%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}


%%begin{figure}
\putgraph{Ex205}
%%\caption{}\label{}
%%end{figure}