Collisionless Boltzmann Equation

A galaxy is a swarm of a hundred billion stars, and the honest way to describe it would be to write down the position and velocity of every one — the gravitational N-body problem, 6×10¹¹ coupled equations that no computer will ever integrate for a Hubble time. The collisionless Boltzmann equation is the escape from that impossibility, and it earns the escape with a remarkable fact: in a galaxy the discrete tug of one star on another almost never matters. Add up all the little deflections a star suffers as it wanders past its neighbours and the time needed to change its orbit appreciably — the two-body relaxation time — comes out longer than the age of the Universe by a factor of order a million. So each star does not feel its neighbours as neighbours at all. It feels only the smooth, collective potential Φ(𝐱,t) that the whole mass distribution sculpts, as a swimmer feels the pool and not the water molecules. Once that mean-field picture is granted, the description becomes statistical. In place of 6N coordinates we carry a single function, the phase-space distribution function f(𝐱,𝐯,t), counting stars per unit volume of the six-dimensional space of positions and velocities. The collisionless Boltzmann equation is one line: f is conserved along the orbit of a star through phase space. It is Liouville's theorem in the clothes of stellar dynamics — the density of representative points flows, shears, and winds into ever-finer filaments, but is never compressed or diluted. From that one line flow the Jeans equations that weigh galaxies, Jeans' theorem that builds equilibrium models, and — relativistically — the kinetic backbone of dark matter, neutrinos, and the microwave background.

Key takeaways

The collisionless Boltzmann equation (CBE) says the phase-space distribution function is conserved along orbits in the smooth mean-field potential: df/dt = ∂ₜf + 𝐯·∇ₓf − ∇Φ·∇ᵥf = 0.
It is the master equation of stellar dynamics, valid whenever the two-body relaxation time greatly exceeds the age of the system — true for galaxies by a factor of order 10⁶ — and follows from Liouville's theorem: the six-dimensional phase flow 𝐰 = (𝐯, −∇Φ) is divergence-free.
Its velocity moments generate the Jeans equations — the stellar-dynamical analogues of the fluid continuity and Euler equations — used to weigh galaxies and dark halos.
Its steady states obey Jeans' theorem: any time-independent f is a function of the integrals of motion (E, L, …) only.
Coupled to Poisson's equation ∇²Φ = 4πG∫f d³v, the CBE is self-consistent and nonlinear — the stars build the potential that in turn steers the stars.
Key idea: along the phase-space orbit of a star moving in the smooth mean-field potential the distribution function is conserved, df/dt = ∂ₜf + 𝐯·∇ₓf − ∇Φ·∇ᵥf = 0 — Liouville's theorem for the mean field, phase-space density flowing but never compressing.

The core insight

Along the phase-space orbit of a star moving in the smooth mean-field potential the distribution function is conserved: df/dt = ∂ₜf + 𝐯·∇ₓf − ∇Φ·∇ᵥf = 0. This is Liouville's theorem specialised to the mean field — phase-space density is incompressible, flowing and shearing but never compressing or diluting.

Where this lives today

The CBE is the equation being solved, in one form or another, by most of computational astrophysics. Every large collisionless N-body simulation — from the Aquarius and IllustrisTNG halos to the billion-particle runs mapping the cosmic web — is a Monte-Carlo integration of the CBE: the particles sample f, their trajectories follow the characteristics, and force softening keeps the artificial relaxation time longer than the run. On the observational side, the Jeans equations are how astronomers weigh dark matter: Gaia's six-dimensional map of a billion Milky Way stars is fit with Jeans and distribution-function models for the Galactic potential and the local dark-matter density, and Jeans modelling of dwarf spheroidals probes halo cusps versus cores. Direct distribution-function modelling — Schwarzschild orbit superposition and made-to-measure methods, both solving the steady CBE — weighs the supermassive black holes at galaxy centres. And the relativistic Boltzmann hierarchy for photons, neutrinos, and dark matter, integrated by codes such as CAMB and CLASS, predicts the cosmic microwave background power spectrum that anchors ΛCDM. One equation, from a galaxy's orbits to the whole observable Universe.

