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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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