What you are looking at
On the left is a 2-D box of gas molecules bouncing off the walls and — crucially — off
each
other, in perfectly elastic collisions that swap energy around. On the right, a live histogram
records how many molecules currently have each speed. Watch it settle into a fixed, lop-sided shape: the
Maxwell–Boltzmann distribution.
Why the speeds spread out
Even if every molecule started at the same speed, collisions would quickly scramble that — one comes away
fast, its partner slow. Energy is conserved overall, but it gets shared unevenly, and the system relaxes to
the most probable arrangement. In two dimensions the resulting distribution of speeds is
f(v) ∝ v · e^(−m v² / 2kT)
It starts at zero (nothing is perfectly still), rises to a peak, then tails off — there are always a few very
fast molecules but no upper limit. Press
"Start all same speed" and watch the single spike
collapse into this smooth curve within seconds: that is a gas
thermalising.
Three special speeds
The distribution has three landmark speeds, and they're always in the same order:
v_p = √(kT/m) < ⟨v⟩ = √(πkT/2m) < v_rms = √(2kT/m)
The
most probable speed v_p sits at the peak; the
average ⟨v⟩ is a little
higher; and the
root-mean-square v_rms — the one tied to the average kinetic energy and
hence the temperature — is higher still, pulled up by the fast tail.
Temperature is just average energy
Temperature is nothing more than the average kinetic energy of the molecules.
Heat the gas
and every molecule speeds up: the whole curve slides to the right and flattens (the fast tail stretches
out).
Cool it and the curve bunches back toward zero. The shape never changes — only its
scale.
Things to try
Heat and cool the gas and watch the histogram and the three markers slide together. Reset to a single speed
and time how fast collisions rebuild the bell curve. Add more molecules for a smoother, less noisy
histogram.