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\sect{Polarization of two-spheres system inside a tube}

Consider two spheres in a very long hollow tube of length ${L}$.
The mass of each ball is ${\mathsf {m}}$, 
the charge of one ball is ${-q}$, 
and the charge of the other one is ${+q}$. 
The ball radius is negligible, 
and the electrostatic attraction between the spheres is also negligible. 
The spheres are rigid and cannot pass through each other. 
The spheres are attached by a drop of water. Due to the surface tension 
there is an attraction force ${\gamma}$ that does not depend on the distance.  
Additionally there is an applied external electric field~$f$.  
The temperature is~${T}$.


\begin {itemize}

\item[(a)]
Write the hamiltonian ${H\left(p_{1},p_{2},x_{1},x_{2}\right)}$ of the system. 
Rewrite it also in terms of center-of-mass and distance ${r=|x_2-x_1|}$ coordinates.

\item[(b)]
Calculate the partition function ${Z(\beta,f)}$ assuming that the 
drop is not teared out. What is the condition for that?

\item[(c)]
Find the probability density function of $\rho(r)$, 
and calculate the average distance ${\langle r \rangle}$.
  
\item[(d)]
Find the polarization ${\tilde{P}}$ as a function of ${f}$.

\item[(e)]
Expand the polarization up to first order in the field, 
namely ${\tilde{P}(f)=\tilde{P}(0)+\chi f + \mathcal{O}\left(f^{2}\right)}$.

\end {itemize}

Express your answers with ${L, \mathsf{m}, q, \gamma, T, f}$.


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