Poincaré (1890): any bounded conservative system will return arbitrarily close to its initial state after a sufficiently long time. This seems to contradict the Second Law — entropy cannot decrease yet gas particles "reset". The resolution: recurrence times are astronomically large (≫age of universe), and the H-theorem describes most-probable, not absolute, behavior.
Time: 0 | Recurrences: 0 | Best match: —
Initial state is marked. When all particles are within ε of their initial positions (green glow), a recurrence is recorded. The histogram shows recurrence time distribution. For N particles in volume V, expected recurrence time ∼ e^N.