>That's not how a mercury thermometer works. The mercury "drops" when the temperature of the mercury goes down. This happens when the is a flow of energy between the thermometer and the matter inside the box. If all the particles were concentrated elsewhere there would be no interaction, no transfer of energy, no change in the mercury level. (Again, I'm considering thermal contact which is the main effect in play and and ignoring radiation energy losses.)

More pedantism. I'm using an example to make a point and I CLEARLY already edited the example enough times so that it's obvious that I'm Not building a physically accurate model. Yes as you said it is completely irrelevant.

>But if only the particles in the vicinity of the thermometer matter for the local temperature number that you take as describing the macrostate you would get the same reading (and the same entropy) if the rest of the particles changed positions as long as they remained out of the vicinity of the thermometer. Is that wrong?

No that's right. The particles are large in number. Any state where no particles are in the viscinity of the thermometer is a low probability. All microstates that encompass this property occupy at the 0 temperature macrostate.

>The main point is that instead of having the balls "positioned near one corner in the cube" they could be positioned in exactly the same relative configuration (the same concentration) near any other corner and the thermometer reading would be different. The entropy is not directly related to the concentration - it depends also on the location of the particles relative to the thermometer. Or maybe the thermometer reading would be same when the particles are in any corners but then it will also be the same for other relative positions so you would also have different levels of concentration corresponding to the same entropy.

In this definition macrostate is related to vicinity. And in a physical thermometer in reality it is also directly related to particles in actual contact with the thermometer. Vicinity matters for both cases.

>I was trying to point out that having a concentration of the particles (different from the homogeneous equilibrium state) doesn't automatically mean lower entropy in your example. The same relative position of the particles among themselves can correspond to different temperatures in your thermometer - including the equilibrium temperature. (The idea was that if the concentrated particles are too fa" the temperature is "too cold". And maybe if they are too close the temperature is "too hot". But there is some goldilocks intermediate distance such that if you put there the initial concentrated state the temperature according to you would be "just right".)

You're not getting it. The entropy states I'm describing are so low in probability that they basically never will happen in your lifetime or the lifetime of this universe.

The reality is, if particles are concentrated in the corner of a box then yes, a thermometer will register 0, but you will never see this.

They don't jive with your intuition of temperature because of this. But also because the word temperature is poorly defined when applied to a box. What does it mean when you say "The temperature of the room." What part of the room at what point in time? That's why I referenced the thermometer, to prevent this type of tangent in the discussion.

There's also ambiguities like what is the temperature of a volume of space smaller than any particle can occupy? But it's pointless to discuss all of this because this isn't a discussion about anything other then linguistics. We are arguing about language and vocabulary here and that to me is not an interesting thing to talk about. It's an illusion. It's like when people argue about "what is life?" without realizing they're just arguing about the definition of a arbitrary vocabulary word. It's sort of what's going on here, I believe.

If you are in agreement with me that entropy is independent of knowledge then we're good; because as far as I was concerned this was the point you were trying to make.