the Maxwell-Boltzmann distribution
How fast are the molecules in a gas moving? Not all at one speed, but spread out over a characteristic, lopsided bell-shaped curve. Some molecules crawl, some race; the distribution tells you what fraction sit in each range of speed at a given temperature. It was the first great triumph of the statistical view of heat.
There are two related statements. First, the velocity components are Gaussian: the probability density for a component v_x is proportional to e^(-m v_x^2 / 2 k_B T), a direct Boltzmann factor for the kinetic energy. Second, the speed distribution multiplies this by the 4 pi v^2 surface of a spherical shell in velocity space: f(v) = 4 pi (m / 2 pi k_B T)^(3/2) v^2 e^(-m v^2 / 2 k_B T). The most probable speed is sqrt(2 k_B T / m), the mean is sqrt(8 k_B T / pi m), and the root-mean-square is sqrt(3 k_B T / m), consistent with (1/2) m <v^2> = (3/2) k_B T.
It underlies effusion, reaction rates, the small tail of very fast molecules that allows evaporation and fusion to proceed, and the Doppler broadening of spectral lines. Honestly, it is the classical, dilute limit. The full quantum result is the Fermi-Dirac or Bose-Einstein distribution; Maxwell-Boltzmann emerges only when the mean occupation of each state is tiny, that is when n lambda^3 << 1, dilute and hot, with the thermal de Broglie wavelength far smaller than the interparticle spacing.
For nitrogen at 300 K the most probable speed is about 420 m/s; the small high-speed tail of the distribution, though rare, drives evaporation and sets chemical reaction rates.
A rising v^2 factor and a falling exponential together give the peaked speed curve.
The peak of the speed distribution comes from the competition between the rising v^2 phase-space factor and the falling Boltzmann exponential; the velocity-component distribution itself peaks at zero.