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\sect{Particle on a ring with electric field}
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[(a)]
Write the Hamiltonian ${H\left(\theta,p\right)}$ of the particle.
\item[(b)]
Calculate the partition function ${Z\left(\beta,{\cal E}\right)}$.
\item[(c)]
Write an expression for the probability distribution
${\rho\left(\theta\right)}$.
\item[(d)]
Calculate the mean position ${\langle x \rangle}$ and ${\langle y
\rangle}$.
\item[(e)]
Write an expression for the probability distribution
${\rho\left(x\right)}$. Attach a schematic plot.
\item[(f)]
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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