“An isolated system is in equilibrium if the sharp constraints and expectation values which define its macrostate pertain to constants of the motion only.”
“Whenever the macroscopic data include an expectation value, a measurement of the pertinent observable may yield a wide range of outcomes.”
I’m not making up the idea of an equilibrium state being associated with a macrostate.
"The equilibrium macrostate is that with the most microstates, and this is the state of greatest entropy. A macroscopic flux is most likely in the direction of increasing entropy."
>I’m not making up the idea of an equilibrium state being associated with a macrostate.
I know you aren't. TBH not even sure why equilibrium needs to be brought up. The topic "is entropy is independent of knowledge?" You disagree with that statement.
Macrostate is defined in terms of microstates and is independent of your knowledge of the exact configuration of the microstates. You have some blunt tool like the thermometer that gives you macroscopic data and that hides the microstates from you.
But if I had some precision tool that reads the position of every particle I can still identify how that configuration of microstates will influence the blunt tool. The blunt tool hides knowledge, but your knowledge of the microstate does not change the reading on that blunt tool.
Because entropy is defined in termms of macrostate it stays the same regardless of which tool you used to do the measurement.
See? We agree that there is an equilibrium macrostate. When I fist asked you to clarify what did you mean by macrostate you told me that “it changes with time; even at equilibrium.”
We agree that the entropy depends on the macrostate. Note that the macrostate is our description of the system and depends on how we choose to describe it which normally depends on what are the constraints, how it was prepared., etc. It’s not just a property of the position of those balls.
It’s because we agree that Gibbs’ entropy is a function of the macrostate that I asked how did you define it in your example. You told me: “Temperature as measured by a thermometer is the macrostate. Every possible configuration of particles (microstates) that causes mercury to rise to a certain degree represents a different macrostate.”
I asked “How does your configuration where the balls were near one corner in the cube cause mercury to rise to a different level than the configuration where they occupy a larger volume near the center?” and the answer “The balls have to touch thermometer” doesn’t cut it. The balls don’t touch the thermometer in either case.
You seemed to imply that the higher concentration means a different macrostate with lower entropy. Or maybe the low entropy in you example is because the balls are near a corner?
Anyway, it would indeed have been easier to say that definition of macrostate included the density of particles in each octant of the cube - or something like that.
I still stand by my statement. Even at equilibrium it can lower in entropy. The equilibrium is simply the highest entropy state.
>I asked “How does your configuration where the balls were near one corner in the cube cause mercury to rise to a different level than the configuration where they occupy a larger volume near the center?” and the answer “The balls have to touch thermometer” doesn’t cut it. The balls don’t touch the thermometer in either case.
I stated this is pedantism. The concept and intuition remain true. I changed the definition so that it's a volume around the thermometer if the particle is in that volume and heading for the thermometer is counts as a collision.
I stated all of this already.
>You seemed to imply that the higher concentration means a different macrostate with lower entropy. Or maybe the low entropy in you example is because the balls are near a corner?
Yes. 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.
>Anyway, it would indeed have been easier to say that definition of macrostate included the density of particles in each octant of the cube - or something like that.
Sure, Divide the box into a bunch of cubes. If 1 or more particles are in the cube then that cube represents 1, otherwise 0. Add those numbers up and that represents a macrostate.
The inuition remains the same. For all particles to be concentrated in 1 cube is a very low probability. And the macrostate will be quite low too. With enough cubes and boxes such a state has a very low probability of occuring.
But all of this is, again, independent of your knowledge of where the particles are in each cube.
