The Ring
White Ball
Black Ball
Marker (flips ball)
Fraction of White Balls Over Time
The Paradox: Each ball moves clockwise and flips color when passing a marker.
The dynamics are deterministic and reversible (system returns to initial state after N steps).
Yet we observe macroscopic relaxation to equilibrium!
Simulation
Initial State
Parameters
Marker Placement
Time Step:
0
White Balls:
32
Fraction White:
100%
Poincaré Period:
—
Boltzmann's H-theorem: In the large N limit, the fraction of white balls
approaches f(t) = ½ + ½(1 - 2n/N)^t·(f₀ - ½) — exponential relaxation despite reversibility!