The state-space (for particles) in statmech is the space of possible positions and momenta for all particles.
The measure that’s used is uniform over each coordinate of position and momentum, for each particle.
This is pretty obvious and natural, but not forced on us, and:
You get different, incorrect predictions about thermodynamics (!) if you use a different measure.
The level of coarse graining is unknown, so every quantity of entropy has an extra “+ log(# microstates per unit measure)” which is an unknown additive constant. (I think this is separate from the relationship between bits and J/K, which is a multiplicative constant for entropy—k_B—and doesn’t rely on QM afaik.)
On the other hand, Liouville’s theorem gives some pretty strong justification for using this measure, alleviating (1) somewhat:
In quantum mechanics, you have discrete energy eigenstates (...in a bound system, there are technicalities here...) and you can define a microstate to be an energy eigenstate, which lets you just count things and not worry about measure. This solves both problems:
Counting microstates and taking the classical limit gives the “dx dp” (aka “dq dp”) measure, ruling out any other measure.
It tells you how big your microstates are in phase space (the answer is related to Planck’s constant, which you’ll note has units of position * momentum).
I wish I had a better citation but I’m not sure I do.
In general it seems like (2) is talked about more in the literature, even though I think (1) is more interesting. This could be because Liouville’s theorem provides enough justification for most people’s tastes.
Finally, knowing “how big your microstates are” is what tells you where quantum effects kick in. (Or vice versa—Planck estimated the value of the Planck constant by adjusting the spacing of his quantized energy levels until his predictions for blackbody radiation matched the data.)
The state-space (for particles) in statmech is the space of possible positions and momenta for all particles.
The measure that’s used is uniform over each coordinate of position and momentum, for each particle.
This is pretty obvious and natural, but not forced on us, and:
You get different, incorrect predictions about thermodynamics (!) if you use a different measure.
The level of coarse graining is unknown, so every quantity of entropy has an extra “+ log(# microstates per unit measure)” which is an unknown additive constant. (I think this is separate from the relationship between bits and J/K, which is a multiplicative constant for entropy—k_B—and doesn’t rely on QM afaik.)
On the other hand, Liouville’s theorem gives some pretty strong justification for using this measure, alleviating (1) somewhat:
https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)
In quantum mechanics, you have discrete energy eigenstates (...in a bound system, there are technicalities here...) and you can define a microstate to be an energy eigenstate, which lets you just count things and not worry about measure. This solves both problems:
Counting microstates and taking the classical limit gives the “dx dp” (aka “dq dp”) measure, ruling out any other measure.
It tells you how big your microstates are in phase space (the answer is related to Planck’s constant, which you’ll note has units of position * momentum).
This section mostly talks about the question of coarse-graining, but you can see that “dx dp” is sort of put in by hand in the classical version: https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)#Counting_of_microstates
I wish I had a better citation but I’m not sure I do.
In general it seems like (2) is talked about more in the literature, even though I think (1) is more interesting. This could be because Liouville’s theorem provides enough justification for most people’s tastes.
Finally, knowing “how big your microstates are” is what tells you where quantum effects kick in. (Or vice versa—Planck estimated the value of the Planck constant by adjusting the spacing of his quantized energy levels until his predictions for blackbody radiation matched the data.)