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

A closed box of colliding particles conserves total energy exactly while its Boltzmann entropy climbs from an ordered start toward equilibrium — a real, numerically grounded demonstration of the second law and the arrow of time, with an optional self-gravity mode showing emergent clustering.

Thermodynamics3DAdvanced60 FPS📱 Mobile-adapted⇄ 2D version

A sealed box of colliding particles is the smallest possible "universe" that still obeys real physics: 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 3D elastic scattering between equal masses, a live Gibbs/Boltzmann entropy computed from the actual speed histogram every frame, and energy and entropy readouts you can watch decouple: one flat, one rising. 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.

⚙ Under the hood

A sealed box of elastically colliding particles conserves total energy exactly while its Boltzmann entropy climbs from an ordered start toward equilibrium, with an optional self-gravity mode showing emergent local clustering.

thermodynamicsentropystatistical-mechanicssecond-lawn-body3D

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install