The Maxwell Boltzmann distribution is the spread of speeds that elastic collisions settle a gas into at temperature T.
In two dimensions each velocity component is Gaussian with variance kT/m, so the speed has density f(v) = (m/kT) v e^(−mv²/2kT), peaking at √(kT/m). Heating the gas or lightening the particles moves the peak right.
In three dimensions the shell of equal speeds grows as v² rather than v, which gives the familiar v² e^(−mv²/2kT).
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