>Ok. I'm glad to read that we finally agree that in your example -using the macrostate you defined- the entropy of the "particles near a corner" state is not different from the entropy of any other "particles not near the thermometer" state.

This has been the definition since I brought it up. That's how thermometers work. It's the obvious consequence of my description and it's how physicall thermometers generally work. You never made it clear that was your problem with it. Well, now that we're both clear there is no problem with it, I think initially it most likely it just didn't jive with your intuition about temperature.

>You could have answered "it doesn't - that's the same macrostate with the same temperature and the same entropy" back then.

I did answer this: "The balls have to touch thermometer. Otherwise the thermometer observes nothing." You failed to pick up on what I meant. If the balls don't touch the thermometer is means "it doesn't" change the thermometer and therefore it doesn't change the macrostate.

You just didn't get it. OK you can accuse me of not explaining it thoroughly. But I can equally at the same time accuse you of not being capable to interpret what I said. I think what's going on here is that you're too stuck in your headspace of thinking that your intuition of entropy was more accurate then mine and you wanted me to come to this realization. I think what happened here is the reverse of this. Your intuition about entropy was less crystallized then mine and most of the communication issues we had was because you thought it was the other way around.

>Instead of answering "I don't agree. Entropy is lower when the current macrostate has a low probability of occuring." you could have said "I agree. I describe the temperature as zero in all those cases, it's the same macrostate and the entropy will be the same."

I told you to refer to my explanation of the macrostate. Yeah in all those cases it will be the same. Ok fine. I could've said that. But I didn't realize that was what you were hung up on. To throw it back at you, you could've said this:

"it doesn't make sense to me that different concentrations of particles in different parts of the box yield the same temperature reading AKA the macrostate. Please explain how this can be possible"

But again I think you didn't realize this was your hang up. I think you thought the my definition of macrostate was a "bad" or something and you were trying to make me realize it?

>I agree! You're the one who picked the reading of that thermometer as the thing that would determine the macrostate and the entropy - and maintained that higher concentrations would mean different thermometer readings and lower entropy.

Yeah I picked it because it's the most intuitive one to illustrate rising entropy. The "2nd law". It's the classic model everyone has in there heads of the "heat death" of the universe.

There are technical issues with the word, but it's the most commonly used one, and we only discussed those technicalities because of a fundamental misunderstanding on your end. IF the intuition was correct, the technicalities do not need to be discussed, or thought about.

>If we agree that the entropy is not a physical property of (the microstate of) the system but a property of a particular (thermodynamical) description of the system -

Sort of. Entropy is directly tied to macrostate, but macrostate is determined by microstates. There is a relationship here from entropy to microstate but not direct.

Also ALL possible macrostate configurations, ARE independent of knowledge of the system. There is no case where knowledge can influence it. There is no case where your initial reply: "If you know the positions and velocities of the balls at all times arguably the entropy is always zero." is correct.

Unless of course you define a macrostate where it's zero all the time. But obviously what you're saying here is that knowledge changes the entropy. Which is not true.