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- #72

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What specifically are you responding to here? Whatever it is, it might help to quote it.You have a misunderstanding here. It's not that Gibbs' H-theorem is inapplicable.

- #73

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No, that's not quite what it shows. What it shows is that ##S(t_1) > S(t_0)##Gibbs' H-theorem only shows that ##S(t_1) > S(t_0)##

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- #74

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In classical physics, yes. (Note, however, that this is not true in quantum physics. But here I take it that we're only discussing classical physics.)The actual state of the system is always a delta state

The coarse-graining is not "artificial". It's a reflection of the fact that we don't knowbut you have to introduce some artifical coarse graining in order to apply the theorem.

- #75

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You are misunderstanding the argument. The argument is not that we apply the H-theorem to explain how ##S_\text{max}## is "reached". The argument is that the H-theorem tell us that,a single application of Gibbs' H-theorem doesn't suffice to reach ##S_\text{max}##

- #76

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Yes, I don't mention such trivial details for the sake of brevity.No, that's not quite what it shows. What it shows is that ##S(t_1) > S(t_0)##ifthe system was not in the thermal equilibrium state (whose ##H## value corresponds to ##S_\text{max}##) at ##t_0##. If the systemwasin the thermal equilibrium state at ##t_0##, then we cannot conclude that ##S(t_1) > S(t_0)##.

Gibbs' H-theorem is classical only, so I don't see why you bring up quantum theory.In classical physics, yes. (Note, however, that this is not true in quantum physics. But here I take it that we're only discussing classical physics.)

The laws of physics don't care about what we know or don't know. The challenge is toThe coarse-graining is not "artificial". It's a reflection of the fact that we don't knowwhichparticular point in phase space represents the actual microscopic state of the system.

Sure, but we need to derive the ensemble from the microscopic details. We can't just make up assumptions that are in contradiction to the axioms. And if we assume that the microscopic distribution isn't a delta, weThat's the whole reason for using ensembles in the first place. If we knew with infinite precision the exact point in phase space that represented the actual microscopic state of the system, we would have no need for thermodynamics or statistical mechanics at all.

No, that's not what the theorem says. Read again what is written in the book. I have also explained it multiple times already. The theorem says that ##S(t_1) > S(t_0)##. It doesn't say that ##S(t_2) > S(t_1)##. You can't concluce this without the additional assumption of an intermediate coarse graining step at ##t_1##.You are misunderstanding the argument. The argument is not that we apply the H-theorem to explain how ##S_\text{max}## is "reached". The argument is that the H-theorem tell us that,unlessthe system is in a state where the entropy isalready##S_\text{max}##, the entropy will increase with time.

Nobody claimed that. You need to know though that the entropy keeps increasing and the theorem doesn't imply this without intermediate coarse graining.We don't need to know specifically how the system "reaches" ##S_\text{max}##.

- #77

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If ##\rho(t_0) = P(t_0)##, where ##\rho## is the fine grained distribution and ##P## is the coarse grained distribution, then for ##t_1 > t_0##, we have ##S(t_1) \geq S(t_0)##

Now if we have ##t_2>t_1##, the theorem explicitely says that ##\rho(t_1) \neq P(t_1)##, so the theorem

So we don't know whether the entropy of ##P## keeps increasing after ##t_1##. In order to conlcude that, we need to coarse grain again by setting ##\rho(t_1):=P(t_1)## and apply the theorem again. The relevant quotations were given in post #65. If you disagree, you should be able to point to the exact sentence you disagree with.

- #78

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Nowhere is that assumed. The ensemble isif we assume that the microscopic distribution isn't a delta

I have, multiple times. It doesn't say what you claim it says.Read again what is written in the book.

You have already said the same thing multiple times. It didn't convince me then and it doesn't convince me now.Let me try to explain it one last time.

I think we're done.

- #79

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I'd like to thank all who have posted here with the hope that some insight will come out of this discussion to resolve the remaining issues of disagreement.

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