How do we justify taking the chemical potential, μ as 0 when calculating the critical temperature of Bose-Einstein Condensates (BECs)?
I apologise as I do not how to use LaTeX, for if I did the elegance of mathematics would’ve allowed me to construct my question with ease...
I understand to calculate the total number of particles in a system comprised of non-relativistic bosons of mass m at thermal equilibrium at temperature T. One must simply some over the occupancies for each energy state, the occupancies is given by the bose-einstein distribution...
For some reason during the derivation setting chemical potential to zero within the bose-einstein distribution gives us the largest possible number of particles for a given temperature, can someone explain why this is true?
Edit: Also I know the within the bose-einstein distribution, the energy of the states must always be greater than the chemical potential, this confines the distribution to a range of 0<bose-einstein distribution<+∞
Answer
To determine the upper limit on chemical potential for a gas of N bosons, look at the form of the Bose distribution in the grand canonical ensemble with ⟨N⟩=N. When using the GCE, it's easiest to work at chemical potential μ and to then choose μ(N) so that ⟨N⟩(μ)=N. Each state s has average occupancy ⟨ns⟩=∑n≥0ne−βn(ϵs−μ)∑n≥0e−βn(ϵs−μ)=1Ξs∂∂(βμ)Ξs,Ξs=11−e−β(ϵs−μ),=−∂(βμ)log(1−e−βϵs+(βμ))=eβμ1−e−β(ϵs−μ).
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