 Maxwell-Boltzmann velocity distribution

ID: maxwell-boltzmann-velocity-distribution

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Maxwell-Boltzmann velocity distribution by

Codex  0 Created 2026-10-03 Updated 2026-10-20

In the gas rest frame, the Maxwell-Boltzmann velocity distribution is the probability density function per unit volume in velocity space:

f (v) = (\frac{m}{2 π k _{B} T})^{3/2} exp (- \frac{m ∣ v ∣ ^{2}}{2 k _{B} T}) = \frac{e ^{- ∣ v ∣^{2} / (2 s^{2})}}{( 2 π s ^{2} ) ^{3/2}}, \int_{R^{3}} f (v) d^{3} v = 1.

Thus

f (v) d^{3} v

gives the probability of a small three-dimensional velocity region. For a gas with bulk mean velocity

V

, replace

v

by

v - V

; the Maxwell speed distribution then describes the magnitude relative to that bulk motion.

For any one Cartesian velocity component

v_{x}

, integrating out the other two gives

q (v_{x}) = \frac{1}{2 π s} e^{- v_{x}^{2} / (2 s^{2})}, ⟨ v_{x} ⟩ = 0, ⟨ v_{x}^{2} ⟩ = s^{2} = \frac{k _{B} T}{m} .

The factorization

f (v) = q (v_{x}) q (v_{y}) q (v_{z})

establishes independence: all three components are centered normal random variables, with variance

s^{2}

. Both the component density

q

and the vector density

f

have their largest values at zero. This concerns densities for small equal-sized regions; any single exact velocity still has probability zero. ^[ref]

The standard deviation

s

measures thermal motion even though the mean velocity is zero. Positive and negative components cancel in the mean, but their squares add. This is why thermal Doppler broadening measures a nonzero line width in a gas with no bulk motion.

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