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> Note that I was the first one to propose a particular way to define the macrostate (energy) of the system to understand what you meant. Your reply was "that's

> Note that I was the first one to propose a particular way to define the macrostate (energy) of the system to understand what you meant. Your reply was "that's not the definition of macrostate. It's just one arbitrary choice you have chosen." And you gave your own arbitrary choice "I'll choose something arbitrary. Temperature as measured by a thermometer". No. You explicitly ASKED for my definition, then I said that. When I said that's not the "definition of macrostate" I did not make an arbitrary choice there. I simply stated that IF you think energy was the definition of macrostate, then you are wrong. > Maybe be can agree that [Gibbs] entropy [which is a property of the distribution of microstates corresponding to our description of the system based one some macroscopic properties] is independent of knowledge [other than about that particular macroscopic description]. This is the point of the ENTIRE thread starting from your INITIAL reply. Technically this argument is over. You weren't in agreement with me, now you are, so you were wrong and I was right. This sentence admits that. The rest of the stuff here is side tangents and it's hard to see the main point. I can entertain it though for a little bit but this here is essentially the end. >The discussion started with me trying to understand what did you mean by "low entropy" when you said: This discussion started with you saying that if I know the microstate of the system entropy is zero. If you wanted to understand What I thought about "low entropy" then this was NEVER stated. The thread is based off what is stated and what is not stated. It is not based off of what your internal thoughts and intentions are. If you want the topic to based off your own thoughts, they need to be expressed explicitly in statements. You did so just now, but way too late. >Note that if for some reason all the particles in the gas go away from the thermometer for a second the temperature of the mercury won't change (ignoring that it will radiate energy over time). The reading of the thermometer doesn't drop to zero just because there is nothing there. This is pedantic. Obviously. I can take it into account. 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. >Let's assume that you are somehow measuring the temperature in a small region somewhere with a very high reaction time. What you measure will not depend in any way on the particles that are elsewhere. The particles in the rest of the box could be well spread or all near one corner and you would have no reason to say that the latter configuration is lower entropy than the former based on your macrostate. I specifically defined it as a thermometer to make location matter. Switching the location of the thermometer is switching the definition as well. I really don't see what your point is. You think I'm wrong about something? What am I wrong about? >But at some intermediate point the reading of the thermometer will be the same as in equilibrium. Your macrostate will be the same as in equilibrium. Your entropy will not be lower than in equilibrium. Even though you said -if I understood you correctly- that the entropy should be lower because the higher concentration has a lower probability of occurring. Your point seems to be buried in here somewhere and I can't parse it. There is a macrostate that is equilibrium, yes. Are you referring to mercury level mid transition? This is pedant-ism to the max if you are. Yes the mercury level will display the WRONG temperature if it's mid transition. Not willing to constantly adjust the model to little flaws you find. Last time: Let's switch to a digital thermometer that displays temperature at time intervals that are equal to the length of time it takes for the maximum speed particle to travel across the box. There is no transition value now. All temperature readings reflect an instantaneous observed truth at an instantaneous point in time, but that temperature is displayed at non-instantaneous intervals. It also seems to me that your definition of macrostate is meaningless. The total energy of the universe is hypothetically the same all the time. If the macrostate was just energy there's no point to it, because entropy would then be an unchanging constant. I think we're done here. I'm just trying to guess what you're driving at. You'll need to clarify your point if you want me to continue. What exactly are you trying to say here?

kgwgk 4y ago
> 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".)

kgwgk 4y ago

> 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".)