## Entropy, expanding space, Noether's theorem, and conservation of free energy

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• 13
For illustration imagine a toy model like the black and white bitmap image discussed earlier. Given that there’s only one white pixel in it and any others are black: it is not at all surprising that that one white pixel would be in the top left corner of a 1x1 image, because that’s the only place it could be, so there’s a 100% chance of it being there; but it only has a 25% chance of being there in a 2x2 image; an 11% chance of being there in a 3x3 image; a 6% chance of being there in a 4x4 image; etc. The bigger the image size, the less likely the one white pixel will be in any particular location, since there are more other locations it could be in. Likewise, any expanding system opens up more possible ways its contents could be arranged, and so makes the way there were already arranged before a less likely possibility, and therefore makes change more probable.
• 18
You're basically saying quantum mechanics can't work. :confused: Okay.
• 4
I thought it has been established that entropy is a feature of the macrophysical world only. It is not meaningful to speak of entropy in the microphysical world which quantum mechanics is a study of.

Much like there is no perpetuum mobile in the macrophysical world, but every motion type is a perpetuum mobile in the microphysical world.

Maybe that's been debunked? I dunno.
• 4
I am sorry, you are talking about things I have already covered. Please try to understand what I said; and not try to explain something to me that I debunked. Sorry. I'll try it one more time to say my opinion in a different way, maybe it will stick, maybe it won't, this is my last chance:

SO: One last time: likelihood, chance, and probability are strictly human constructs, to fill gaps where we don't know the outcome, but have only statistical probability to predict. IN the real world, things happen only one way, and not many ways, so the chance of anything happening is 100%, not any less. Reality and events in reality are a straight path; there are no branches, and chance is an illusion. If you want branches, then reality ought to happen more than one way. But it does not, does it?
• 18
Entropy is originally a thermodynamic concept. Thermodynamics is reducible to statistical mechanics, which has its own definition of entropy which approaches that of thermodynamics. Statistical mechanics is entirely reducible to quantum mechanics, which has _its_ definition of entropy which yields statistical mechanics' definition.

In quantum mechanics, entropy is basically the number of microstates -- exact and degenerate quantum states -- that make up a macroscopic state.
• 13
If you want branches, then reality ought to happen more than one way. But it does not, does it?

Doesn't it? Why would you think it didn't?

Anyway, the point of my explanation was to undermine your claim that probability is entirely a human construct. If there's only one possibility, the probability of that possibility being realized is 100%. If there are many possibilities, the probability of any particular one of them being realized is low. That's what alethic probability (rather than epistemic probability like you're talking about) is, definitionally: what percent of the various ways things could be are like so? That's the probability of things being like so.
• 41
@Gnomon
In quantum mechanics, entropy is basically the number of microstates -- exact and degenerate quantum states -- that make up a macroscopic state.
And so it goes from one microstate (minimum disorder, simplicity) to N microstates (maximum disorder, complexity) of a macrostate, no?
• 4
Doesn't it? Why would you think it didn't?

Because it's obvious it does not. Can you name two different events that happen to the same object at the same time and in the same respect? Can a car go forward and backward at the same time and in the same respect? Can an electron reverse its spin and keep the same spin at the same time and in the same respect? Can a decay to a nucleus happen and not happen at the same time and in the same respect, in a mass of nuclei of the same atom, where the nuclei decay in random order? My answer to my four questions is a resounding "no". If you say yes, a car CAN go both forward and backward at the same time and in the same respect, please demonstrate. Thanks.
• 4
Thanks. The first paragraph has been known to me, but I am glad you clarified it anyway. The second paragraph is intuitive, but it does not explain what entropy is in the microphysical level. If you would care, please explain -- in terms that a person with Grade 12 physics can comprehend. If no, you won't explain, that's okay too. Won't lose sleep over that.
• 13
You're applying the wrong standard of justification. "This definitely doesn't happen unless you can prove that it definitely does" would require a retreat to abject nihilism because nothing can be conclusively proven. The correct standard is "this might possibly happen unless you can prove that it can't possibly". So I'm asking you to prove that it can't possibly, or else admit that it might. Just saying "prove that it definitely can" doesn't prove that it can't possibly.
• 4
You're applying the wrong standard of justification. "This definitely doesn't happen unless you can prove that it definitely does" would require a retreat to abject nihilism because nothing can be conclusively proven. The correct standard is "this might possibly happen unless you can prove that it can't possibly". So I'm asking you to prove that it can't possibly, or else admit that it might. Just saying "prove that it definitely can" doesn't prove that it can't possibly.

I disagree. But that's okay. If you were right in your criticism, then the law of excluded middle would never be applicable. But it's always applicable. So your criticism fails.
• 13
The law of the excluded middle is still applicable. Everything is still either true or false. But that's a separate issue from epistemic possibility or necessity.

• 18
• 18
Sure thing.

You're probably aware of the idea of energy levels in quantum mechanics, often depicted as concentric orbits which is inaccurate but might do for now.

The higher the energy of an electron in an atom, the higher up these energy levels it is. Energy levels are labelled n=1,2,3,... In addition to these energy levels, for each value of n there are possible different angular momenta available, essentially the shape of the orbit in the concentric picture.

