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

Toy Universe

Particle count 150 Applies on Reset / Reseed

Initial order 90% Low = all particles start with the same speed (low entropy). Applies on Reset / Reseed.

Time scale 1.0×

Self-gravity (emergence)

Live Readouts

Kinetic energy Σ½v²

0.00

Total energy (E)

0.00

Entropy S (nats)

0.00

S / S_max

Entropy vs. simulated timet = 0.0

How it works

N point particles bounce elastically off the walls of a box and off each other. Every collision is a real 3D elastic scatter between equal masses:

v1' = v1 − [(v1−v2)·(x1−x2) / |x1−x2|²] (x1−x2)
v2' = v2 − [(v2−v1)·(x2−x1) / |x2−x1|²] (x2−x1)

This exchange conserves total momentum and total kinetic energy exactly — nothing in the microscopic rule is irreversible. The box starts in a low-entropy state: every particle launched with (nearly) the same speed, just random directions. As collisions randomize the speeds, they spread toward the classic Maxwell–Boltzmann distribution.

Entropy is measured the way Boltzmann and Gibbs defined it — coarse-grain the particle speeds into K bins, read off the occupation fractions p_k, and sum:

S = −Σ p_k ln p_k   (k_B = 1, this is Boltzmann's H-function with a sign flip: S = −H)
Second law:  dS/dt ≥ 0   for an isolated system

Energy stays flat while entropy climbs — that is the whole point. Reversing every velocity would, in principle, walk the system back toward its ordered start, but the odds of that happening by chance are astronomically small once N is large: that asymmetry between a reversible rule and an irreversible outcome is the statistical arrow of time.

Self-gravity adds a soft pairwise attraction, integrated with a semi-implicit (symplectic) Euler step so total energy (kinetic + gravitational potential) stays approximately conserved even as the gas is no longer uniform. Switching it on lets you watch order re-emerge locally — particles clump into transient clusters — without violating the second law globally: local order can still grow as long as total entropy of the whole system does not fall.

Slow particle Fast particle

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