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\sect{Fermi gas in 2D+3D connected boxes with gravitation}

Given a 'mezoscopic' box with ${L\times L\times \ell}$. ${L}$ is a
macroscopic size while ${\ell}$ is a mezoscopic size and s
${\ell<<L}$. We insert ${N}$ fermions with spin ${\frac{1}{2}}$ and
mass ${\mathsf {m}}$ in to a box. in paragraphs  a-d assume the
temperature is ${T=0}$. (mezoscopic size is very small in relation
to a large macroscopic size, in relation to microscopic size).


\begin {itemize}

\item[(a)]
Describe the uniparticle states density and note what is the energy
field which the uniparticle states density in it are like a particle
in a two dimensional box.

\item[(b)]
Find the fermi level ${E_{F}}$ in condition it's possible to relate
to the box as a two dimensional one. What is the maximum number of
${N_{max}}$ fermions it's possible to insert in to the box in this
condition.

\item[(c)]
For ${N<N_{max}}$, as above, find the pressure ${P}$ on the sided
walls of the box, and the force  ${F}$ on the horizontal walls. The
box, as above,  is attached to a tank with the dimensions ${L\times
L\times L}$ which is ${D}$ higher from the box. consider
gravitation. (assume a very strong  gravitation field ${\mathsf
{g}}$, so the question will be reasonable from the order of
magnitude).

\item[(d)]
In the conditions of paragraph b', i.e. ${N=N_{max}}$, What is the
minimum hight ${D_{min}}$ to place the tank so all of the fermions
will stay in the box?

\item[(e)]
The temperature of the system was raised a little bit. Assume
temperature ${T}$ and also ${N=N_{max}, D=D_{min}}$. as a result,
some of the particles that were in the mezoscopic box, transferred
to the tank. Estimate the number ${N'}$ of these particles. For
that, assume that the chemical potential of the system is almost not
changing as a result of raising the temperature. Express your
answers using ${T,\mathsf {g},\mathsf {m},L,\ell}$ only.

\end {itemize}

given:
%
%
\[
\int_{0}^{\infty}\frac{x^{\frac{1}{2}}}{e^{x}+1}dx=\Gamma\left(\frac{3}{2}\right)\left(1-\frac{1}{\sqrt{2}}\right)\zeta\left(\frac{3}{2}\right)=0.678
\]


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