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.

 

[paranormal]

As a scientist, your attempt at a solution is a good start. However, there are a few things that need to be clarified and addressed.

Firstly, the formula for the normal BEC temperature of an ideal Bose gas that you have provided is for a system that is not under the influence of gravity. In this case, we are dealing with a system in a uniform gravitational field, which will affect the energy levels of the particles and therefore, the density of states. So, we need to modify the formula to take into account the gravitational potential energy.

Secondly, in order to calculate the volume of phase space, we need to consider the constraints of the system. In this case, the particles are confined to a container of height L, so we need to take that into account when integrating over the phase space.

Lastly, it is important to remember that the Bose-Einstein condensation occurs when the chemical potential becomes equal to the ground state energy. So, we need to find the temperature at which this condition is satisfied.

To calculate the density of states for a system in a gravitational field, we need to take into account the energy levels of the particles. The energy levels for a particle in a uniform gravitational field are given by E_n = \frac{p^2}{2m} + mgh_n, where p is the momentum, m is the mass of the particle, g is the gravitational acceleration, and h_n is the height of the energy level.

Using this, we can modify the formula for the normal BEC temperature to be: T = \frac{h^2}{2 \pi m k}\left(\frac{N}{V\xi(3/2)}\right)^{2/3} \exp\left(-\frac{mgh_0}{kT}\right), where h_0 is the height of the ground state energy level.

To calculate the volume of phase space, we need to integrate over the phase space with the constraints of the system. In this case, the particles are confined to a container of height L, so the integral becomes: \int_0^L \cdot\cdot\cdot\int_0^L (d^{3N}q \,d^{3N}p) = \frac{1}{h^{3N}}\left(\frac{L^3}{V}\right)^{3N}.

Now, to find the temperature at which Bose