= Maxwell-Boltzmann distribution
{c}
{title2=$s=\sqrt{k_BT/m}$}
{wiki=Maxwell–Boltzmann_distribution}
The Maxwell-Boltzmann distribution describes the random <velocities> and <speeds> of particles in a classical <ideal gas> at <thermal equilibrium>. Individual particles keep changing <velocity> through <collisions>, while the equilibrium <probability distribution> stays fixed. For one species of particle <mass> $m$ at positive <temperature> $T$, the characteristic scale is
$$
s=\sqrt{\frac{k_BT}{m}},
$$
where $k_B$ is the <Boltzmann constant>. A signed <velocity> component follows a <normal distribution>; the nonnegative magnitude follows the <Maxwell speed distribution>. Their peaks differ because a <speed> measurement groups together all directions of motion.
Use the gas rest frame, three spatial dimensions, and <nonrelativistic particles>, with $k_BT\ll mc^2$ where $c$ is the <speed of light>. Assume a dilute gas whose interactions between encounters can be neglected in the <ideal gas> approximation and whose translational motion is classical. A sufficient condition for negligible quantum exchange effects is $n\lambda^3\ll1$, where $n$ is the <number density> and $\lambda$ the <thermal de Broglie wavelength>. Diluteness alone does not guarantee <thermal equilibrium>; the gas must have equilibrated. These are the assumptions behind the classical <Maxwell-Boltzmann velocity distribution>. https://farside.ph.utexas.edu/teaching/sm1/lectures/node72.html[Fitzpatrick, The Maxwell distribution]{ref} https://farside.ph.utexas.edu/teaching/sm1/lectures/node82.html[Fitzpatrick, Quantum statistics in the classical limit]{ref}
The connection to <Maxwell–Boltzmann statistics> is the <Boltzmann factor> for translational <kinetic energy>, $E_{\rm kin}=m|\mathbf v|^2/2$. Normalizing $e^{-E_{\rm kin}/(k_BT)}$ over <velocity> gives a probability density; an occupation factor $e^{(\mu-E)/(k_BT)}$ instead specifies a mean occupation, with <chemical potential> $\mu$. Their normalizations serve different purposes. https://farside.ph.utexas.edu/teaching/sm1/lectures/node81.html[Fitzpatrick, Maxwell-Boltzmann statistics]{ref}
This connects microscopic motion to measurements: a <gas thermometer> uses the <ideal gas law>, <collisions> and the <mean free path> govern transport, and measurements of <spectral lines> use <thermal Doppler broadening> to infer <temperature>. Each probes a different aspect of the same thermal motion.
In 1860, <James Clerk Maxwell> presented the component and <speed> laws in Part I, Proposition IV of \i[Illustrations of the Dynamical Theory of Gases], pp. 22–23. His argument combined <independence> of the three <velocity> components with <isotropy>, and he explicitly compared the component law with the <normal distribution> of observational errors. Independence is an assumption of that argument; perpendicular coordinate axes alone do not establish it. https://sites.pitt.edu/~jdnorton/teaching/2559_Therm_Stat_Mech/handouts/Maxwell_1860_annot.pdf#page=5[Maxwell (1860), original article scan with annotations]{ref}
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