Distribution Function and Phase Space

A galaxy is a crowd of a hundred billion stars, and no one will ever integrate a hundred billion orbits. Yet the galaxy is not chaos: watch it long enough and its shape, its rotation, the spread of its stellar velocities all hold steady. What matters is never any single star but the density of stars — not in ordinary space alone, but in the six-dimensional space whose axes are the three positions $\mathbf{x}$ and three velocities $\mathbf{v}$ of a star. That density is the distribution function $f(\mathbf{x},\mathbf{v},t)$ , the central object of this article and of all stellar dynamics. Give me $f$ and I can hand back everything a telescope could measure: where the stars are, how fast they stream, how their motions correlate, how much unseen mass their orbits betray. Two ideas make the picture powerful. The first is a separation of scales: over a galaxy's life the deflections from individual star–star encounters are utterly negligible — the two-body relaxation time exceeds the age of the universe millions of times over — so each star simply falls freely in the smooth mean field $\Phi(\mathbf{x},t)$ of the whole system. The second is a conservation law: because stars are conserved and their flow through phase space is incompressible (Liouville's theorem), $f$ is constant along every orbit. That single sentence is the collisionless Boltzmann equation, the master equation of the subject — and the same machinery governs dark matter, photons, and neutrinos across the whole history of the cosmos.

Key takeaways

The distribution function $f(\mathbf{x},\mathbf{v},t)$ is the density of stars in six-dimensional phase space: $\mathrm{d}N = f\,\mathrm{d}^3x\,\mathrm{d}^3v$ .
Every observable — number density, mean streaming, velocity dispersion — is a velocity moment of $f$ ; the DF holds more than any finite set of moments.
For a collisionless system the phase-space flow is incompressible, so $f$ is conserved along orbits: the collisionless Boltzmann equation $\mathrm{d}f/\mathrm{d}t=0$ .
In equilibrium, Jeans theorem forces $f$ to depend on phase-space coordinates only through isolating integrals such as $E$ and $L$ .
A Maxwellian energy DF builds the singular isothermal sphere $\rho=\sigma^2/2\pi G r^2$ , whose flat rotation curve $v_c=\sqrt{2}\,\sigma$ was an early dark-halo template.
Key idea: a stellar system is completely specified by one scalar field on six-dimensional phase space, $f(\mathbf{x},\mathbf{v},t)$ ; everything observable is a moment of $f$ , and its evolution is fixed by the conservation of phase-space number along orbits.

The core insight

A stellar system is completely specified by a single scalar field on six-dimensional phase space, $f(\mathbf{x},\mathbf{v},t)$ ; everything observable is a moment of $f$ , and its evolution is fixed by the requirement that phase-space number be conserved along orbits.

Where this lives today

The distribution function stopped being an abstraction the day a satellite began measuring it. ESA's Gaia mission has charted positions and velocities for nearly two billion Milky Way stars — the first empirical map of the Galactic $f(\mathbf{x},\mathbf{v})$ — and it revealed the disk is not in equilibrium: the "Gaia snail," a spiral winding through the $(z,v_z)$ slice discovered in 2018, is phase mixing caught in the act, a bundle of stars shearing exactly as the Liouville figure predicts after a recent perturbation. On galaxy scales, Schwarzschild orbit-superposition and distribution-function modelling invert projected moments of $f$ to weigh supermassive black holes and map dark halos. Cold stellar streams such as GD-1 are Liouville's theorem written across the sky — debris that stays cold in phase space while stretching into thin arcs, constraining the Galactic potential and even dark subhalos. In cosmology, $N$ -body and Vlasov codes solve the collisionless Boltzmann–Poisson system in an expanding universe to grow the cosmic web, while the Tremaine–Gunn bound turns the maximum relic phase-space density into a lower limit on the dark-matter particle mass — helping rule out light neutrinos as the dark matter. Every one is a statement about how a phase-space distribution function is measured, modelled, or evolved.