The physical picture

Represent the galaxy not as stars in space but as a cloud of points in a six-dimensional phase space with axes 𝐱 and 𝐯. Each star is one point; the local density of points is f. As time runs, every point glides along a trajectory fixed by Hamilton's equations — position drifts at rate 𝐯, velocity at rate −∇Φ. The collisionless Boltzmann equation is the single statement that if you ride along with the flow, inside a small parcel of these points, the density you measure never changes. The parcel stretches, shears, and winds into a thin ribbon, but never bunches up or thins out. It is an incompressible fluid in phase space.

Figure

Phase-space incompressibility. A parcel of stars at time t shears into a thin filament by a later time, but its phase area Δx Δv is exactly preserved because the six-dimensional flow 𝐰 = (𝐯, −∇Φ) is divergence-free, ∇₆·𝐰 = 0, so df/dt = 0

The dye analogy is exact. Stir a drop of non-diffusing ink into water: it is drawn into finer and finer swirls, its total area frozen, its concentration along the flow unchanged. Galactic phase mixing works identically — a clump of stars filaments into a spiral without ever changing f along any orbit. What is emphatically not happening is genuine collision: in a real gas, molecular encounters would erase the fine filaments and drive the system toward a Maxwellian, adding a collision term to the right-hand side. In a galaxy those encounters are too rare to matter, so the right-hand side is exactly zero. "Collisionless" is thus a statement about timescales, which the second figure makes quantitative for a galaxy and a globular cluster.

Figure

Why galaxies are collisionless. Using t_relax ≈ 0.1 N/ln N × t_cross, a galaxy (N ≈ 10¹¹) relaxes only after ≈ 2×10¹⁶ yr — about 10⁶ Hubble times — so it is collisionless, whereas a globular cluster (N ≈ 10⁵) relaxes in ≈ 3×10⁸ yr, well within its age, so it is collisional

Finally, the potential in that force term is not handed to us from outside: the stars themselves are the mass that sources Φ through Poisson's equation, so f determines Φ and Φ evolves f — a closed, nonlinear system in which a galaxy forever sculpts the landscape down which it slides.

The core equations

The equation is most transparent in Cartesian coordinates with an explicit gravitational force. Three terms: the local rate of change of f, the streaming of stars through space at velocity 𝐯, and their acceleration through velocity space by −∇Φ. Their sum is zero.

\frac{\partial f}{\partial t}+\mathbf{v}\cdot\frac{\partial f}{\partial \mathbf{x}}-\nabla\Phi\cdot\frac{\partial f}{\partial \mathbf{v}}=0 \qquad (1)

where: $f(\mathbf{x},\mathbf{v},t)$ — phase-space distribution function, stars (or mass) per unit d³x d³v; $\mathbf{v}$ — velocity coordinate; $\Phi(\mathbf{x},t)$ — smooth mean-field potential; $-\nabla\Phi$ — gravitational acceleration. Reading: the Eulerian change ∂ₜf at a fixed phase point is exactly cancelled by what streaming and acceleration carry through it. The same content is cleaner as a total derivative. If 𝐱(t), 𝐯(t) trace a star's orbit, the convective (Lagrangian) derivative of f is zero — f is a constant of the motion. Equation (1) is just this with the orbit equations 𝐱̇ = 𝐯, 𝐯̇ = −∇Φ substituted in.

\frac{\mathrm{d}f}{\mathrm{d}t}=\frac{\partial f}{\partial t}+\dot{\mathbf{x}}\cdot\frac{\partial f}{\partial \mathbf{x}}+\dot{\mathbf{v}}\cdot\frac{\partial f}{\partial \mathbf{v}}=0,\qquad \dot{\mathbf{x}}=\mathbf{v},\ \ \dot{\mathbf{v}}=-\nabla\Phi \qquad (2)

where: d/dt — total derivative following the orbit; 𝐱̇ = 𝐯 — phase flow in position; 𝐯̇ = −∇Φ — phase flow in velocity. This form names the physics: f is conserved along characteristics, and the characteristics are the stellar orbits. In canonical coordinates the structure is even more universal. With coordinates and conjugate momenta (qᵢ, pᵢ) and a single-particle Hamiltonian H, the streaming-plus-force operator is exactly the Poisson bracket [f, H], so the CBE is ∂ₜf + [f, H] = 0 — Liouville's equation, valid for any Hamiltonian system, not just gravity.

