Boltzmann's H-Theorem 2D: A Toy Universe's Arrow of Time

A sealed 2D box of colliding disks is the smallest possible "universe" that still obeys real statistical mechanics: total energy is exactly conserved by every elastic collision, yet the Boltzmann entropy of the speed distribution climbs from an ordered starting condition toward its equilibrium maximum and stays there. This simulation runs that system for real — real 2D elastic scattering between equal masses, a live Gibbs/Boltzmann entropy computed from the actual speed histogram every frame, and a theoretical Rayleigh equilibrium curve (the correct 2D analogue of the Maxwell–Boltzmann distribution) drawn alongside it so you can watch the gas converge. An optional self-gravity mode adds a softened pairwise attraction so you can also see how local structure (clusters) can emerge from chaotic motion without breaking the second law, since it is the entropy of the whole system, not any one region, that must never decrease. Drag to pan and scroll to zoom the box view; every slider changes the physics live.