Back

Maxwell-Boltzmann Emergence from Hard-Disk Collisions

What you are seeing: 80 hard disks in a 2D box, all starting at the same speed v0v_0 but with random directions (top panel). Pairwise elastic collisions redistribute the speeds. Despite starting with a delta function δ(∣v∣−v0)\delta(|v| - v_0), the distribution relaxes to the 2D Maxwell-Boltzmann form p(v)=(v/σ2)exp⁡(−v2/2σ2)p(v) = (v / \sigma^2) \exp(-v^2 / 2\sigma^2) where σ=v0/2\sigma = v_0 / \sqrt{2}. The bottom panel shows the running speed histogram with the analytic MB curve overlaid.

The energy is exactly conserved by elastic collisions; only the distribution shape evolves. Some particles slow down close to zero while others speed up to higher than the initial v0v_0. The mean speed converges to σπ/2≈0.886 v0\sigma \sqrt{\pi / 2} \approx 0.886 \, v_0, lower than the initial mean because the distribution develops a long high-speed tail. The most probable speed is σ=v0/2≈0.707v0\sigma = v_0 / \sqrt{2} \approx 0.707 v_0.

Figure 1. Hard-disk gas in 2D. Method: time-stepped pairwise elastic collisions with reflecting walls; histogram updated each render.
speed3

WHAT TO TRY

  • Let the disks collide: they start at a single speed, but elastic collisions reshuffle energy until the histogram settles onto the 2D Maxwell-Boltzmann curve, with no thermostat imposing it.
  • Compare the live mean speed against the prediction: equilibrium emerges purely from momentum-conserving collisions, the microscopic origin of temperature.
  • Speed it up and watch the tail fill in: the rare fast disks of the high-energy tail are exactly the ones that drive reaction rates and evaporation.