\frac{\partial f}{\partial t}+\sum_{i=1}^{3}\left(\frac{\partial f}{\partial q_i}\dot q_i+\frac{\partial f}{\partial p_i}\dot p_i\right)=\frac{\partial f}{\partial t}+[f,H]=0 \qquad (3)

where: (qᵢ, pᵢ) — canonical coordinates and momenta; H(q,p,t) — one-particle Hamiltonian; [f,H] — Poisson bracket. Reading: in a steady state ∂ₜf = 0, so [f,H] = 0 — f is a constant of motion, the germ of Jeans' theorem. The equation is closed by Poisson's equation, which turns f back into the potential in the force term. The zeroth velocity moment of f is the mass density ρ, and ρ sources Φ.

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

where: G — gravitational constant; ρ(𝐱,t) — mass density, the zeroth moment ∫f d³v. This coupling makes the whole problem nonlinear: f sets ρ, ρ sets Φ, Φ steers f. Taking velocity moments of the CBE — multiplying by 1, then vⱼ, then integrating over velocities — generates the Jeans equations, the working tools of galaxy modelling. The first moment is the momentum equation, the stellar-dynamical Euler equation, in which an anisotropic velocity-dispersion tensor plays the role of pressure.

\frac{\partial(\nu\,\bar v_j)}{\partial t}+\frac{\partial(\nu\,\overline{v_i v_j})}{\partial x_i}+\nu\frac{\partial\Phi}{\partial x_j}=0 \qquad (5)

where: ν(𝐱,t) — number density ∫f d³v; v̄ⱼ — mean streaming velocity; $\overline{v_i v_j}$ — second velocity moment, splitting into streaming plus the dispersion tensor σ²ᵢⱼ. Reading: the divergence of the stellar "pressure" plus the gravitational body force drives the change in momentum, exactly as in a fluid — but with no equation of state to close it. Finally, for a time-independent potential Jeans' theorem collapses the search for equilibria into a search for functions of the constants of motion. Any f built from integrals Iₖ solves the steady CBE automatically.

f=f(I_1,I_2,\ldots),\qquad \frac{\mathrm{d}I_k}{\mathrm{d}t}=0 \qquad (6)

where: Iₖ(𝐱,𝐯) — isolating integrals of motion, e.g. energy E = ½v² + Φ and angular momentum L. Reading: rather than solve a partial differential equation in six variables, one picks a function of E (and perhaps L) and imposes Poisson self-consistency.

Derivation AThe collisionless Boltzmann equation from Liouville's theorem

The goal is to derive equation (1) from two ingredients only — the bare conservation of stars (none are created or destroyed) and the geometry of the phase flow — and to expose that the whole content of the CBE is the incompressibility of that flow, ∇₆·𝐰 = 0, Liouville's theorem. We treat the cloud of star-points as a continuum of density f, write its phase-space continuity equation, and show the divergence of the phase velocity vanishes identically for any potential Φ(𝐱,t), so the flux term collapses to the convective derivative.

Step 1. Define the six-dimensional phase velocity 𝐰 = (𝐱̇, 𝐯̇) = (𝐯, −∇Φ), the vector field along which every star-point drifts. Because stars are conserved, f obeys a continuity equation in phase space, ∂ₜf + ∇₆·(f𝐰) = 0, where ∇₆ = (∂/∂𝐱, ∂/∂𝐯). Why: this is nothing but "what flows in minus what flows out equals the build-up", applied to the six-dimensional volume — the only physical input is that star number is conserved.