The physical picture

Represent each star not by a dot in the sky but by a point in a six-dimensional space whose coordinates are its three positions and three velocities. A globular cluster becomes a cloud of $10^5$ such points; a galaxy, a cloud of $10^{11}$ . The distribution function is the local density of that cloud, smoothed over cells large enough to hold many stars yet small enough to resolve structure. Because stars are conserved, the cloud behaves like an incompressible fluid: it can be stirred, sheared, and folded, but its six-dimensional density is never squeezed or diluted. That is Liouville's theorem, and the first figure shows it in the slice $(x, v_x)$ we can draw — a patch of phase space carried by the flow, stretched thin in one direction exactly as much as it is compressed in the other, so its area, and hence $f$ , is unchanged.

Figure

Liouville's theorem in the x–vₓ slice of phase space: the flow carries a cell of ΔN = 40 stars from an upright square into a sheared, phase-mixed sliver, but the area Δx·Δv is conserved, so the phase-space density f = ΔN / Δx Δv stays fixed — squeeze in position and you must stretch in velocity

The distribution function is a translator between two languages. Sum it over velocities and you recover the number density $n(\mathbf{x})$ ; weight it by velocity first and you recover streaming and the random-velocity spread. But the DF holds strictly more than any of its moments: two galaxies with identical density and dispersion profiles can be built from completely different orbits — one nearly circular, the other plunging and radial — a difference invisible to the low-order moments and legible only in the full $f$ . This is why the distribution function, not the density, is the true state of a stellar system. The second figure shows what that buys: choose $f$ to depend on energy alone in Boltzmann form and Poisson's equation returns the singular isothermal sphere, whose rotation curve is dead flat at $v_c=\sqrt{2}\,\sigma$ — the earliest DF model of a dark halo.

Figure

The singular isothermal sphere as a distribution-function model of a flat rotation curve: a Maxwellian energy DF with dispersion σ = 150 km/s gives ρ ∝ r⁻² and enclosed mass M ∝ r, so the circular speed is v_c = √2 σ = 212 km/s at every radius — independent of r, the hallmark of a dark halo

The core equations

The distribution function is defined by counting: it is the number of stars per unit six-dimensional volume of phase space. Everything that follows is either a moment of this definition (an observable) or a law for how it evolves. We adopt the number convention here; many texts, including Binney & Tremaine, use a mass DF with $\rho=\int f\,\mathrm{d}^3v$ , differing only by the stellar mass $m$ .

\mathrm{d}N = f(\mathbf{x},\mathbf{v},t)\,\mathrm{d}^3x\,\mathrm{d}^3v \qquad (1)

where: $\mathrm{d}N$ — stars in the phase-space cell; $f$ — the phase-space number density; $\mathbf{x},\mathbf{v}$ — position and velocity; $\mathrm{d}^3x\,\mathrm{d}^3v$ — the six-dimensional volume element. Reading: $f$ is a non-negative density in six dimensions, of units (length·velocity) $^{-3}$ . Integrating $f$ over all velocities at a fixed position discards the velocity information and leaves configuration-space density — the zeroth moment, the bridge from the unobservable $f$ to what a telescope records.

n(\mathbf{x},t) = \int f(\mathbf{x},\mathbf{v},t)\,\mathrm{d}^3v \qquad (2)

where: $n$ — spatial number density of stars. Reading: summing over the whole velocity axis collapses the phase-space density to number per unit volume; $\rho = m\,n$ . The next two moments extract kinematics: weighting by velocity gives the mean streaming, and by the squared deviation from that mean the velocity-dispersion tensor, the stellar analogue of a pressure.

\overline{v_i}(\mathbf{x},t) = \frac{1}{n}\int v_i\, f\,\mathrm{d}^3v \qquad (3)

where: $\overline{v_i}$ — mean (bulk) velocity along direction $i$ ; $v_i$ — the $i$ -th velocity component. Reading: the number-weighted mean velocity is the streaming seen as a Doppler shift.

\sigma_{ij}^2(\mathbf{x},t) = \overline{(v_i-\overline{v_i})(v_j-\overline{v_j})} = \frac{1}{n}\int (v_i-\overline{v_i})(v_j-\overline{v_j})\,f\,\mathrm{d}^3v \qquad (4)

where: $\sigma_{ij}^2$ — the velocity-dispersion (stress) tensor, in general anisotropic. Reading: $\sigma_{rr}\neq\sigma_{\theta\theta}$ signals orbits biased toward radial or tangential motion — information no density profile can supply. Now the dynamics. In a collisionless system the DF is carried unchanged along each orbit, so its convective derivative vanishes; written with the mean-field acceleration $-\nabla\Phi$ , this is the collisionless Boltzmann equation, the master equation of stellar dynamics.

