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\exnumber{5980}
\heading{BEC regarded as a phase transition}
\auname{Edo Hogeg}
{\bf The problem:}
\Dn
Consider $N$ bosons that each have mass $M$ in a box of volume $V$.
The overall density of the particles is ${\rho=N/V}$.
The temperature is $T$.
Denote by $m$ the number of particles that occupy the ground state orbital of the box.
The canonical partition function of the system can be written as
%
\[ Z \ \ = \ \ \sum_{m=0}^{N} Z_{N-m}
\ \ = \ \ \sum_{m=0}^{N} e^{-\tilde{A}(m)}
\ \ = \ \ \int d\varphi \ e^{-N\,A(\varphi) + \text{const}} \]
%
In this question you are requested to regard the the Bose-Einstein condensation as phase transition that can be handled within the framework of the canonical formalism where $m$ is the order parameter. Whenever approximations are required assume that $1 \ll m \ll N $ such that ${\varphi=(m/N)}$ can be treated as a continuous variable. In the first part of the question assume that the gas is ideal, and that $Z_{N-m}$ can be calculated using the Gibbs prescription. In item~5 you are requested to take into account the interactions between the particles. Due to the interactions the dispersion relation in the presence of $m$ condensed bosons is modified as follows:
%
\[ E_k \ = \sqrt{ \left(\epsilon_k + 2g\frac{m}{V} \right) \epsilon_k} \]
%
where $ \epsilon_k $ are the one-particle energies in the absence of interaction, and ${ g }$ is the interaction strength. For the purpose of evaluating ${ Z_{N-m} }$ for large ${ m }$ assume that the above dispersion relation can be approximated by a linear function ${ E_k \propto k }$
\begin {itemize}
\item[(1)] Write an explicit expression for the probability ${ p_m }$ of finding ${ m }$ particles in the ground state orbital. Calculation of the overall normalization factor is not required.
\item[(2)] Find the most probable value ${ \bar{m} }$. Determine what is the condensation temperature ${T_c }$ below which the result is non-zero.
\item[(3)] Assuming ${ T < T_c }$ write a Gaussian approximation for ${ p_m }$
\item[(4)] Using the Gaussian approximation determine the dispersion ${ \delta m }$
\item[(5)] Correct your answer for ${ p_m }$ in the large ${ m }$ range where the interactions dominate.
\item[(6)] On the basis of your answer to item3, write an expression for ${ A(\varphi;f) }$
that involves a single parameter ${f}$ whose definition should be provided using ${\rho,M,T}$.
\item[(7)] On the basis of your answer to item5, write an expression for ${ A(\varphi;a) }$
that involves a single parameter ${a}$ whose definition should be provided using ${\rho,M,T}$ and $g$.
\end {itemize}
\Dn
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{\bf The solution:}
\Dn
(1)
By Gibbs prescription the probability $p_{m}$ of finding $m$ atoms in the ground state is given by
\[
p_{m}\propto \frac{1}{(N-m)!} z^{N-m}=\frac{1}{(N-m)!}\left(V\left(\frac{MT}{2\pi}\right)^{\frac{3}{2}}\right)^{N-m}
\]
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\Dn
(2)
To find the most probable $\bar{m}$ we donote $\tilde{A}(m)$ and find its extemum
\[
\tilde{A}(m)=\ln{((N-m)!)}-(N-m)\ln{z}
\]
\[
\tilde{A}^{`}(m)=-\ln{(N-m)}+\ln{z}=0
\]
\[
\bar{m}=N-V\left(\frac{MT}{2\pi}\right)^{\frac{3}{2}}
\]
We find the condensation temperature by demanding $\bar{m}>0$
\[
N-V\left(\frac{MT}{2\pi}\right)^{\frac{3}{2}}>0
\]
\[
T_{c}=\frac{2\pi}{M} \left(\frac{N}{V}\right)^{\frac{2}{3}}
\]
and so if $T