= Maxwell-Boltzmann velocity distribution
{c}
{title2=$f(\mathbf v)$}
= Maxwell velocity distribution
{c}
{synonym}
In the gas rest frame, the <Maxwell-Boltzmann velocity distribution> is the <probability density function> per unit volume in <velocity> space:
$$
f(\mathbf v)=
\left(\frac{m}{2\pi k_BT}\right)^{3/2}
\exp\left(-\frac{m|\mathbf v|^2}{2k_BT}\right)
=\frac{e^{-|\mathbf v|^2/(2s^2)}}{(2\pi s^2)^{3/2}},
\qquad \int_{\mathbb R^3}f(\mathbf v)\,d^3\mathbf v=1.
$$
Thus $f(\mathbf v)\,d^3\mathbf v$ gives the <probability> of a small three-dimensional <velocity> region. For a gas with bulk <mean velocity> $\mathbf V$, replace $\mathbf v$ by $\mathbf v-\mathbf V$; 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
$$
q(v_x)=\frac{1}{\sqrt{2\pi}s}e^{-v_x^2/(2s^2)},
\qquad \langle v_x\rangle=0,
\qquad \langle v_x^2\rangle=s^2=\frac{k_BT}{m}.
$$
The factorization $f(\mathbf v)=q(v_x)q(v_y)q(v_z)$ 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. https://farside.ph.utexas.edu/teaching/sm1/lectures/node72.html[Fitzpatrick, component distribution and factorization]{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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