\frac{\mathrm{d}f}{\mathrm{d}t} = \frac{\partial f}{\partial t} + \mathbf{v}\cdot\nabla f - \nabla\Phi\cdot\frac{\partial f}{\partial \mathbf{v}} = 0 \qquad (5)

where: $\Phi(\mathbf{x},t)$ — the smooth mean gravitational potential; $-\nabla\Phi$ — the acceleration; $\mathrm{d}f/\mathrm{d}t$ — the derivative following a star's orbit. Reading: an observer drifting with the flow sees a fixed local density; only $\partial f/\partial t$ at a fixed point changes. The potential is not external — it is generated self-consistently by the very mass whose motion it governs, closing the system through Poisson's equation. The pair (5)+(6) is the collisionless Boltzmann–Poisson system every equilibrium galaxy model must solve.

\nabla^2 \Phi = 4\pi G \int f\, m\,\mathrm{d}^3v = 4\pi G\,\rho \qquad (6)

where: $\rho = m\,n$ — mass density; $G$ — Newton's constant. Reading: the DF sources the field through its zeroth moment, and the field steers the DF through (5) — a nonlinear feedback loop. In a steady state $\mathrm{d}f/\mathrm{d}t=0$ forces $f$ to be built only from quantities conserved along orbits. This is Jeans theorem, the organizing principle for equilibrium models.

f = f(I_1, I_2, \ldots, I_k) \qquad (7)

where: $I_k$ — an isolating integral of motion (energy $E$ , angular momentum $L$ or $L_z$ , a third integral $I_3$ ). Reading: rather than guess a function of six variables, one writes $f$ from a few conserved labels — $f(E)$ for isotropic spheres, $f(E,L)$ for anisotropic ones. Finally, the inverse problem — recovering $f$ from an observed density — has a closed form for isotropic spheres. Eddington's formula is an Abel inversion turning $\rho(\Psi)$ into $f(E)$ .

f(E) = \frac{1}{\sqrt{8}\,\pi^2}\frac{\mathrm{d}}{\mathrm{d}E}\int_{0}^{E}\frac{\mathrm{d}\rho}{\mathrm{d}\Psi}\frac{\mathrm{d}\Psi}{\sqrt{E-\Psi}} \qquad (8)

where: $\Psi = -\Phi$ — the relative potential (positive, increasing inward); $E = \Psi - \tfrac{1}{2}v^2$ — the relative energy per unit mass (positive for bound stars); $\rho(\Psi)$ — density as a function of relative potential. Reading: this returns the unique isotropic DF reproducing a given $\rho(r)$ — physical only if $f\geq0$ , the realizability test.

Derivation AThe collisionless Boltzmann equation from number conservation and Liouville's theorem

The goal is to derive equation (5) from two ingredients only: that stars are conserved, and that their phase-space flow is generated by Hamilton's equations. We write a continuity equation for $f$ in six-dimensional phase space, prove the flow is divergence-free (Liouville's theorem), and combine the two to show the convective derivative of $f$ vanishes, so that $f$ is carried unchanged along every orbit.

Step 1. The six coordinates $w_\alpha = (\mathbf{x},\mathbf{v})$ move with six-velocity $\dot{w}_\alpha = (\dot{\mathbf{x}},\dot{\mathbf{v}}) = (\mathbf{v},-\nabla\Phi)$ , and conservation of star number gives the phase-space continuity equation $\partial f/\partial t + \sum_\alpha \partial(f\,\dot{w}_\alpha)/\partial w_\alpha = 0$ . Why: exactly as mass conservation gives $\partial\rho/\partial t + \nabla\cdot(\rho\mathbf{u})=0$ in a fluid, star conservation gives the same in six dimensions.

Step 2. Expand with the product rule: $\partial f/\partial t + \sum_\alpha \dot{w}_\alpha\,\partial f/\partial w_\alpha + f\sum_\alpha \partial\dot{w}_\alpha/\partial w_\alpha = 0$ . The middle term is convective transport; the last measures whether the flow compresses phase-space volume. Why: separating transport from compression isolates the piece Liouville's theorem will kill.

