Measurements
Cumulative A→B vs B→A imbalance0.0%
Position and direction
The funnel trades a narrower range of positions for a wider range of directions.
Before and after the funnel
Reversibility and equilibrium
Specular billiard dynamics is time-reversible. Reverse every velocity and the same microstate retraces its path.
Liouville's theorem says the phase-space distribution is carried without compression. A collimated ensemble may spread and look more uniform in position, but it does not literally become the equilibrium distribution in fine-grained phase space.
Under the ergodic assumption, long-time averages sample the equilibrium measure. Convergence of an instantaneous ensemble toward equilibrium is the stronger property of mixing, which this polygonal example does not assume.
Model: point particles at fixed speed, static specular walls, no friction or gravity, and no particle-particle collisions.