Step 2. Expand the flux by the product rule: ∂ₜf + 𝐰·∇₆f + f (∇₆·𝐰) = 0. The middle term is the convective transport of f; the last measures how much the flow compresses the phase volume. Why: separating advection from compression is the crux — if the compression term vanishes, the density simply rides along with the flow.

Step 3. Evaluate the divergence term by term: ∇₆·𝐰 = Σᵢ ∂vᵢ/∂xᵢ + Σᵢ ∂(−∂Φ/∂xᵢ)/∂vᵢ. The first sum is zero because position and velocity are independent coordinates (vᵢ does not depend on xᵢ); the second is zero because Φ = Φ(𝐱,t) has no dependence on 𝐯 at all. Hence ∇₆·𝐰 = 0. Why: this is Liouville's theorem — a Hamiltonian flow preserves phase-space volume — and it holds for the mean-field force precisely because that force is velocity-independent.

Step 4. With the compression term gone, the surviving equation is ∂ₜf + 𝐯·∇ₓf − ∇Φ·∇ᵥf = 0, which is df/dt = 0. Why: the entire equation is the single geometric fact that gravitational phase flow neither bunches nor dilutes the star-points, so their local density is frozen into the motion. Verified against Binney and Tremaine §4.1, Eq. (4.11), and Carroll and Ostlie Ch. 24.

Go deeper · Derivation B · 2 minThe first Jeans equation from the velocity moment

Derivation BThe first Jeans equation from the velocity moment

The aim is to show that although the CBE for f is intractable, multiplying it by a velocity component vⱼ and integrating over all velocities yields a momentum equation — the stellar-dynamical Euler equation — involving only observable moments: the number density ν, the mean streaming velocity v̄ⱼ, and the velocity-dispersion tensor. This is what makes the CBE useful to observers, connecting the unmeasurable f to quantities read off spectra. We take the vⱼ-moment of each CBE term, integrate the force term by parts, and discard the vanishing surface terms.

Step 1. Multiply equation (1) by vⱼ and integrate over d³v. The time term gives ∂ₜ∫vⱼf d³v = ∂ₜ(νv̄ⱼ), and the streaming term gives ∂ₓᵢ∫vᵢvⱼf d³v = ∂ₓᵢ(ν $\overline{v_i v_j}$ ), since vᵢ and the x-derivative pass freely through the velocity integral. Why: every moment of the CBE turns derivatives of f into derivatives of measurable moments — the point of the whole moment method.

Step 2. Treat the force term −∂ₓᵢΦ ∫vⱼ ∂ᵥᵢf d³v by parts in velocity. The boundary term vanishes because f → 0 as |𝐯| → ∞ (no infinitely fast stars), leaving +∂ₓᵢΦ ∫f ∂vⱼ/∂vᵢ d³v = ∂ₓᵢΦ · δᵢⱼ · ν = ν ∂ₓⱼΦ. Why: the integration by parts is where the gravitational force re-emerges as a body force on the stellar "fluid", with no derivative of f left over.

Step 3. Collect the three pieces: ∂ₜ(νv̄ⱼ) + ∂ₓᵢ(ν $\overline{v_i v_j}$ ) + ν ∂ₓⱼΦ = 0. Splitting $\overline{v_i v_j}$ = v̄ᵢv̄ⱼ + σ²ᵢⱼ exposes the dispersion tensor as an anisotropic pressure. Why: this is Euler's equation for stars, but the "pressure" is a full tensor with no equation of state — which is why stellar systems can be flattened and anisotropic while a gas cannot.

Step 4. The zeroth moment (multiply by 1, integrate) gives the companion continuity equation ∂ₜν + ∂ₓᵢ(νv̄ᵢ) = 0 by the same boundary argument. Together they are the first two Jeans equations. Why: continuity plus momentum is precisely the fluid pair, so a collisionless star system obeys fluid-like bookkeeping without ever being a fluid. Verified against Binney and Tremaine §4.8 (the Jeans equations).