Step 3. Evaluate the compression term. Position rates depend only on velocity and velocity rates only on position, so $\partial\dot{x}_i/\partial x_i = 0$ and $\partial\dot{v}_i/\partial v_i = 0$ . Their sum vanishes identically: the phase flow is incompressible. Why: this is Liouville's theorem — in canonical variables it follows from the equality of mixed partials, $\partial^2 H/\partial q\,\partial p = \partial^2 H/\partial p\,\partial q$ , so phase-space volume is conserved.

Step 4. Drop the vanishing term and read off $\partial f/\partial t + \mathbf{v}\cdot\nabla f - \nabla\Phi\cdot\partial f/\partial\mathbf{v} = 0$ , that is $\mathrm{d}f/\mathrm{d}t = 0$ . The distribution function is constant along orbits. Why: this is why $f$ can be written as a function of the integrals of motion (Jeans theorem) — anything conserved along an orbit is a legal argument of a steady-state DF.

Go deeper · Derivation B · 1 minFrom the distribution function to the Jeans equations

Derivation BFrom the distribution function to the Jeans equations

The goal is to convert the collisionless Boltzmann equation for the unobservable six-dimensional $f$ into equations for the observable moments $n$ , $\overline{v_i}$ , and $\sigma_{ij}^2$ . The method mirrors how the fluid equations follow from the kinetic Boltzmann equation: multiply the master equation by successive powers of velocity and integrate, so that each moment yields one fluid-like conservation law, with the higher moment always sourcing the lower.

Step 1. Integrate equation (5) over $\mathrm{d}^3v$ . The acceleration term is a velocity-space divergence and integrates to a surface term at $|\mathbf{v}|\to\infty$ , where $f\to0$ , so it vanishes, leaving $\partial n/\partial t + \partial(n\overline{v_i})/\partial x_i = 0$ . Why: this is the continuity equation — the zeroth moment conserves star number, with $n\overline{v_i}$ as the flux.

Step 2. Now multiply equation (5) by $v_j$ before integrating: $\int v_j\big(\partial f/\partial t + v_i\,\partial f/\partial x_i - \partial\Phi/\partial x_i\,\partial f/\partial v_i\big)\mathrm{d}^3v = 0$ . The first two terms become $\partial(n\overline{v_j})/\partial t$ and $\partial(n\overline{v_i v_j})/\partial x_i$ . Why: weighting by momentum turns the master equation into a statement about momentum flow — the stellar analogue of Euler's equation.

Step 3. Integrate the potential term by parts in velocity: $-\partial\Phi/\partial x_i\int v_j\,\partial f/\partial v_i\,\mathrm{d}^3v = n\,\partial\Phi/\partial x_j$ , the boundary term dying at infinity and $\partial v_j/\partial v_i = \delta_{ij}$ selecting a single gradient. Why: gravity enters the fluid picture as a body force $-n\nabla\Phi$ per unit volume, identical in form to a pressure gradient.

Step 4. Assemble the pieces into $\partial(n\overline{v_j})/\partial t + \partial(n\overline{v_i v_j})/\partial x_i = -n\,\partial\Phi/\partial x_j$ , then split the second moment as $n\overline{v_i v_j} = n\overline{v_i}\,\overline{v_j} + n\sigma_{ij}^2$ . Why: the dispersion tensor $n\sigma_{ij}^2$ appears as an anisotropic pressure — this is the Jeans equation used to weigh galaxies, and the unknown $\sigma_{ij}^2$ trading off against enclosed mass is the mass–anisotropy degeneracy.

Go deeper · Derivation C · 1 minThe singular isothermal sphere from a Maxwellian energy DF

Derivation CThe singular isothermal sphere from a Maxwellian energy DF

The goal is to show how a single choice of distribution function — the Boltzmann form $f\propto e^{E/\sigma^2}$ , depending on energy alone — generates a complete self-consistent galaxy model with a flat rotation curve, purely by feeding its density into Poisson's equation. This is the simplest nontrivial solution of the collisionless Boltzmann–Poisson system, and it was the first DF template for a dark-matter halo.

Step 1. Posit the isotropic Maxwellian $f(E) = \dfrac{\rho_1}{(2\pi\sigma^2)^{3/2}}\,e^{E/\sigma^2}$ with $E = \Psi - \tfrac12 v^2$ and constant $\sigma$ . This is a legal steady-state DF because it depends only on the integral $E$ . Why: a velocity-space Gaussian of fixed width $\sigma$ everywhere is the maximum-entropy choice, the stellar echo of a gas in thermal equilibrium — hence "isothermal".

