[jbay808]

Can you elaborate on the difference between a "fine-grained" macrostate and a macrostate that is not fine-grained?

I think you will find it hard to separate the concept of a macrostate from the state of knowledge (or ignorance) of an individual subjective observer.

[oh_my_goodness]

Sure. A fine-grained macro-state contains fewer micro-states. A coarse-grained macro-state contains more micro-states.

Say I flip 8 coins and I don't look at the results. A fine-grained macro state is TTTT TTTT. A coarser-grained macro state is TTTT xxxx. The one has 4 bits more entropy than the other. It works the same way in statistical mechanics. Call them spins.

We're just talking about some ensemble of micro states, and then we divide the ensemble up into macro-states. To do statistical mechanics at all, I think I have to define some macro-states according to which micro-states they contain. That doesn't mean I necessarily have any information about which macro-state the system is actually in.

[jbay808]

> A fine-grained macro state is TTTT TTTT.

This macrostate seems fine enough to be a microstate, but sure. For the macrostates with at least one 'x' in it, that 'x' seems to be a placeholder for the concept of subjective ignorance.

> That doesn't mean I necessarily have any information about which macro-state the system is actually in.

But the entire purpose of the exercise of assigning microstates to macrostates is so that you can match up a description of some system to a microstate ensemble and calculate its entropy! Otherwise there's no point to arbitrarily labelling various groups of microstates.

To follow your example more practically, let's say you have an 8-spin system, whose net spin is zero. (You know because you've measured its overall magnetic moment or something). I've just described a system that is in one of the following possible microstates:

TTTT HHHH, TTTH HHHT, ..., HHHH TTTT

Now you can go ahead and define the macrostates as fine-grained as you want, where TTTT HHHH and HHHH TTTT are in different macrostates, but to calculate the entropy of this system, you're going to have to sum up all of those macrostates anyway to get the one that's consistent with the described system.

[oh_my_goodness]

Good review of common ground. At this point hopefully the active folks in the discussion can see that we're all describing statistical mechanics exactly the same way.

What I'm saying is pedagogical. We need to define our macro-states. We don't need to go on and talk about our definitions being information or 'knowledge.' We could just use the definitions and calculate. We can talk about 'knowledge' but we don't need to.

The exception is when we actually have some information about what macro-state some system is really in. Obviously we then have to build that information into our model, and the entropy changes. What I'm saying is this: it's not necessary to mix that into our definition of entropy. That definition is not going to help folks who don't understand entropy, and it's unnecessary.

[kgwgk]

How is a macrostate TTTTxxxx different from having the information about the TTTT part and not about the xxxx part?

Talking bout the entropy of the macrostate TTTTxxxx makes sense only conditional on the TTTT information.

[oh_my_goodness]

It's different because I can define a macro-state without any information about which macro-state any system is actually in. As I think you're also saying, the only information I need is information about how I've defined my own macro-states.

If we just define the macro-states, we're good to go. We don't need to talk about 'knowledge'. We can talk about 'knowledge', it's fine, but that lets in unnecessary woo.

[kgwgk]

I'm not sure I see what's the point of that distinction.

The entropy of a macrostate is a measure of the indetermination about the microstate conditional on the macrostate. If you don't want to call that 'knowledge' the substance of the matter doesn't change.

A macrostate is not an intrinsict property of a physical system. It's related to our description of the system. In general, the same microstate of the system of interest may be compatible with multiple macrostates.

Given the thermodynamical description of some system I can calculate the entropy if T=T_1 and the entropy if T=T_2 without knowing what's the actual temperature specifying the macrostate. But in the first case the calculation is conditional on the hypothetical information T=T_1 and in the second case conditional on T=T_2.

[oh_my_goodness]

The point is pedagogical. Entropy takes a lot of time for people to understand clearly. That is the discussion from the OP.

Adding "knowlege" to the definition (or to an initial explanation) of entropy makes that learning process even more difficult. And it's unnecessary. It's better than the older talk about "disorder" but it's distracting. We can bypass 'knowledge' and come back later, with no penalty and plenty of time savings.

Apart from that single pedagogical point, we seem to be saying the same things back and forth to each other in different words. I'm not sure why.

[kgwgk]

I think that the "microstate counting" approach - if that's what you are defending - doesn't allow to understand entropy clearly because only works for the microcanonical description. It doesn't make sense to count the microstates for a volume of gas at some pressure and temperature. (Which is the standard thermodynamics problem.)

The concept of how much can we tell about the microstate given only the pressure and temperature seems quite natural and a better starting point. Boltzmann's entropy is a nice illustration but there is no reason to avoid the general concept.

[oh_my_goodness]

Of course you can count micro states of a gas within dE or delta-E of some total energy. The density-of-states approach is exactly that.

I thought we were discussing statistical mechanics.

[im3w1l]

I think in practice we select macrostates based on stuff we can easily measure. And stuff we can easily measure gets pretty close to intrinsic properties.