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\sect{State equations derived from $Z(T)$}

Make sure you'r well aware of the basic equations of the canonical
ensemble, and knows how to prove those equations for the state
functions.
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\[\left(*\right) Z\left(\beta,X\right)\equiv \sum_{r}e^{-\beta
E_{r}}\]
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\[E=-\frac{\partial \ln Z}{\partial \beta}\]
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\[y=\frac{1}{\beta}\,\frac{\partial \ln Z}{\partial X}\]
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\[F\left(T,X\right)\equiv -\frac{1}{\beta}\ln
Z\left(\beta,X\right)\]
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\[S=-\frac{\partial F}{\partial T}\]
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More definitions

(Heat capacity) ${C_{x}\equiv \frac{\partial E}{\partial T}|_{X}}$

(Generalized susceptibility) ${\chi\equiv \frac{\partial y}{\partial
X}}$

(*) for a classical particle
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\[\sum_{r}\mapsto \int \frac{dxdp}{2\pi}\,\,E_{r}\mapsto H\left(X
P\right)\]
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