PX284 - E1b - entropy

S = - k_{B} \sum_{i} P_{i} \ln P_{i}

\begin{aligned} P_{i} & = \frac{\exp (- β E_{i})}{Z} \\ ∴ \ln P_{i} & = - β E_{i} - \ln Z \end{aligned}

⟹ S = k_{B} \sum_{i} P_{i} (β E_{i} + \ln Z) = k_{B} (β U + \ln Z)

note: $U = \sum_{i} E_{i} P_{i}$ and $\sum_{i} P_{i} = 1$