Boltzmann entropy

The Boltzmann entropy of a macrostate is

where $\Omega$ is the number of microstates compatible with the macroscopic constraints, and $k_B$ is Boltzmann’s constant.

Interpretation

Entropy measures the logarithm of the phase space volume (or state count) accessible to the system. Higher entropy means more microscopic configurations are compatible with the observed macroscopic state.

Properties

• Extensivity: For independent subsystems, $\Omega_{\text{tot}} = \Omega_1 \Omega_2$, so $S_{\text{tot}} = S_1 + S_2$.
• Non-negativity: $S \geq 0$ (since $\Omega \geq 1$).
• Maximum at equilibrium: Isolated systems evolve toward states of maximum entropy.

Relation to Gibbs entropy

For the microcanonical ensemble with uniform distribution, the Gibbs entropy $S = -k_B \sum_i p_i \ln p_i$ reduces to the Boltzmann form.

Historical note

Boltzmann’s formula $S = k \log W$ is inscribed on his tombstone in Vienna.