Step 2. Integrate over velocities: $\rho = \int f\,\mathrm{d}^3v = \rho_1 e^{\Psi/\sigma^2}\!\int \dfrac{e^{-v^2/2\sigma^2}}{(2\pi\sigma^2)^{3/2}}\mathrm{d}^3v = \rho_1\,e^{\Psi/\sigma^2}$ , since the normalized Gaussian integrates to one. Why: the density inherits an exponential dependence on the relative potential, which makes Poisson's equation tractable.

Step 3. Insert this into the spherical Poisson equation $\dfrac{1}{r^2}\dfrac{\mathrm{d}}{\mathrm{d}r}\!\Big(r^2\dfrac{\mathrm{d}\Psi}{\mathrm{d}r}\Big) = -4\pi G\rho_1 e^{\Psi/\sigma^2}$ , using $\Psi=-\Phi$ . Why: the self-consistency loop between field and matter is now a single closed ODE for the potential.

Step 4. Try the power law $\rho = \dfrac{\sigma^2}{2\pi G r^2}$ . Then $\Phi = 2\sigma^2\ln r + \text{const}$ , so $\nabla^2\Phi = r^{-2}\,\mathrm{d}(r^2\cdot 2\sigma^2/r)/\mathrm{d}r = 2\sigma^2/r^2 = 4\pi G\rho$ , satisfied exactly. Why: the scale-free $r^{-2}$ profile is the unique power law consistent with constant dispersion; it is singular at the centre and its mass diverges outward, flaws cured by cored and truncated variants.

Step 5. Compute the circular speed: $M({<}r) = \int_0^r 4\pi r'^2\rho\,\mathrm{d}r' = 2\sigma^2 r/G$ , so $v_c^2 = GM/r = 2\sigma^2$ , and $v_c = \sqrt{2}\,\sigma$ at every radius. Why: enclosed mass rising linearly with radius is exactly what a flat rotation curve demands — the signature that first betrayed dark halos. *Note: $\rho=\sigma^2/2\pi G r^2$ solves the spherical Poisson equation exactly, and $v_c=\sqrt{2}\,\sigma$ ; numerically, $\sigma=150$ km/s gives $v_c=212$ km/s.*

Limiting cases

The smooth DF is a modelling choice whose validity depends on the regime — when it holds, and when it must be repaired.

Case	Result	Meaning
Collisionless (galaxies)	$t_{\mathrm{relax}} \gg t_{\mathrm{age}}$	Encounters negligible; $f$ evolves by the collisionless Boltzmann equation and is conserved along orbits.
Collisional (cluster cores)	$t_{\mathrm{relax}} \lesssim t_{\mathrm{age}}$	Two-body encounters redistribute energy; a collision term $\partial f/\partial t\|_{\mathrm{coll}}$ must be added, driving core collapse and evaporation.
Steady state	$\partial f/\partial t = 0$	Jeans theorem applies: $f$ is a function only of isolating integrals of motion, e.g. $f=f(E,L)$ .
Spherical isotropic	$f=f(E)$	Eddington inversion recovers a unique $f(E)$ from $\rho(r)$ ; physical only if $f\geq0$ .
Decoupled cosmological species	$\Gamma \ll H$	Interaction rate below the expansion rate: $f$ free-streams, its shape frozen while momenta redshift as $p\propto a^{-1}$ .

Common trapThe distribution function is the real system, not a smeared-out approximation to it

It is tempting to think of $f$ as a convenient blur over the "true" system of $N$ discrete point masses, valid only when you squint. That inverts the logic. For a galaxy the smooth mean-field description is the accurate dynamics: the granularity of individual stars matters only on the two-body relaxation time, which exceeds the age of the universe millions of times over, so the collisionless $f$ is essentially exact, not an approximation. The blurring becomes a genuine approximation only where the relaxation time falls to the system age, as in dense globular-cluster cores, and there the fix is not to abandon $f$ but to restore a collision term $\partial f/\partial t|_{\mathrm{coll}}$ . A subtler trap: the fine-grained $f$ is conserved by Liouville, but the coarse-grained density you actually measure is not — phase mixing lets it fall while the fine-grained value is preserved, which is how a system relaxes without any collisions at all.

