Question

My approach:

1. Write down the partition function Z₁ of a single particle, approximate the summation with integral so Z₁ = ∫ e − β E g(E) dE where g(E) is the density of states.
2. Mark the ground state with a dummy variable μ so Z′₁ = Z₁ − 1 + μ.
3. The partition function Z of N bosons is 1 / N! Z′^N₁.
4. The number of particles in the ground state is ∂ / ∂μ logZ ∣∣ _μ _=₁.
However, my result is quite different from BEC, specifically, the ratio of particles in the ground state is independent of the particle number N.

I suspect this derivation must contain some logical mistakes but I am unable to find them. Any help is appreciated.

Answer 1

There may be logical mistakes in the derivation provided in the question. The ratio of particles in the ground state may be independent of the particle number N in certain conditions. Further review of the derivation and consulting reliable sources is recommended.

Step-by-step explanation:

In the derivation you provided, one possible logical mistake could be the assumption that the number of particles in the ground state is equal to the derivative of the partition function with respect to the dummy variable μ.

To understand the behavior of the BEC (Bose-Einstein Condensate), it is necessary to consider the nature of the system and the specific conditions in which it is being studied. The ratio of particles in the ground state may indeed be independent of the particle number N under certain conditions.

I would recommend reviewing the derivation and assumptions made to identify any errors or inconsistencies. Additionally, consulting reliable sources on the topic of Bose-Einstein statistics and BEC can provide further insights.