I Bose-Einstein statistics for μ>ε

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The Gibbs sum for a system with chemical potential μ greater than energy ε simplifies to Z ≈ [λ exp(-ε/τ)]^N, leading to an average particle number ⟨N⟩ equal to N. However, this model becomes ineffective as it suggests that for any parameter X, ⟨X⟩ approaches X_N, rendering the model practically useless. Additionally, the Bose-Einstein distribution indicates that f(ε) becomes negative when μ exceeds ε, which is physically unacceptable. The discussion highlights that for bosons, the lowest energy state ε_0 imposes an upper limit on the chemical potential, with the phenomenon of Bose-Einstein condensation occurring as μ approaches ε_0. These insights raise questions about the physical validity of the model under the condition μ > ε.
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Flaws in Bose-Einstein statistics for μ>ε
The Gibbs sum is given by

$$Z=\sum[\lambda \exp(-\varepsilon/\tau)]^N$$

where ##\lambda\equiv\exp(\mu/\tau)##. Since we are assuming ##\mu>\varepsilon##, we take only the last term of the sum because all others can be neglected.

thus

$$Z\approx[\lambda \exp(-\varepsilon/\tau)]^N$$

Now

$$\langle N\rangle =\lambda\frac{\partial}{\partial\lambda}\ln Z=\lambda\frac{\partial}{\partial\lambda}(N\ln\lambda-N\varepsilon/\tau)=N$$


But in general, we see that for any parameter ##X##

$$\langle X\rangle=\sum X_N[\lambda \exp(-\varepsilon/\tau)]^N/Z\approx X_N$$

where ##X_N## is the value at ##N\rightarrow\infty##, thus making the whole model useless.

But is this the only flaw in taking ##\mu>\varepsilon##?

That is, for the usual Bose-Einstein distribution

$$f(\varepsilon)=\frac1{\exp[(\varepsilon-\mu)/\tau]-1}$$

we get ##f(\varepsilon)<0## for ##\mu>\varepsilon##, which is physically wrong.

Are any such "physical" conditions present for the above model?
 
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For bosons the energy ##\epsilon_0## of the lowest energy state sets an upper limit for the chemical potential. In case when the chemical potential ##\mu## approaches the lowest energy level ##\epsilon_0## from below (##\mu\rightarrow\epsilon_0##), the occupation of the lowest energy level diverges; this phenomenon is called the Bose-Einstein condensation.
 
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