The H-functional monotonically decreases toward equilibrium — and the velocity distribution converges to Maxwell-Boltzmann
Boltzmann defined the H-functional: H = ∫ f(v) ln f(v) dv where f(v) is the velocity distribution. He proved from the Boltzmann equation (using the Stosszahlansatz — molecular chaos assumption) that dH/dt ≤ 0, and H = 0 only for the Maxwell-Boltzmann distribution.
Note: −H is essentially Boltzmann's entropy. Loschmidt's objection: since mechanics is time-reversible, how can H always decrease? Boltzmann's answer: it uses the molecular chaos assumption which breaks time-reversal symmetry by assuming pre-collision correlations are absent.