[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.

[Jensson]

> It doesn't make sense to count the microstates for a volume of gas at some pressure and temperature.

But nobody does that since the total value of entropy isn't important. What you do is count the factor difference in count of microstates between two volumes, that is what you care about, and it is easy to see how the number of microstates changes when you double the volume or other similar changes.

[kgwgk]

Is it easy to see how the number of microstates changes when you increase the temperature - everything else being equal?

How would you say that it changes then?

I'd say that the number of compatible microstates doesn't change. The probability of each microstate does change though.

[Jensson]

Your statement doesn't make sense, temperature is defined in terms of entropy changes, you can't calculate temperature without first calculating entropy changes.

[kgwgk]

And of course the "knowledge" part - the entropy being a function of the probability distribution for the microstate conditional on the macrostate - is there just the same in the microstate counting approach. (Where the latter is applicable!)

If given the macrostate all microstates are equally probable we can just count them. The more there are the higher the entropy.

In general we have a probability distribution for microstates conditional on the macrostate. To have a clear understanding of entropy that should be at least mentioned.

[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.

[kgwgk]

> I thought we were discussing statistical mechanics.

This is from the message that you first replied to in this thread:

"Typically when you calculate the entropy of a system at temperature X, that means all you know is that you stuck a thermometer in it and measured X. You don't know anything more than the average temperature. It could be in any state consistent with that temperature."

Will you tell students to count the microstates consistent with the temperature?

[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.