Maxwell-Boltzmann emergence from hard-disk collisions

What you are seeing: 80 hard disks in a 2D box, all starting at the same speed $v_0$ but with random directions (top panel). Pairwise elastic collisions redistribute the speeds. Despite starting with a delta function $\delta(|v| - v_0)$, the distribution relaxes to the 2D Maxwell-Boltzmann form $p(v) = (v / \sigma^2) \exp(-v^2 / 2\sigma^2)$ where $\sigma = v_0 / \sqrt{2}$. 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 $v_0$. The mean speed converges to $\sigma \sqrt{\pi / 2} \approx 0.886 \, v_0$, lower than the initial mean because the distribution develops a long high-speed tail. The most probable speed is $\sigma = v_0 / \sqrt{2} \approx 0.707 v_0$.