context, see the earlier post on entropy, which I put a lot more care into). Specifically, it is the logarithm of the number of such states, multiplied by a factor called the Boltzmann constant to make the units come out right and to scale the numbers down, because in real systems the numbers are ridiculously large (though not as large as some of these numbers), and even their logarithms are quite large. Boltzman's constant is 1.380649×10−23 Joules per Kelvin.
The natural logarithm of 5 is about 1.6 and the natural logarithm of 53,125 is about 10.9. Multiplying by Boltzmann's constant doesn't change their relative size: The scattered macrostate has about 6.8 times the entropy of the lined-up macrostate.
If you start with the marbles in the low-entropy lined-up macrostate and give the box a good shake, 10,625 times out of 10,626 you'll end up in the higher-entropy scattered macrostate. Five marbles in 25 compartments is a tiny system, considering that there are somewhere around 10,800,000,000,000,000,000,000,000 molecules in a milliliter of water. In any real system, except cases like very low-temperature systems with handfuls of particles, the differences in entropy are large enough that "10,625 times out of 10,626" turns into "always" for all intents and purposes.
This distinction between microstates and macrostates gives a rigorous basis for the intuition that going from lined-up marbles to scattered-wherever marbles is a significant change, while going from one particular scattered state to another isn't.
In both cases, the marbles are going from one microstate to another, possibly but very rarely the one they started in. In the first case, the marbles go from one macrostate to another. In the second, they don't. Macrostate changes are, by definition, the ones we consider significant, in this case, between lined-up and scattered. Because of how we've defined the macrostates, the first change is significant and the second isn't.
Let's slice this a bit more finely and consider a scenario where only part of a system can change at any given time. Suppose you don't shake up the box entirely. Instead, you take out one marble and put it back in a random position, including, possibly, the one it came from. In that case, the chance of going from lined-up to scattered is 20 in 21, since out of the 21 positions the marble can end up in, only one, its original position, has the marbles all lined up, and in any case it doesn't matter which marble you choose.
What about the other way around? Of the 53,120 microstates in the scattered macrostate, only 500 have four of the five marbles in one row. For any microstate, there are 105 different ways to take one marble out and replace it: Five marbles times 21 empty places to put it, including the place it came from.
For the 500 microstates with four marbles in a row, only one of those 105 possibilities will result in all five marbles in a row: Remove the lone marble that's not in a row and put it in the only empty place in the row of four. For the other 52,615 microstates in the scattered macrostate, there's no way at all to end up with five marbles lined up by moving only one marble.
So there are 500 cases where the scattered macrostate becomes lined-up, 500*104 cases where it might but doesn't, and 52,615*105 cases where it couldn't possibly. In all, that means that the odds are 11,153.15 to one against scattered becoming lined-up by removing and replacing one marble randomly.
Suppose that the marbles are lined up at some starting time, and every time the clock ticks, one marble gets removed and replaced randomly. After one clock tick, there is a 104 in 105 chance that the marbles will be in the high-entropy scattered state. How about after two ticks? How about if we let the clock run indefinitely -- what portion of the time will the system spend in the lined-up macrostate?
The there are tools to answer questions like this, particularly Markov chains and stochastic matrices (that's the same Markov Chain that can generate random text that resembles an input text). I'll spare you the details, but the answer requires defining a few more macrostates, one for each way to represent the number five as the sum of whole numbers: [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].
The macrostate [5] comprises all microstates with five marbles in one row, the macrostate [4, 1] comprises all microstates with four marbles in one row and one in another row, the macrostate [2, 2, 1] comprises all microstates with two marbles in one row, two marbles in another row and one marble in a third one, and so forth.
Here's a summary
| Macrostate | Microstates | Entropy |
|---|
| [5] | 5 | 1.6 |
| [4,1] | 500 | 6.2 |
| [3,2] | 2,000 | 7.6 |
| [3,1,1] | 7,500 | 8.9 |
| [2,2,1] | 15,000 | 9.6 |
| [2,1,1,1] | 25,000 | 10.1 |
| [1,1,1,1,1] | 3,125 | 8.0 |
The Entropy column is the natural logarithm of the Microstates column, without multiplying by Boltzmann's constant. Again, this is just to give a basis for comparison. For example [2,1,1,1] is the highest-entropy state, and [2,2,1] has four times the entropy of [5].
