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It can be shown that in the high temperature exploration of the Fermi gas, the Fermi function may be expanded to second order in $e^{\beta \mu}$, where $\beta = 1/kT$ and $\mu$ is the chemical potential like so $$f_+ \approx e^{-\beta(\epsilon_i - \mu)} - e^{-2\beta(\epsilon_i - \mu)}$$

Using the following results $$\sum_i \epsilon_i e^{-\beta \mu} = \frac{3}{2} \frac{Z(1)}{\beta}\,\,\,\,\,\,\,\sum_i \epsilon_i e^{-2\beta\epsilon_i} = \frac{3}{2} \frac{Z(1)}{2^{5/2}\beta}$$ we obtain $$e^{\beta \mu} \approx \frac{N}{Z(1)} + \frac{1}{2^{3/2}} \left(\frac{N}{Z(1)}\right)^2$$

My question is what is the physical interpretation of the fugacity $e^{\beta \mu}$ in the small regime?

Many thanks.

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The wikipedia article on fugacity is helpful in this regard. –  user7757 Jun 17 at 14:00

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