“Is the water colder, because we know more about it? …Yes! Yes it is!”
You’re kidding, right? Knowing something about a system doesn’t change the system (neglecting quantum, of course). The statistical way to define entropy (as you mentioned) is the log of the number of microstates. The fact that you know all the trajectories/positions couldn’t matter less to the glass of water, the only thing that matters is (using your jargon) the phase space volume it occupies.
Reshape the space for a second. Call it 6-D, with each particle a point, instead of 6N-D. Now the entropy would correspond to the volume actually occupied in 6-D space, rather than the possible volume among which your single point can choose. W
With the single point, you get sucked into the fallacy that because you know where the point is at one time, that’s the only possible location it can have, and you’re tricked into believing the entropy is much smaller than it is.
Statistical physics assumes exact particle trajectories are random and unknowable, although this was never believed to be fundamental. It was just a convenient way to ignore things nobody cared about, and take averages. Restricting yourself to that one point in phase space, you violate that assumption.
“Is the water colder, because we know more about it? …Yes! Yes it is!”
You’re kidding, right? Knowing something about a system doesn’t change the system (neglecting quantum, of course). The statistical way to define entropy (as you mentioned) is the log of the number of microstates. The fact that you know all the trajectories/positions couldn’t matter less to the glass of water, the only thing that matters is (using your jargon) the phase space volume it occupies.
Reshape the space for a second. Call it 6-D, with each particle a point, instead of 6N-D. Now the entropy would correspond to the volume actually occupied in 6-D space, rather than the possible volume among which your single point can choose. W
With the single point, you get sucked into the fallacy that because you know where the point is at one time, that’s the only possible location it can have, and you’re tricked into believing the entropy is much smaller than it is.
Statistical physics assumes exact particle trajectories are random and unknowable, although this was never believed to be fundamental. It was just a convenient way to ignore things nobody cared about, and take averages. Restricting yourself to that one point in phase space, you violate that assumption.