Worked examples

1 · An isothermal-sphere halo from its dispersion (confidence builder)

A dark-matter halo is a singular isothermal sphere with 1-D velocity dispersion $\sigma = 150$ km/s. Find its circular speed, the mass within 10 kpc, and the local density there. Take $G = 6.674\times10^{-11}$ SI, $1\ \mathrm{kpc}=3.086\times10^{19}$ m, $M_\odot = 1.989\times10^{30}$ kg.

Step 1. The circular speed follows directly from the derivation above and is independent of radius: $v_c = \sqrt{2}\,\sigma = \sqrt{2}\times150 = \mathbf{212\ km/s}$ . The rotation curve is flat — the defining kinematic signature of a dark halo.

Step 2. The enclosed mass grows linearly, $M({<}r) = 2\sigma^2 r/G = v_c^2 r/G$ . With $\sigma = 1.5\times10^5$ m/s and $r = 10\ \mathrm{kpc} = 3.086\times10^{20}$ m, $M(<10\,\mathrm{kpc}) = 2(1.5\times10^5)^2(3.086\times10^{20})/(6.674\times10^{-11}) = 2.08\times10^{41}\ \mathrm{kg} = \mathbf{1.0\times10^{11}\ M_\odot}$ — a Milky-Way-scale mass.

Step 3. The local density is $\rho = \sigma^2/(2\pi G r^2)$ . Numerically $\rho = (2.25\times10^{10})/[2\pi(6.674\times10^{-11})(3.086\times10^{20})^2] = 5.6\times10^{-22}\ \mathrm{kg\,m^{-3}}$ , which is $\mathbf{8.3\times10^{-3}\ M_\odot\,pc^{-3}}$ — a few thousandths of a solar mass per cubic parsec, the faint but decisive weight of dark matter. v_c = 212 km/s, M within 10 kpc = 1.0×10¹¹ M_⊙, ρ = 8.3×10⁻³ M_⊙ pc⁻³.

2 · Why a galaxy is collisionless but a globular-cluster core is not

Estimate the two-body relaxation time for (a) a galaxy, $N=10^{11}$ , $t_{\mathrm{cross}}\approx10^8$ yr, and (b) a globular cluster, $N=2\times10^5$ , $t_{\mathrm{cross}}\approx7\times10^5$ yr, using $t_{\mathrm{relax}}\approx \dfrac{N}{8\ln N}\,t_{\mathrm{cross}}$ ; compare with the age of the universe, $1.38\times10^{10}$ yr.

Step 1. Galaxy: $\ln N = 25.3$ , so $N/(8\ln N) = 4.9\times10^{8}$ crossing times, giving $t_{\mathrm{relax}} = 4.9\times10^8\times10^8\ \mathrm{yr} = \mathbf{4.9\times10^{16}\ yr}$ — roughly $3.6\times10^{6}$ times the age of the universe. Encounters are utterly negligible: the galaxy is collisionless and $f$ obeys the collisionless Boltzmann equation to superb accuracy.

Step 2. Globular cluster: $\ln N = 12.2$ , so $N/(8\ln N) = 2.0\times10^{3}$ crossing times, giving $t_{\mathrm{relax}} = 2.0\times10^3\times7\times10^5\ \mathrm{yr} = \mathbf{1.5\times10^{9}\ yr}$ — about a tenth of the age of the universe.

Step 3. Interpretation: the cluster has lived through roughly ten relaxation times, so two-body encounters have reshaped its core — energy equipartition, mass segregation, core collapse, evaporation of the lightest stars. Its central dynamics need a collision term. The same formula gives collisionless dynamics for the galaxy and collisional dynamics for the cluster, purely because relaxation scales as $N/\ln N$ and $N$ differs by six orders of magnitude. Galaxy t_relax ≈ 5×10¹⁶ yr (collisionless); cluster t_relax ≈ 1.5×10⁹ yr (collisional).

3 · The frozen phase space of relic neutrinos

Neutrinos decouple at $T\sim1$ MeV and their DF free-streams unchanged; later $e^+e^-$ annihilation heats the photons but not the neutrinos, so entropy conservation gives $T_\nu/T_\gamma = (4/11)^{1/3}$ . Find today's neutrino temperature from $T_\gamma = 2.725$ K.

