Hello PF. I won't just lurk around today, I will pose a question.(adsbygoogle = window.adsbygoogle || []).push({});

I was looking at a dilute ultracold bosonic gas and was trying to see how one can predict the existance of a BEC and got stuck on this:

I was comparing probabilities between finding the system in the lowest state (energetically) and the first excited state through the use of the Boltzmann (and also Gibbs) factor.

Let E_{0}be the energy of the lowest state and also let me set this as my zero on the energy scale (E_{0}= 0), and let E_{1}be the energy of the first excited state. Now let's check the ratio of probability

P(E_{0})/P(E_{1}) = exp(-E_{0}/kT)/exp(-E_{1}/kT) = exp(E_{1}/kT), ok!

However, energy between these levels can be (for example) of the order of E_{1}/k = 10^{-14}(temperature units) and thus exp(E_{1}/kT) ≈ 1.

This gives (from above) the relation P(E_{0}) ≈ P(E_{1}). So the system is as likely to be in the lowest state as the first excited one.

Now I turn my headand look at the Bose distribution function

f_{BE}= 1/(exp[(E_{x}-μ)/kT] - 1)

which describes the mean occupation of bosons in a state of energy E_{x}at temperature T, k and μ are Boltzmanns constant and the chemical potential (negative here!) respectivley.

Now I get something completely different from before (at least that's how it appears to me)! Let's say I have a system consisting of around N = 10^{22}bosons, and as above the energy difference (in temperature units) is of the order 10^{-14}. I set the temperature to about T = 1 mK and let E_{x}= E_{1}.

When I now check the ratio f_{BE}/N ≈ 5*10^{-12}.

So there's not a substantial fraction of particles at all in the first excited state! They must (nearly) all be in the lowest state. Since at this temperature the chemical potential is very close to E_{0}and thus f_{BE}is really large. E_{0}dominates!

Now how do I resolve this?From looking at f_{BE}I can see that there's an onset of condensation at low temperatures. But when looking at probabilities the lowest state is as good as the first excited one. What differs between these viewpoints!?

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# Regarding BEC: Bose distribution and probabilities

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