What Temperature Initiates Bose-Einstein Condensation in a Gravitational Field?

[qbslug]

Homework Statement

For an ideal Bose gas in a uniform gravitational field, at what temperature does Bose-Einstein condensation set in. Gas is in a container of height L.

Homework Equations

Normal BEC temperature of an ideal Bose gas not under the influence of gravity is
$$T = \frac{h^2}{2 \pi m k}\left(\frac{N}{V\xi(3/2)}\right)^{2/3}$$

The Attempt at a Solution

I think one must first calculate the density of states for such a system by calculating the volume of phase space and dividing by h^(3N). But I don't know how to calculate the volume of phase space for this situation. I could be wrong of course in this attempt.
$$\omega = \int^\prime\cdot\cdot\cdot\int^\prime (d^{3N}q \,d^{3N}p) = ?$$

 

[kloptok]

I have an exam tomorrow in statistical mechanics so I should be able to solve this!

Normally you calculate the condensation temperature assuming that the particles are free, i.e. $$E = p^2 / 2m$$. Now you have a potential energy term in your hamiltonian, $$V=mgz$$, where z is the height of the particle in the container. This will influence your density of states and you will get something like:

$$N = \int^{\infty}_{p=0} \int^h_{z=0} <n(p,z)> f(p,z) dz dp$$

for the number of particles, where <n> is the mean occupation number in BE statistics (expressed in terms of p and z), and f(p,z) the density of states, also expressed in terms of p and z.

Do the appropriate approximations in that integral and solve it. Then you can find the condensation temperature.

(Note: I haven't done the calculations but I'm guessing this would be one way to solve the problem, alert me if something seems to be wrong)

 

[qbslug]

yeah the hard part is deriving the density of states for a bose gas under gravity. It doesn't seem to be so trivial.

 

[kloptok]

No, I sat some time trying to solve it and had some serious trouble. How about the density of states as:
$$f(p,z)=\frac{4 \pi A zdz p^2dp}{h^3}$$
?

(The 4pi comes from integrating out angular dependence in p (spherical coordinates) and the A from the spatial part, except z (x and y, or $$\rho$$ and $$\varphi$$ in cylindrical coordinates).)

Chief concern for me is then how to solve the integral. Using that $$\mu \rightarrow 0$$ at condensation we have
$$<n>= \frac{1}{e^{p^2/2m + mgz} -1}$$
, right?

So how is the upper limit in the integral for z, L, translated into something dependent on p? We have $$E=p^2/2m + mgz$$. I think that can be used to find the upper limit L in the z-integral in terms of p.