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\sect{One dimensional hard sphere gas}

${N}$ spheres with diameter ${a}$ are threaded over a wire of length ${L}$. 
Assume ${N\gg1}$ but ${Na \ll L}$. The system is in thermic equilibrium, temperature ${T}$. 
Find the force ${F}$ that operates on the edges of the wire. Write the result
in the shape ${F=NT/L_{eff}$. Express ${L_{eff}}$ using the
data and explain it's physical meaning. Hints:

\begin {itemize}

\item[(a)]
While calculating the distribution function, notice that if the
beads permutation were permitted, it was causing  ${Z\rightarrow N!Z}$.

\item[(b)]
Assume that a typical distance between two beads is much bigger than ${a}$.

\item[(c)]
To calculate a product ${A=\prod_{n=1}^{N}a_{n}}$ 
look at the sum ${\ln A}$, and use reasonable approximations.

\end {itemize}


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