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\sect{Pressure by a 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~${x=X}$.
Consequently the force that acts of the piston is ${F_0=-K(X-L)}$.
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[(1)]
Write the Hamiltonian (be careful).
\item[(2)]
Write an expression for the partition function ${Z\left(\beta,X\right)}$.
The answer is an expression that may contain a definite integral.
\item[(3)]
Write an expression for the force ${F}$ on the piston.
The answer is an expression that may contain a definite integral.
\item[(4)]
Find a leading order (non-zero) expression for ${F-F_0}$
in the limit of high temperature.
\item[(5)]
Find a leading order (non-zero) expression for ${F-F_0}$
in the limit of low temperature.
\end {itemize}
Your answers should not involve exotic functions,
and should be expressed using $(X,L,K,\mass,T)$.
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