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\sect{BEC regarded as a phase transition}

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 
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\[ Z \ \ = \ \ \sum_{m=0}^{N} Z_{N-m} 
\ \ = \ \ \sum_{m=0}^{N} e^{-\tilde{A}(m)} 
\ \ = \ \ \int d\varphi \ e^{-N\,A(\varphi) + \text{const}}  \]
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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: 
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\[ E_k \ = \sqrt{ \left(\epsilon_k + 2g\frac{m}{V} \right) \epsilon_k} \]  
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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}

{\bf Tip:} Only Stirling's approximation for $\ln((N-m)!)$ is requested/required/allowed in this question.

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