Step 1. Once $\Gamma\ll H$ the neutrino DF keeps its Fermi–Dirac shape $f(p) = [\exp(p/T_\nu)+1]^{-1}$ , momenta redshifting as $p\propto a^{-1}$ : the functional form is frozen though the neutrinos no longer interact — Liouville applied to the cosmological phase space.

Step 2. With $(4/11)^{1/3} = 0.7138$ , $T_\nu = 0.7138\times2.725\ \mathrm{K} = \mathbf{1.945\ K}$ — a relic sea colder than the CMB by exactly this factor, predicted purely by tracking a phase-space distribution through decoupling. T_ν = 1.945 K.

Physics of the portrait

The distribution function carries the name of the man who first described a gas — or a galaxy — not star by star but as a density in the space of positions and velocities.

Origins · 1 minLudwig Boltzmann and how this was discovered

Ludwig Boltzmann

1844–1906 · kinetic theory and the statistical description of many-body systems

Ludwig Boltzmann gave physics the distribution function. Working on the kinetic theory of gases in Vienna and Graz through the 1860s and 1870s, he asked how to describe $10^{23}$ molecules without following any of them, and answered with $f$ : the density of particles in the space of positions and velocities — the phase space of this article. His 1872 transport equation, $\mathrm{d}f/\mathrm{d}t = (\partial f/\partial t)_{\mathrm{coll}}$ , is the direct ancestor of every equation here; strip away the collision term, as a galaxy allows, and it becomes the collisionless Boltzmann equation exactly. He also supplied the equilibrium form $f\propto e^{-E/kT}$ that reappears here as the Maxwellian DF of the isothermal sphere, and his entropy $S=k\ln W$ — the logarithm of phase-space volume — is why coarse-grained phase-space density can only decrease. He fought bitterly for the reality of atoms and took his own life in 1906, before the atomic hypothesis he championed was vindicated. When stellar dynamicists write $f(\mathbf{x},\mathbf{v},t)$ for a galaxy, or cosmologists a Boltzmann equation for dark matter, they use his language wholesale — the distribution function is his single most consequential invention.

History

1838

— Liouville proves that phase-space volume is conserved along a Hamiltonian flow

1872

— Boltzmann writes the transport equation for the distribution function of a gas

1915

— Jeans applies the collisionless Boltzmann equation to stellar systems, deriving the Jeans equations and theorem

1916

— Eddington inverts a spherical density profile to its isotropic distribution function f(E)

1942

— Chandrasekhar systematizes stellar dynamics, two-body relaxation, and dynamical friction

Going further

Texts. Binney & Tremaine, Galactic Dynamics, 2nd ed., Ch. 4 (DF, collisionless Boltzmann and Jeans equations, Jeans theorem, Eddington inversion) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 24–25 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 5 · Dodelson, Modern Cosmology, Ch. 3 · Ryden, Introduction to Cosmology, 2nd ed. Key results to keep in mind: (i) $\mathrm{d}f/\mathrm{d}t=0$ follows from phase-space continuity plus Liouville's theorem; (ii) the Jeans equations are the zeroth and first moments of the CBE; (iii) the Maxwellian DF gives $\rho=\sigma^2/2\pi G r^2$ with $v_c=\sqrt{2}\,\sigma$ , so $\sigma=150$ km/s $\Rightarrow v_c=212$ km/s and $M$ within 10 kpc $=1.0\times10^{11}\,M_\odot$ ; (iv) $t_{\mathrm{relax}}\approx(N/8\ln N)t_{\mathrm{cross}}$ gives $\sim5\times10^{16}$ yr (galaxy), $\sim1.5\times10^{9}$ yr (globular); (v) $T_\nu=(4/11)^{1/3}T_\gamma=1.945$ K.

Prerequisites. Gravitational N-Body Problem · Two-Body Relaxation Time · Galactic Potential Models. Leads to. Collisionless Boltzmann Equation · Jeans Equations · Jeans Theorem · Action-Angle Variables.

Ask the tutor: Ask me to build an $f(E)$ for a real density profile by Eddington inversion, or to show exactly how the Jeans equations weigh a galaxy's dark matter.