Go deeper · Derivation C · 1 minJeans' theorem: steady states are functions of the integrals of motion

Derivation CJeans' theorem: steady states are functions of the integrals of motion

The objective is to prove the theorem that underpins all equilibrium galaxy modelling: in a time-independent potential, any f built solely from isolating integrals of motion solves the steady CBE, and — for regular orbits — every steady solution can be written this way. This turns an impossible partial differential equation into a recipe: choose f(E) or f(E,L) and impose Poisson self-consistency. We start from the steady CBE, recall what defines an integral of motion, and apply the chain rule.

Step 1. In a steady state ∂ₜf = 0, so the CBE reduces to 𝐯·∇ₓf − ∇Φ·∇ᵥf = df/dt = 0: f is constant along every orbit. By definition an integral of motion Iₖ(𝐱,𝐯) satisfies the same condition, dIₖ/dt = 𝐯·∇ₓIₖ − ∇Φ·∇ᵥIₖ = 0. Why: the steady CBE operator is identical to the operator that defines a constant of motion — the whole coincidence the theorem exploits.

Step 2. (Sufficiency.) Let f = F(I₁,…,Iₙ) be any smooth function of integrals. By the chain rule df/dt = Σₖ (∂F/∂Iₖ)(dIₖ/dt) = 0, since each dIₖ/dt = 0. So any such F is a steady solution. Why: this is the constructive half — it licenses writing down f(E) = anything decreasing in energy and knowing it solves the CBE before any further work.

Step 3. (Necessity.) Conversely, a steady f has df/dt = 0, so f is itself constant along orbits — it is an integral of motion and can be expressed through a complete set of isolating integrals. The strong form requires the orbits to be regular (quasiperiodic on tori); phase-space-filling chaotic orbits can support extra dependences. Why: the caveat matters in real galaxies, where box and chaotic orbits mean f need not be a clean function of E and L alone — the theorem is a guide, not a straitjacket. Verified against Binney and Tremaine §4.2 (Jeans' theorems).

Limiting cases

The CBE is the leading term of a hierarchy, and each limit sharpens its meaning by showing what it keeps and what it throws away.

Case	Result	Meaning
Collisionless limit (galaxies)	t_relax ≫ t_age ⇒ df/dt = 0	For N ≈ 10¹¹ the relaxation time exceeds the Hubble time by ≈ 10⁶, so the collision term is utterly negligible and the pure CBE holds.
Collisional systems (globular clusters, cluster cores)	df/dt = (∂f/∂t)_coll	When t_relax ≲ t_age a Fokker–Planck collision term must be restored; encounters drive relaxation, mass segregation, and core collapse.
Steady state	∂ₜf = 0 ⇒ 𝐯·∇ₓf − ∇Φ·∇ᵥf = 0	Time-independent equilibria; by Jeans' theorem f is then a function of the integrals of motion only.
Cold / single-stream limit (dark matter)	f = ρ(𝐱,t) δ³(𝐯 − 𝐮(𝐱,t))	A pressureless, single-velocity component; the CBE reduces to continuity plus pressureless Euler — valid only until orbits cross (shell crossing) and the stream multivalues.
Fluid closure	$\overline{v_i v_j}$ → (P/ρ)δᵢⱼ	An equation of state closes the moment hierarchy, and the Jeans equations become the Euler equations of a gas — the isotropic special case.

Common trapThe CBE does not say the distribution function is constant in time.

The most common misreading is to see df/dt = 0 and conclude ∂f/∂t = 0 — that f is frozen at every phase point. It is not. The CBE says the total (Lagrangian) derivative vanishes: f is constant along each orbit. The local Eulerian value at a fixed (𝐱,𝐯) generally changes with time, because different orbits carry different values of f through that point. Equally, "collisionless" does not mean "force-free": the smooth mean-field force −∇Φ is the whole second-and-third-term structure of the equation — it is discrete star-on-star scattering that is absent, not gravity. And "conserved f" does not mean "no evolution": a galaxy phase-mixes, forms bars and spiral arms, and violently relaxes, all with df/dt = 0 — the fine-grained f is conserved while the coarse-grained f̄ smooths out.

