> Let's say the greatest average speed measured of a particle and the time it takes for a particle at this speed to travel across the box is the time it takes for the thermometer to drop to zero degrees from the maximum temperature. It's quite fast but particles within the vicinity will keep the mercury level stable but if they were concentrated in a corner, the mercury drops fast enough to change the reading of the thermometer.

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 temperature of the mercury, no change in its level. It's not a question of waiting a second to see it go to zero in a vacuum. (Again, I'm considering thermal contact which is the main effect in play and and ignoring radiation energy losses.)

But that's an irrelevant aside and I had already accepted that you can get that kind of reading with some kind of thermometer. I'll concentrate on what I think is the main open point:

> I specifically defined it as a thermometer to make location matter. Switching the location of the thermometer is switching the definition as well.

I'm not switching the location of the thermometer. I'm switching the location of the particles. I'm trying to understand how the entropy - that you calculate from the temperature reading in one thermometer placed in some specific place - depends on the location of the particles.

You've written that: "The higher concentration has a lower probability of occurring. And occupies a different temperature reading on the thermometer. Each temperature reading is a different macrostate."

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 -being more or less concentrated- as long as they remained out of the vicinity of the thermometer. Is that wrong?

> Your point seems to be buried in here somewhere and I can't parse it.

Maybe it was a mistake to talk again about equilibrium... I was trying to show that there are also concentrated configurations of the particles that nevertheless you will consider maximum entropy - using your determination of the macrostate as "what this thermometer says" if the temperature is equal to the equilibrium temperature then the entropy is the same, right?

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 will 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.

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 far the temperature is "too cold". And maybe if they are too close the temperature is "too hot". But there is at least some goldilocks intermediate distance such that if you put there the initial concentrated state the temperature according to you would be "just right".)