Maxwell Velocity Distribution

The Maxwell-Boltzmann distribution describes the distribution of particle speeds in an ideal gas at thermal equilibrium. Derived independently by James Clerk Maxwell (1860) and Ludwig Boltzmann (1868), it is a cornerstone of statistical mechanics and kinetic theory.

The Distribution Function

The probability that a molecule has a speed between $v$ and $v + dv$ is:

$f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2}\, v^2 \, \exp\!\left(-\frac{mv^2}{2k_B T}\right)$

The shape of this curve comes from two competing effects:

Factor	Expression	Meaning
Phase-space factor	$v^2$	More ways to arrange a velocity vector at higher speeds (surface area of a sphere in velocity space)
Boltzmann factor	$e^{-mv^2/2k_BT}$	Exponential penalty for high kinetic energy

The peak arises where these two factors balance.

Three Characteristic Speeds

• Most probable speed $v_p = \sqrt{2k_BT/m}$ — the peak of the distribution
• Mean speed $\langle v \rangle = \sqrt{8k_BT/\pi m}$ — the arithmetic average, always slightly above $v_p$
• RMS speed $v_{\text{rms}} = \sqrt{3k_BT/m}$ — root-mean-square speed, related to the average kinetic energy via $\langle E_k \rangle = \tfrac{1}{2}m v_{\text{rms}}^2 = \tfrac{3}{2}k_BT$

They always satisfy $v_p < \langle v \rangle < v_{\text{rms}}$ regardless of gas or temperature.

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Things to Try

1. Increase temperature from 100 K to 1000 K — watch the peak flatten and shift right. Higher temperature means molecules are faster on average, but the distribution also broadens.
2. Switch between gases at the same temperature — lighter molecules (H₂, He) move much faster than heavier ones (N₂, O₂).
3. Compare the three speed lines — notice they always appear in the same order: $v_p$ (green) < $\langle v \rangle$ (cyan) < $v_{\text{rms}}$ (yellow).
4. Ghost reference — the faint dotted curve always shows the same gas at 300 K, making it easy to see how your settings differ from room temperature.

Real-World Applications

• Atmospheric escape: On small, warm bodies (like the Moon), the tail of the speed distribution for light gases exceeds escape velocity, which is why the Moon has essentially no atmosphere. This speed distribution is essential to understanding stellar distances and atmospheric physics.
• Thermal neutrons: Nuclear reactors moderate fast neutrons to thermal energies; the resulting speed distribution is Maxwell-Boltzmann at the moderator temperature.
• Chemistry: Reaction rates depend on the fraction of molecules with kinetic energy above the activation energy — the high-speed tail of this distribution (Arrhenius equation).
• Stellar atmospheres: Spectral line broadening due to thermal Doppler shifts follows a Gaussian derived from this distribution.

Assumptions of the model — The Maxwell-Boltzmann distribution assumes an ideal gas: no intermolecular forces, elastic collisions only, and thermal equilibrium. For very dense gases, low temperatures, or quantum particles, one must use the Fermi-Dirac (fermions) or Bose-Einstein (bosons) distributions instead.