Boltzmann's entropy formula

Remarks

The proof is one line: $S = -k_B \sum_{i=1}^{W} \frac{1}{W} \ln \frac{1}{W} = k_B \ln W$. Boltzmann used $S = k_B \ln W$ as the definition of the entropy of a macrostate, and the formula is engraved on his tombstone in Vienna. Example: $N$ independent spins with exactly $K$ pointing up give $W = \binom{N}{K}$. By Stirling's formula, $S \approx -N k_B (x \ln x + (1 - x) \ln (1 - x))$ for large $N$, where $x = K/N$. The formula needs the microstates to be equally likely; for a non-uniform distribution the Gibbs entropy is strictly smaller than $k_B \ln W$.

Going deeper

For a student

The formula links mechanics to thermodynamics: from the count $W(E, V, N)$ one gets $S$, and then temperature by $1/T = \partial S / \partial E$ and pressure by $p/T = \partial S / \partial V$. It is the special case of the Gibbs entropy for the uniform distribution, and the uniform distribution maximizes the Gibbs entropy on $W$ states. For continuous systems, $W$ is a phase-space volume divided by $N! \, h^{3N}$; this choice gives the Sackur–Tetrode equation for the ideal gas.