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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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