In classical statistical mechanics, the partition function is usually defined by: $$ Z = \int \prod dx_i \int \prod dp_i e^{-\beta H(x_i,p_i)} $$ The standard justification for this definition is that we are coarse graining the phase space into little cells and we add up the Boltzmann weight. Hence the standard Riemann integration measure on $x-p$ space is proper.

However, one could imagine that we coarse grain the phase space differently, into cells of $dr \cdot d\theta$ instead of $dx \cdot dp$. Then the partition function would become (in the simple case of 2D phase space): $$ Z = \int dr d \theta e^{-\beta H(r, \theta)} $$ Where $r = \sqrt{x^2 + p^2}$ and $\theta = \tan^{-1}(\frac{p}{x})$. Notice that this partition function is different from the standard definition since it is lacking a Jacobian factor. My question is then: why did we choose to do coarse graining in $x-p$ space instead of $r-\theta$ space in the first place?


I would say this question is not so much about coarse graining as it is about choosing the right measure on phase space -- say, the measure $dx \, dp$ (equal to $r \,dr\, d\theta$ in your example) versus the measure $dr \, d\theta$. The measure you choose will affect how you coarse-grain into equal-volume cells, or conversely your choice of equal-volume cells will affect what you'd like to define as the continuum measure.

What's special about the standard measure $dx \, dp$ on phase space? Here's one nice property: it's preserved under Hamiltonian (time) evolution on phase space, by Liouville's theorem. Now let's go back and see why that's relevant.

We're looking for the measure on phase space which will give us the correct result when calculating the partition function in statistical mechanics. The partition function comes up when considering the canonical ensemble. We might instead ask, in the canonical ensemble, why is $dx \, dp \,e^{-\beta H(x,p)}$ the right probability measure over phase space? (Then the definition of the partition function follows.)

Meanwhile, the canonical ensemble is usually derived by assuming the system is part of a larger system+environment super-system, where the super-system is in the microcanonical ensemble at fixed energy. So for some general system at fixed energy, what probability density should we choose? That is, how should we define the microcanoical ensemble? Well, it's basically up to us. We're looking for a good empirical description of equilibrium systems.

Because we're assuming the system is at equilibrium, the measure we choose on the energy shell shouldn't change under time-evolution. Let's say we know our system has energy in some small window $(E,E+\Delta E)$. That corresponds to some thickened energy shell in phase space. I'm only using a thickened energy shell so that the shell is not measure zero (infinitely thin) with respect to the measure on the ambient phase space. This is a subtlety related to the question According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general. I want to simplify the discussion by just saying: assume $\Delta$ is very small but nonzero.

If we choose the measure $dx \, dp$ on the energy shell, then we know this measure won't change under time evolution, by Liouville's theorem. So it's a good candidate for our equilibrium measure. Are there any other measures left invariant by Hamiltonian flow? The answer is no, so long as we make a crucial assumption: we assume any state in the energy shell will -- at some point in the future -- come arbitrarily close to every other state in the energy shell. (Something like this assumption is necessary because if there were more conserved quantities -- if certain regions of the energy shell never reached other regions under time evolution -- then the microcanonical ensemble would likely be a poor model to begin with.)

To see why there is a unique measure on the energy shell that's invariant under Hamiltonian flow, imagine some other measure $f(x,p) dx \, dp$. If this measure is invariant under time-evolution, and time evolution takes every point $(x,p)$ in the energy shell arbitrarily close to every other point, then $f(x,p)$ must be constant.

So $dx \, dp$ is the right measure for the microcanonical ensemble. When you derive the canonical ensemble on some system from the microcanonical ensemble on a super-system, the measure $dx \, dp$ on the super-system will ultimately yield the measure $dx \, dp e^{-\beta H(x,p)}$ on the system.

All of this discussion has been classical, which is fine, because we'd like to be able to do statistical mechanics classically. However, you could also ask why this choice of measure on classical phase space yields statistical results close to what we'd derive using quantum statistical mechanics. There are a few different approaches to the foundations of quantum statistical mechanics, e.g. https://arxiv.org/pdf/quant-ph/0511225.pdf. Anyway, I think the key feature of the measure $dx \, dp$ that makes classical statistical mechanical results agree with quantum statistical mechanical results is that the number of orthogonal quantum states corresponding to some small region of phase space is approximately proportional to the volume of the region as measured using $dx \, dp$.


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