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.
• 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.