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\sect{The Helmholtz function}

We define ${F\equiv -\frac{1}{\beta} \ln Z_{\beta}}$

Prove:
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\[E=F+TS\]
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\begin{displaymath}
\left\{\begin{array}{ll}
S=-\frac{\partial F}{\partial T}& \textrm{}\\
y=-\frac{\partial F}{\partial X}& \textrm{}\\
\end{array}\right.
\end{displaymath}
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It is possible to rely on the expressions that express ${X,S,y}$ by
the distribution function and by the ${F}$ definition.



