Maxwell-Boltzmann velocity distribution ($f(\mathbf{v})$)

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

The Maxwell speed distribution is the probability density function for the magnitude $v=|\mathbf{v}|$. Unlike a signed velocity component, speed cannot be negative:For $0\le a<b$, the probability of a speed in $[a,b]$ is ${\int }_a^{b}g(v)dv$, and ${\int }_0^{\infty }g(v)dv=1$. The density has units of inverse speed, so its height alone is not a probability. ^[ref]

All velocities with magnitudes from $v$ to $v+dv$ occupy a thin spherical shell of volume $4\pi v^{2}dv$ in velocity space. The isotropy of the Maxwell-Boltzmann velocity distribution makes its density constant over each shell, giving $g(v)=4\pi v^{2}f(v{\mathbf{e}}_x)$ for any fixed unit direction ${\mathbf{e}}_x$. A larger radius offers more directions and more volume. The increasing shell factor initially wins over the decreasing Boltzmann factor, so $g(0)=0$ and the most probable speed is positive. This is the geometric reason that the peaks of the vector and speed densities disagree.

With the dimensionless quantity $u=v/s$, every such gas has the same rescaled speed density:For comparison, $w=v_x/s$ has the standard normal distribution density $\varphi (w)=e^{-w^2/2}/\sqrt{2\pi }$ on the whole real line. The Maxwell speed distribution is neither the positive half of this curve nor the distribution of $|v_x|$: it combines three components.

The first two moments of the Maxwell speed distribution give the mean and root mean square:Consequently the mean translational kinetic energy is $m⟨v^2⟩/2=3k_{B}T/2$, as required by the equipartition theorem. ^[ref] The mean speed weights fast particles more than the density peak does, and the root mean square weights them more strongly still.

Changing temperature or particle mass only changes $s\propto \sqrt{T/m}$. At fixed mass, four times the temperature doubles every characteristic speed; at fixed temperature, four times the mass halves them. A stretched density becomes lower so that its area stays one. For two species at the same temperature, equal mean translational kinetic energy therefore means different typical speeds.

The most probable speed is the mode, or density maximum, of the Maxwell speed distribution:For small equal-width speed bins, the bin near $v_{\mathrm{mp}}$ contains the largest expected fraction of particles. This does not mean that any exact speed has positive probability, or that every particle has this speed. The three common summaries obeyso $v_{\mathrm{mp}}<⟨v⟩<v_{\mathrm{rms}}$. The first describes a density peak, the second the mean speed, and the third the root mean square associated with kinetic energy. ^[ref]

For a concrete scale $s=300\mathrm{m}{\mathrm{s}}^{-1}$, these are approximately $424$, $479$, and $520\mathrm{m}{\mathrm{s}}^{-1}$. The most probable velocity vector and each most probable component remain zero in the gas rest frame. A zero vector labels one small region of velocity space, whereas a positive speed labels an entire spherical shell.

A short check uses the derivative of the logarithm of the Maxwell speed distribution for $v>0$:It is positive below $\sqrt{2}s$ and negative above it, locating the unique peak. The factor $v^2$ from the spherical shell supplies the term $2/v$; omitting it would incorrectly move the speed peak to zero.

Doppler broadening is the spread of an observed spectral line caused by the distribution of emitter velocities along the line of sight.

A Maxwellian emitter has Gaussian Doppler frequency FWHM $\Delta {\nu }_D/\nu =\sqrt{8k_{B}T\ln 2/(m{c}^2)}$. Equating this width to the minimum adjacent weak-field sodium component separation $2{\mu }_{B}B/(3h)$ gives the title resolution scale. A Gaussian blend has no unique resolution threshold: disappearance of individual components and disappearance of the whole broadened multiplet are different criteria. Instrumental resolution, pressure broadening, selected polarization and line strengths must be specified for a sharper prediction.

If nonrelativistic emitters of mass $m$ have rest-frame angular frequency ${\omega }_0$, then the one-dimensional Maxwell-Boltzmann velocity distribution produces a Gaussian line with standard deviationMeasuring the line width therefore determines the gas temperature when nonthermal broadening is negligible or separately modeled.

A purely thermal hydrogen absorption line has velocity-space optical depth proportional to $\exp [-(v-v_0{)}^2/b^2]$ with the displayed width. Additional independent Gaussian broadening contributes positive terms to $b^2$. Fitting the lower envelope of width against neutral-hydrogen column density, with numerical calibration for density and pressure smoothing, estimates the temperature-density relation of photoionized intergalactic gas. A single broad line only bounds the temperature from above when its nonthermal width is unknown.