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It is shown that the statistical entropy coincides with the thermodynamic expression for entropy.
We note note that, for a system in contact with heat bath, the probability of a microstate with energy \(E_k\) is $P(E_k) = Z^{-1} e^{-\beta E_k}$. The expression \begin{equation} S=-k \sum_k P(E_k) \ln P(E_k). \end{equation} for entropy will be used to obtain an expression for the entropy in terms of the canonical partition function \(Z\) and energy \(U\).
\begin{eqnarray}
S &=& -k \sum_k P(E_k) \ln P(E_k)\\ &=& - k \sum_k P(E_k) \beta\Big(-\ln Z - E_k \Big)\\ &=& k \ln Z + k \beta \sum_k P(E_k) E_k\\
&=& k \ln Z + k\beta U\end{eqnarray}which coincides with another expression for entropy which can derived using \(TdS\) equations.
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