Why six dimensions — is that just a bookkeeping trick, or is it physical?

It is physical, forced on us by Newtonian dynamics. A star's future is fixed only once you know both where it is and how fast it is moving: two stars at the same place with different velocities end up in different places. The state of one star is therefore a point in a six-dimensional space — three positions, three velocities — and a galaxy is a cloud of such points whose density is $f$ . The six dimensions are not extra spatial directions you could travel in; they are the minimum coordinates needed to specify a star's dynamical state, exactly what Hamilton's equations evolve.

If Liouville's theorem keeps f constant, how does a galaxy ever relax to equilibrium?

Through phase mixing, not any change in the fine-grained $f$ . Liouville keeps the fine-grained density — measured with infinitely sharp resolution, following a bundle of stars — constant. But that bundle stretches into an ever thinner filament winding through phase space, like cream stirred into coffee. Any real measurement averages over a finite cell, and that coarse-grained density falls as the filament thins and interleaves with empty space. The system reaches a smooth, steady coarse-grained state that looks relaxed while the fine-grained $f$ is secretly conserved. This phase mixing (and its violent-relaxation cousin) is how collisionless systems settle without collisions — and the Gaia snail is a picture of it happening.

What is the difference between the collisionless Boltzmann equation and the Jeans equations?

The same physics at two levels of detail. The collisionless Boltzmann equation governs the full six-dimensional $f$ and contains everything, but $f$ is not directly observable. The Jeans equations are its velocity moments — multiply by $1$ , $v_j$ , $v_iv_j$ and integrate — governing the observables $n$ , $\overline{v_i}$ , $\sigma_{ij}^2$ directly. The price is that the hierarchy never closes: each moment's equation contains the next-higher one, so density needs streaming, streaming needs dispersion, and so on. To close it you assume something — an anisotropy profile, or the full DF — which is the origin of the mass–anisotropy degeneracy that plagues galaxy mass estimates.

Why must a steady-state distribution function depend only on integrals of motion?

Because in a steady state $\partial f/\partial t=0$ , and the collisionless Boltzmann equation then reads $\mathbf{v}\cdot\nabla f - \nabla\Phi\cdot\partial f/\partial\mathbf{v}=0$ , saying $f$ is constant along every orbit. A quantity constant along orbits is, by definition, an integral of motion — so $f$ can depend on phase-space coordinates only through such integrals: energy $E$ always, angular momentum $L$ or $L_z$ when the potential is symmetric, sometimes a third integral $I_3$ . This is Jeans theorem, and it is enormously practical: instead of an unknown function of six variables you write $f$ from two or three conserved labels and solve Poisson for consistency — $f(E)$ for isotropic spheres, $f(E,L_z)$ or $f(E,L_z,I_3)$ for flattened, rotating systems.

The isothermal sphere has infinite mass — why is it taken seriously?

Because it captures the feature that mattered most historically: a flat rotation curve. Its density falls as $r^{-2}$ , so the enclosed mass $M({<}r)\propto r$ grows without bound and the total mass diverges — unphysical for a real galaxy. But out to observed radii, $M\propto r$ is exactly what produces the flat $v_c=\sqrt2\,\sigma$ seen in spirals, which is why it was the first DF model of a dark halo. In practice one uses cured versions — a cored sphere flattens the central singularity, a truncated model cuts off the outer mass. A model can be wrong in its extremes yet exactly right about the physics: a constant dispersion demands an $r^{-2}$ halo, and an $r^{-2}$ halo demands a flat rotation curve.

Does the same distribution function really describe both stars and dark matter and neutrinos?

Yes, with the same equation and different collision terms. Any collection of particles moving under gravity — stars in a galaxy, dark matter in a halo, neutrinos in the early universe — is a phase-space DF obeying a Boltzmann equation. When interactions are negligible (a galaxy, or any decoupled species) the collision term vanishes and $f$ is conserved along orbits or free-streams and redshifts; when they matter (a cluster core, or a species coupled to the plasma) you add a collision operator. That universality is the power of the viewpoint: the CMB keeping a blackbody spectrum, relic neutrinos keeping a Fermi–Dirac shape at $T_\nu=1.945$ K, dark matter growing the cosmic web, and stars settling into a galaxy are all one statement — a distribution function evolving under the Boltzmann equation.