It's straightforward, but tedious, to count the number of ways one macrostate can transition to another. For example, of the 105 transitions for [3,2], 4 end up in [4,1], 26 end up back in [3,2] (not always by putting the removed marble back where it was), 30 end up in [3, 1, 1] and 45 end up in [2, 2, 1]. Putting all this into a matrix and taking the matrix to the 10th power (enough to see where this is converging) gives
| Macrostate | % time | % microstates |
|---|
| [5] | .0094 | .0094 |
| [4,1] | .94 | .94 |
| [3,2] | 3.8 | 3.8 |
| [3,1,1] | 14 | 14 |
| [2,2,1] | 28 | 28 |
| [2,1,1,1] | 47 | 47 |
| [1,1,1,1,1] | 5.9 | 5.9 |
The second column is the result of the tedious matrix calculations. The third column is just the size of the macrostate as the portion of the total number of microstates. For example, there are 500 microstates in [4,1], which is 0.94% of the total, which is also the portion of the time that the matrix calculation says system will spend in [4, 1]. Technically, this means the system is ergodic, which means I didn't have to bother with the matrix and counting all the different transitions.
Even in this toy example, the system will spend very little of its time in the low-entropy lined-up state [5], and if it ever does end up there, it won't stay there for long.
Given some basic assumptions, a system that evolves over time, transitioning from microstate to microstate, will spend the same amount of time in any given microstate (as usual, that's not quite right technically), which means that the time spent in each macrostate is proportional to its size. Higher-entropy states are larger than lower-entropy states, and because entropy is a logarithm, they're actually a lot larger.
For example, the odds of an entropy decrease of one millionth of a Joule per Kelvin are about one in e(1017). That's a number with somewhere around 40 quadrillion digits. To a mathematician, the odds still aren't zero, but to anyone else they would be.
For all but the tiniest, coldest systems, the chance of entropy decreasing even by a measurable amount are not just small, but incomprehensibly small. The only systems where the number of microstates isn't incomprehensibly huge are are small collections of particles near absolute zero.
I'm pretty sure I've read about experiments where such a system can go from a higher-entropy state to a very slightly lower-entropy state and vice versa, though I haven't had any luck tracking them down. Even if no one's ever done it, such a system wouldn't violate any laws of thermodynamics, because the laws of thermodynamics are statistical (and there's also the question of definition over whether such a system is in equilibrium).
So you're saying ... there's a chance? Yes, but actually no, in any but the tiniest, coldest systems. Any decrease in entropy that could actually occur in the real world and persist long enough to be measured would be in the vicinity of 10−23 Joules per Kelvin, which is much, much too small to be measured except under very special circumstances.
For example, if you have 1.43 grams of pure oxygen in a one-liter container at standard temperature and pressure, it's very unlikely that you know any of the variables involved -- the mass of the oxygen, its purity, the size of the container, the temperature or the pressure, to even one part in a billion. Detecting changes 100,000,000,000,000 times smaller than that is not going to happen.
But none of that is what got me started on this post. What got me started was that the earlier post tried to define some sort of notion of "statistical symmetry", which isn't really a thing, and what got me started on that was my coming to understand that higher-entropy states are more symmetrical. That in turn was jarring because entropy is usually taken as a synonym for disorder, and symmetry is usually taken as a synonym for order.
Part of the resolution of that paradox is that entropy is a measure of uncertainty, not disorder. The earlier post got that right, but evidently that hasn't stopped my for hammering on the point for dozens more paragraphs and a couple of tables in this one, using a slightly different marbles-in-compartments example.
The other part is that more symmetry doesn't really mean more order, at least not in the way that we usually think about it.
From a mathematical point of view, a symmetry of an object is something you can do to it that doesn't change some aspect of the object that you're interested in. For example, if something has mirror symmetry, that means that it looks the same in the mirror as it does ordinarily.
It matters where you put the mirror. The letter W looks the same if you put a mirror vertically down the middle of it -- it has one axis of symmetry. The letter X looks the same if you put the mirror vertically in the middle, but it also looks the same if you put it horizontally in the middle -- it has two axes of symmetry.
Another way to say this is that if you could draw a vertical line through the middle of the W and rotate the W out of the page around that line, and kept going for 180 degrees until the W was back in the page, but flipped over, it would still look the same. If you chose some other line, it would look different (even if you picked a different vertical line, it would end up in a different place). That is, if you do something to the W -- rotate it around the vertical line through the middle -- it ends up looking the same. The aspect you care about here is how the W looks.
