Undergrad Bose-Einstein statistics for μ>ε

[LightPhoton]

TL;DR

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?

 

[Lord Jestocost]

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.