Statistical Mechanics Part II: The Ideal Gas - Comments

[NFuller]

Greg Bernhardt submitted a new PF Insights post

Statistical Mechanics Part II: The Ideal Gas

Continue reading the Original PF Insights Post.

 

[Greg Bernhardt]

Great work! Will there be a part 3?

 

[fresh_42]

Another example of an Insight that made me a real fan of this PF feature. Is it the practice here or the special mentality, that makes them readable in a way, that all crucial keywords are mentioned and the explanation is still somewhere above the boring of a textbook and the out-of-context of a Wiki article, i.e. close to what an actual lecture would be. Absolute an appetizer for non specialists like me!

I wonder if it were possible to make an index of all of them.

 

[jedishrfu]

Great article!

I haven’t read much about Stat Mech in over forty years. I think we used only one ensemble in our undergrad coursework.

Your article was very insightful. Thanks!

 

[vanhees71]

I think it should be mentioned that here tacitly the Bose distribution was used to count the states without giving the argument for why to use Planck's constant $h=2\pi \hbar$, which is of course a measure of phase space not spatial volume.

That's definitely not the way Boltzmann did it. Of course, it's correct, but the classical result comes from doing a rather tough approximation using Stirlings formula, which is in fact a bit tricky anyway. Another problem is that in the given expression of the entropy there are dimensionful quantities as arguments in logarithms. This should be corrected. Otherwise it's a very nice derivation in the microcanonical ensemble.

For a treatment (however in the context of kinetic theory), see section 1.8 of

https://th.physik.uni-frankfurt.de/~hees/publ/kolkata.pdf

 

[NFuller]

Great work! Will there be a part 3?

Yes, I intend to do a few more parts as time permits.

 

[NFuller]

I think it should be mentioned that here tacitly the Bose distribution was used to count the states without giving the argument for why to use Planck's constant h=2πℏh=2πℏh=2\pi \hbar, which is of course a measure of phase space not spatial volume.

You're right that I never proved that using Planck's constant yields the correct discritization of the space space. Rather, I assumed it a priori as a reasonable guess which yields the right answer. It's also true that dividing the volume by Planck's constant doesn't really mean anything physical on its own. The point is to discritize the phase space and tacking on the $h^{3}$ term accomplishes this in an intuitive, although not exactly correct, way.

there are dimensionful quantities as arguments in logarithms. This should be corrected.

Are you referring to the logarithms after applying Stirling's approximation?

 

[vanhees71]

It's not "space space" (I guess you mean "configuration space") but "phase space" that's to be divided into cells of the size $(2 \pi \hbar)^{3f}$!

For the case of an ideal gas, it's easily derived by solving the problem for the single-particle phase space. See my kinetic-theory manuscript for that:

https://th.physik.uni-frankfurt.de/~hees/publ/kolkata.pdf

And yes, there are logarithms with dimensionful quantities in their arguments. As far as I can see you can very easily repair it by taking the logarithms together since I think the formulae are in principle right!

 

[NFuller]

It's not "space space" (I guess you mean "configuration space")

Yes, that was a typo.

And yes, there are logarithms with dimensionful quantities in their arguments. As far as I can see you can very easily repair it by taking the logarithms together since I think the formulae are in principle right!

These should be fixed now. I was originally doing the derivation with dimensionless variables but then later switched to dimensionful variables, but forgot to change this in the article.

 

[vanhees71]

Now it's better. Only in the third-last formula you should again combine the two ln terms :-).

 

[Grinkle]

The point is to discritize the phase space and tacking on the h3h^{3} term accomplishes this in an intuitive, although not exactly correct, way.

Why is it intuitive? Is the assertion that h-bar is a reasonable lower bound based on uncertainty, and because there are three dimensions in the volume it needs to be cubed?

 

[vanhees71]

First of all, phase space is not descritized in quantum theory. Second, the choice of the natural phase-space volume to be able to define phase-space distribution functions has been the greatest enigma for Maxwell and Boltzmann when formulating classical statistical theory.

Quantum theory resolves this problem without further thought since it implies that the single-particle phase-space-cell measure indeed is $h^3=(2 \pi \hbar)^3$. Indeed to establish statistical mechanics, you have to define a phase-space distribution function for a single particle $f(t,\vec{x},\vec{p})$ which must have the dimension of inverse action cubed, because that's the dimension of phase-space volume $\mathrm{d}^3 x \mathrm{d}^3 p$.

Now suppose you have a particle in a spatial volume $\mathrm{d}^3 x$. Then imposing periodic boundary conditions you find that there are $\mathrm{d}^3 x \mathrm{d}^3 p/(2 \pi \hbar)^3$ quantum states. This implies that the phase-space distribution function has to be normalized such that
$$\int_{\mathbb{R}^3} \mathrm{d}^3 x \int_{\mathbb{R}^3} \mathrm{d}^3 p \frac{1}{(2 \pi \hbar)^3} f(t,\vec{x},\vec{p})=1,$$
i.e., QT uniquely defines the appropriate single-particle phase-space measure as $(2 \pi \hbar)^3$. For details see

https://th.physik.uni-frankfurt.de/~hees/publ/kolkata.pdf