Take \(x = 1/(Wp_i)\) where \(W\) is the number of microstates: \[\begin{align} \begin{aligned} \ln \left[\frac{1}{Wp_i}\right] &\le \frac{1}{Wp_i} - 1 \quad \textrm{for}\;\textrm{all}\; i = 1,\dots, ...Take \(x = 1/(Wp_i)\) where \(W\) is the number of microstates: \[\begin{align} \begin{aligned} \ln \left[\frac{1}{Wp_i}\right] &\le \frac{1}{Wp_i} - 1 \quad \textrm{for}\;\textrm{all}\; i = 1,\dots, W. \\ \sum_{i=1}^W p_i \ln \left[\frac{1}{Wp_i}\right] &\le \sum_{i=1}^W \left(\frac{1}{W} - p_i\right) \\ - \sum_{i=1}^W p_i \ln W - \sum_{i=1}^W p_i \ln p_i &\le 1 - 1 = 0 \\ - k_b \sum_{i=1}^W p_i \ln p_i &\le k_b \ln W. \end{aligned}\end{align}\] Moreover, the equality holds if and only if \(p_…
