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A magneto-optical trap combines laser cooling with a position-dependent magnetic field and suitable beam electromagnetic polarization to produce a restoring force. Its spatial confinement is additional to the velocity damping of uniform optical molasses.

The mass flow rate through an oriented cross-section is the integral of mass density times normal velocity. For constant density it is density times the volumetric flow rate. In a steady pipe without leaks, conservation of mass makes this rate independent of cross-section.

Porous-matrix drag depends on fluid velocity relative to the matrix, so it is not invariant under adding a uniform fluid velocity while leaving the matrix fixed. In a swimmer frame with stationary laboratory matrix, the drag is proportional to $\mathbf{u}-U{\mathbf{e}}_x$. A constant shift can alternatively be absorbed into a linear pressure term, whose physical interpretation must then be retained.

For a nonrotating axisymmetric galactic distribution function even in $L_z$, reverse every retrograde azimuthal velocity. The resulting distribution is $F_{+}(E,L_z)=2F(E,L_z)\Theta (L_z)$: it preserves density and even velocity moments but makes the streaming velocity $⟨|v_{\varphi }|⟩$. Dependence on conserved $E,L_z$ preserves stationarity. A centered Gaussian distribution of width $\sigma$ becomes a half-normal distribution, with mean $\sqrt{2/\pi }\sigma$.

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 iswhere $k_B$ 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 $k_{B}T\ll m{c}^2$ 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 $n{\lambda }^3\ll 1$, 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. ^{[ref] [ref]}

The connection to Maxwell–Boltzmann statistics is the Boltzmann factor for translational kinetic energy, $E_{\mathrm{kin}}=m|\mathbf{v}{|}^2/2$. Normalizing $e^{-E_{\mathrm{kin}}/(k_{B}T)}$ over velocity gives a probability density; an occupation factor $e^{(\mu -E)/(k_{B}T)}$ instead specifies a mean occupation, with chemical potential $\mu$. Their normalizations serve different purposes. ^[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 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. ^[ref]

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.

For a charged particle minimally coupled to a magnetic vector potential, the mechanical momentum is $\mathbf{\pi }=\mathbf{p}-q\mathbf{A}$, with $\mathbf{p}$ the canonical momentum. It is the mass times velocity in nonrelativistic mechanics.

At fixed position a two-integral distribution depends on $v_R,v_z$ only through $v_R^2+v_z^2$. Swapping these integration variables makes their second moments equal. Their odd moments and mixed second moments vanish by reflection symmetry. For an even function of $L_z$ the azimuthal velocity is also centered, so all mixed entries of the velocity ellipsoid vanish.