Boltzmann entropy formula (S = k log W)

The Boltzmann entropy formula, S = k_B ln Ω, defines the entropy of a macrostate as Boltzmann's constant times the natural logarithm of its multiplicity Ω, the number of microstates that realise it. It is the bridge between the microscopic world of molecular configurations and the macroscopic entropy that Clausius had defined through heat and temperature, dS = dQ_rev/T.

The logarithm is forced by the requirement that entropy be extensive (additive): when two independent systems combine, their multiplicities multiply (Ω₁Ω₂) but their entropies must add, and only the logarithm turns a product into a sum. Boltzmann's constant supplies the units, converting a dimensionless count into joules per kelvin. From this single formula one recovers the Sackur–Tetrode equation, the ideal-gas law, and the second law as a statement that isolated systems evolve toward macrostates of larger Ω.

The headstone version uses W (for the German Wahrscheinlichkeit, 'probability') in place of the modern Ω. It is carved on Boltzmann's tomb in the Vienna Zentralfriedhof.

Boltzmann developed the statistical interpretation of entropy in papers of 1872 and 1877; the compact form S = k log W was crystallised by Planck around 1900. Boltzmann died in 1906, and the formula was later inscribed on his monument.