Neglecting Inertia

I was thinking about the differences between classical and quantum stat mech...

When we talk about a bunch of particles diffusing under the influence of an external force we use the Smoluchowski equation, with the force corresponding to the negative gradient of some potential energy. We then proceed to say that the concentration at equilibrium is proportional to exp(-U/kT). I had always thought that by doing this we were essentially assuming that the particles have negligible inertia i.e. we are operating in the low-Re regime (since I would have thought that the concentration would be proportional to exp(-(K+U)/kT) if this weren't the case, K being the kinetic energy).

It turns out that I was wrong:
concentration = integral of the distribution function over all momenta,
so the momentum is `integrated out', contributing only to the constant sitting out the front of exp(-U/kT).

For those doing statmech: why don't I need a degeneracy function i.e. the density of states?
The answer lies in the interpretation;
N = integral of g*f over energies = integral of concentration over space = integral of f over phase space;
so the degeneracy is automatically accounted for by the fact that one integrates over a volume in phase space, rather than over a line (in semi-classical mechanics this is where the density of states, g, comes from in the first place: it is the volume of a thin slice of phase space with energy e to e+de, divided by the number of uncertainty-limited cells that fit in that slice).

It's worth noting that we will still be in the low-Re regime anyway: if the force-velocity response is linear it implies the drift velocity is less than or comparable to the thermal velocity, which means that for most things (proteins in water, etc.) Re will be small.

Good luck with exams!