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\sect{Particle on a ring with electric field}

A particle of mass ${\mathsf{m}}$ and charge ${e}$ 
is free to move on a ring of radius ${R}$.
The ring is located in the ${(x,y)}$ plan. 
The position of the particle on the ring is
${x=R\cos\left(\theta\right)}$ and ${y=R\sin\left(\theta\right)}$.
There is an electric field ${\cal E}$ is the $x$ direction. 
The temperature is ${T}$.

\begin {itemize}

\item[(1)]
Write the Hamiltonian ${H\left(\theta,p\right)}$ of the particle.

\item[(2)]
Calculate the partition function ${Z\left(\beta,{\cal E}\right)}$.

\item[(3)]
Write an expression for the probability distribution ${\rho\left(\theta\right)}$.

\item[(4)]
Calculate the mean position ${\langle x \rangle}$ and ${\langle y \rangle}$.

\item[(5)]
Write an expression for the probability distribution ${\rho\left(x\right)}$. 
Attach a schematic plot.

\item[(6)]
Write an expression for the polarization. 
Expand it up to first order in ${{\cal E}}$, 
and determine the susceptibility.

\end {itemize}

\[\frac{1}{2\pi}\int_{0}^{2\pi} \exp\left(z\cos\left(\theta\right)\right)d\theta = I_{0}\left(z\right)\]
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\[I_{0}'\left(z\right) = I_{1}\left(z\right)\]
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\[I_{0}\left(z\right) = 1+\left(\frac{1}{4}\right)z^{2}+\left(\frac{1}{64}\right)z^{4+...}\]

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