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\sect{Pressure by particle in a spring-box system}
A spring that has an elastic constant ${K}$ and
natural length ${L}$ is connected between a wall
at $x=0$ and a piston at~${x=X}$.
A classical particle of mass ${\mathsf{m}}$ is attached
to the middle point of the spring.
The system is at equilibrium, the temperature is~${T}$.
\begin {itemize}
\item[(a)]
Write the Hamiltonian (careful!!!).
\item[(b)]
Write the integral that defines the partition function ${Z\left(\beta,X\right)}$.
\item[(c)]
Write a formal expression for the force ${F}$ on the piston.
\item[(d)]
Find elementary expressions (that do not involve exotic functions) in the limits
of high and low temperatures. Explain the results that you get.
\end {itemize}
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