
\sect{Baruch's  A22.}

Consider a one-dimensional classical gas of ${N}$ particles in a
length ${L}$ at temperature ${T}$. The particles have mass $m$ and
interact via a 2-body "hard sphere" interaction (${x_{i}}$ is the
position of the $i$-th particle):
\begin{eqnarray}
V(x_{i}- x_{j}) &=& \infty  \qquad \qquad    |x_{i}-x_{j}|<a \nonumber\\
&=& 0 \qquad \qquad \,\,\, |x_{i}- x_{j}|>a \nonumber
\end{eqnarray}

\begin{itemize}
\item [(a)]
Evaluate the exact free energy F(T,L,N).
\item [(b)]
Find the equation of state and identify the first virial
coefficient; compare with its direct definition.
\item [(c)]
Show that the energy is ${E=Nk_{B}T/2}$. Why is there no effect of
the interactions on ${E}$ ?
\item [(d)]
In three dimensions $V(|{\bf r}_{i}-{\bf r}_{j}|)$ is defined as
above with ${\bf r}$ the position vector. Explain why should the
effective volume satisfy $V>V_{eff}>V-v_0N$ where $v_0$ is the excluded
volume for each particle (i.e. $4\pi a^3/3$, where $a$ is the particle's diameter) and $Z=V_{eff}^N/(\lambda^{3N}N!)$. Find
$E(T)$ and explain how can it be consistent with the presence of a
phase transition.\\
\end{itemize}

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