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\sect{Ideal Fermi gas in 2D space}

Consider ${N}$ mass ${\mathsf {m}}$ Fermions that have spin $1/2$, 
that are are held in a two dimensional box that has an area~${A}$.
Show that:
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\[
N\left(\beta,\mu\right) \ \ = \ \ A\frac{\mathsf{m}}{\pi}T \ln\left(1+e^{\frac{\mu}{T}}\right)
\]
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Tip: Define ${X=e^{\beta \left(E-\mu \right)}}$ 
and use the integral ${\int_{1}^{\infty}\frac{dx}{x\left(x+1\right)}=\ln\left(\frac{1+X_{1}}{X_{1}}\right)}$.\\

Write and explain what is the ${T=0}$ result. \\

Find the chemical potential ${\mu\left(T,N\right)}$. \\

Find the Fermi energy ${E_{F} \equiv \mu\left(T\rightarrow 0,N\right)}$. \\

Show that at low temperatures
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\[\mu\left(T\right) \ \ \approx \ \ E_{F}-Te^{-\frac{E_{r}}{T}}\]


Show that at high temperatures the result is consistent with the Boltzmann approximation. \\

Find ${E\left(\beta,\mu\right)}$ and ${{P}\left(\beta,\mu\right)}$ at zero temperature. \\

Derive the following results:
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\[E=A\frac{\pi}{m}\frac{1}{2}\left(\frac{N}{A}\right)^{2}, \ \ \ \ \ 
{P}=\frac{\pi}{m}\frac{1}{2}\left(\frac{N}{A}\right)^{2}\]


Clarify why at zero temperature ${{P}\propto{1}/{A^{2}}}$, 
while at high temperatures ${{P}\propto{1}/{A}}$.