Those possible values are finite for finite n, and go L=0,1,2,...,n-1. In the absence of an external magnetic field, each of these orbit shapes have the same energy.

So for a hydrogen atom in its lowest-energy state (n=1), there is precisely one orbital shape: L=0. No angular momentum. Straight through the goddam middle. For the next highest energy level n=2, L=0 or 1: straight through the goddam middle or a tight orbit. For the next highest, L=0, 1 or 2: adding a slightly less tight orbit.

There are other parameters than L. There's spin S, and how L and S are coordinated, and these follow similar rules: for a higher energy level n, there are an increased number of possible values of these parameters. These are the electron shells you would have been taught in chemistry: 2, 8, 18, etc. that underpins the periodic table.

Thus as you add energy to a bound electron, you increase the number of equivalent states it could end up in, which is how entropy is defined in QM, from which you can derive statistical mechanics and its definition of entropy (and then thermodynamics and its version of entropy).
• 4
Thus as you add energy to a bound electron, you increase the number of equivalent states it could end up in, which is how entropy is defined in QM
This is a very good explanation except there is no definition involved. Could you please rephrase this to make it into a proper definition?

Thanks for the rest of the explanation, too. Yes, I remember from Grade 13 chemistry the bound electron paths, as they go 2, 8, 18, 32, ... 2*(n^2), where n is the cardinal number of the "shell", that is, a path that electrons take as the atomic number (the protons in the nucleus) dictate. Don't ask me why they take these shell structures... too many equations involved, but please do tell me the definition of entropy in QM. In my high school days the only electron path that had been found or calculated specific energy levels were the electron of the Hydrogen atom, which, as we all know, has only one electron. Exciting this electron could be calculated in those years to what L level it jumps to; maybe QM has advanced now to know and calculate the energy levels of other electron paths.
• 4
I think you are overcomplicating things, and got lost in the quagmire of your own thoughts. My opinion only, don't take this to heart, please, I know you wouldn't anyway. But I still think you are not clear in understanding my original proposition, despite several clear examples with irrefutable logic accompanying them. I imagine (no proof) that as you are reading my earlier explanations, you apply your previously learned analytical tests to test them, but the tests you apply are not applicable. That's about the size of it.

I think there are quite a number of post-graduate students or just other formal students of logic and philosophy on this site who take their previously learned knowledge and fail to apply it properly to new and emerging thoughts. Especially where pure reason is involved. I don't know why this has to be a problem, but the phenomenon of being over-learned and thus influenced to not understand simple explanations is certainly pervasive on this site.

Think about it again: I said that reality can't happen more than one way. To this your replied, "Doesn't it? Why would you think it didn't?" There is no historical evidence that something happened to anything anywhere at the same time and in the same respect both ways that contradict each other, and there is the a priori knowledge that something can't both happen and not happen at the same time and in the same respect. You are arguing against these, not against my argument that I derived from these two. You must first convince me that reality can happen more than one way, that is, to a specific body a specific event can happen and not happen at the same time and in the same respect.

I am sorry, but I close this discussion with you. If you don't see the point in the bolded part, then there is something fundamentally not in alignment in our thinking, and all further arguments would be futile. I don't like futile. It's a personal preference. Sorry.
• 18
This is a very good explanation except there is no definition involved. Could you please rephrase this to make it into a proper definition?

S = k ln(W)

Entropy S is Boltzmann's constant k times the natural log of the number of microstates W.
• 4
S = k ln(W)

Entropy S is Boltzmann's constant k times the natural log of the number of microstates W.

Theoretically there is an infinite number of microstates. So entropy is a constant, and furthermore it is infinite.

Did you not want to write dW, that is, the change in microstates? And changing in which direction?

We'll get there yet.
• 18
Theoretically there is an infinite number of microstates.

Degenerate microstates for a particular macrostate.
• 4
Degenerate microstates for a particular macrostate.
Thank you. What is the unit of entropy in QM? This definition only yields a number. A number alone is not a physics value.
• 18
What is the unit of entropy in QM?

Boltzmann's constant has an SI unit of Joules (energy) per degree Kelvin (temperature), so entropy is the same. These are easily Googlable questions btw.
• 4
Boltzmann's constant has an SI unit of Joules (energy) per degree Kelvin (temperature), so entropy is the same. These are easily Googlable questions btw.

wonderful. Thanks. I shall turn my attention to google now, and ask it "What's the meaning of life", "What is the purpose of existence" and "who created the world, and what was he wearing at the time".
• 18
You can also Google "substitution fallacy" and conclude something about pretending that simple matters of recorded fact and deep mysteries of philosophy are equivalent.
• 4
Sort of like an old newspaper cartoon I read. A kid, eight-year-old, and supersmart computer whiz, types into his Commodore 64 (cutting edge tech then), "What's the nature of god?" To which the computer answers, "Two eggs, a pound of flour, baking powder and a cup of butter". The kid's hair stands on air, then he calms down and says, "I gotta mark these floppy disks more carefully."
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