To put it somewhat more rigorously: if f is the particular mapping that takes each point to its mirror image across the axis, then f takes the set of points in the W to the exact same set of points. Any point on the axis maps to itself, and any point off the axis maps to its mirror image, which is also part of the W. The map f is defined for every point on the plane and it moves all of them except for the axis. The aspect we care about, which f doesn't change, is whether a particular point is in the W.
If you look at all the things you can do to an object without changing the aspect you care about, you have a mathematical
group. For a W, there are two things you can do: leave it alone and flip it over. For an X, you have four options: leave it alone, flip it around the vertical axis, flip it around the horizontal axis, or do both. Leaving an object alone is called the
identity transformation, and it's always considered a symmetry, because math. An asymmetrical object has
only that symmetry (it's symmetry group is
trivial).
In normal speech, saying something is symmetrical usually means it has the same symmetry group as a W -- half of it is a mirror image of the other half. Technically, it has bilateral symmetry. In some sense, though, an X is more symmetrical, since its symmetry group is larger, and a hexagon, which has 12 elements in its symmetry group, is more symmetrical yet.
A figure with 19 sides, each of which is the same lopsided squiggle, would have a symmetry group of 19 (rotate by 1/19 of a full circle, 2/19 ... 18/19, and also don't rotate at all). That would make it more symmetrical than a hexagon, and quite a bit more symmetrical than a W, but if you asked people which was most symmetrical, they would probably put the 19-sided squigglegon last of the three.
Our visual system is mostly trained to recognize bilateral symmetry. Except for special situations like reflections in a pond, pretty much everything in nature with bilateral symmetry is an animal, which is pretty useful information when it comes to eating and not being eaten. We also recognize rotational symmetry, which includes flowers and some sea creatures, also useful information.
It would make sense, then, that in day to day life, "more symmetrical" generally means "closer to bilateral symmetry". If a house has an equal number of windows at the same level on either side of the front door, we think of it as symmetrical, even though the windows may not be exactly the same, the door itself probably has a doorknob on one side or the other and so forth, so it's not quite exactly symmetrical. We'd still say it's pretty symmetrical, even though from a mathematical point of view it either has bilateral symmetry or it doesn't (and in the real world, nothing we can see is perfectly symmetrical).
That should go some way toward explaining why, along with so many other things, symmetry doesn't necessarily mean the same thing in its mathematical sense as it does ordinarily. The mathematical definition includes things that we don't necessarily think of as symmetry.
Continuing with shapes and their symmetries, you can think of each shape as a macrostate. You can associate a microstate with each mapping (technically, in this case, any rigid transformation of the plane) that leaves the shape unchanged. The macrostate W has two microstates: one for the identity transformation, which leaves the plane unchanged, and one for the mirror transformation around the W's axis.
The X macrostate has four microstates, one for the identity, one for the flip around the vertical axis, one for the flip around the horizontal axis, and one for flipping around one axis and then the other (in this case, it doesn't matter what order you do it in). The X macrostate has a larger symmetry group, which is the same as saying it has more entropy.
In this context, a symmetry is something you can do to the microstate without changing the macrostate. A larger symmetry group -- more symmetry -- means more microstates for the same macrostate, which means more entropy, and vice-versa. They're two ways of looking at the same thing.
In the case of the marbles in a box, a symmetry is any way of switching the positions of the marbles, including not switching them around at all. Technically, this is a permutation group.
For any given microstate, some of the possible permutations just switch the marbles around in their places (for example, switching the first two marbles in a lined-up row), and some of them will move marbles to different compartments. For a microstate of the lined-up macrostate [5], there are many fewer permutations that leave the marbles in the same macrostate (all in one row, though not necessarily the same row) than there are for [2, 1, 1, 1]. Even though five marbles in a row looks more symmetrical, since it happens to have bilateral visual symmetry, it's actually a much less symmetrical macrostate than [2, 1, 1, 1], even though most of its microstates will just look like a jumble.
In the real world, distributing marbles in boxes is really distributing energy among particles, generally a very large number of them. Real particles can be in many different states, many more than the marble/no marble states in the toy example, and different states can have the same energy, which makes the math a bit more complicated. Switching marbles around is really exchanging energy among particles, and there are all sorts of intricacies about how that happens.
Nonetheless, the same basic principles hold: Entropy is a measure of the number of microstates for a given macrostate, and a system in equilibrium will evolve toward the highest-entropy macrostate available, and stay there, simply because the probability of anything else happening is essentially zero.
And yeah, symmetry doesn't necessarily mean what you think it might.