Worked examples

1 · Why a galaxy is collisionless — but a globular cluster is not

Estimate the two-body relaxation time for the Milky Way (N ≈ 10¹¹, R ≈ 10 kpc, v ≈ 200 km/s) and a globular cluster (N ≈ 10⁵, R ≈ 3 pc, v ≈ 10 km/s), and compare each to the Hubble time. Use t_relax ≈ (0.1 N / ln N) × t_cross.

Step 1. Crossing times first. For the galaxy, t_cross = R/v = (10 × 3.086×10¹⁶ km)/(200 km/s) = 1.54×10¹⁵ s ≈ 4.9×10⁷ yr. For the cluster, t_cross = (3 × 3.086×10¹³ km)/(10 km/s) = 9.3×10¹² s ≈ 2.9×10⁵ yr. A star crosses a galaxy in about 50 Myr and a globular cluster in about 0.3 Myr.

Step 2. The Coulomb-logarithm factors: ln N = ln(10¹¹) = 25.3 (galaxy) and ln(10⁵) = 11.5 (cluster). The relaxation multiplier 0.1 N/ln N is therefore 0.1×10¹¹/25.3 = 3.9×10⁸ crossings for the galaxy but only 0.1×10⁵/11.5 = 870 for the cluster — how many crossings before encounters reshape an orbit is the whole story.

Step 3. Multiply through. Galaxy: t_relax = 3.9×10⁸ × 4.9×10⁷ yr ≈ 1.9×10¹⁶ yr. Cluster: t_relax = 870 × 2.9×10⁵ yr ≈ 2.5×10⁸ yr.

Step 4. Compare to the Hubble time t_H ≈ 1.4×10¹⁰ yr. For the galaxy, t_relax/t_H ≈ 1.4×10⁶ — a million Hubble times away, so the collision term is negligible and the pure CBE applies. For the cluster, t_relax/t_H ≈ 0.02 — it relaxes many times over within its age, so it is collisional and needs a Fokker–Planck term (hence globular clusters mass-segregate and undergo core collapse, while galaxies do not). The same formula, with N differing by 10⁶, cleanly separates the collisionless galaxy from the collisional cluster.

2 · From a distribution function to a galaxy mass: the singular isothermal sphere

Jeans' theorem lets us pick f = f(E). Take the isothermal f(E) = ρ₁(2πσ²)^(−3/2) exp(−E/σ²) with E = ½v² + Φ and constant σ. Show it gives ρ ∝ r⁻², and weigh a galaxy with σ = 200 km/s inside r = 10 kpc.

Step 1. Integrate f over all velocities. With E = ½v² + Φ, ρ(r) = ∫f d³v = ρ₁ e^(−Φ/σ²) (2πσ²)^(−3/2) ∫ e^(−v²/2σ²) d³v = ρ₁ e^(−Φ/σ²), since the Gaussian velocity integral is exactly (2πσ²)^(3/2). The density is set entirely by the potential — the hallmark of an isothermal system.

Step 2. Impose Poisson self-consistency. Substituting ρ = ρ₁ e^(−Φ/σ²) into ∇²Φ = 4πGρ admits the scale-free solution ρ(r) = σ²/(2πG r²) with Φ(r) = 2σ² ln r — the singular isothermal sphere, density falling as r⁻², the profile behind flat rotation curves.

Step 3. Integrate the density for the enclosed mass: M(r) = ∫₀ʳ 4πr′²ρ dr′ = ∫₀ʳ (2σ²/G) dr′ = 2σ²r/G. Equivalently v_c² = GM/r = 2σ² = constant — a flat rotation curve, exactly what the CBE-plus-Jeans-theorem model predicts.

Step 4. Put in numbers: σ = 200 km/s = 2×10⁵ m/s, r = 10 kpc = 3.086×10²⁰ m, G = 6.674×10⁻¹¹ SI. M = 2(2×10⁵)²(3.086×10²⁰)/(6.674×10⁻¹¹) = 3.70×10⁴¹ kg = 1.9×10¹¹ M_⊙. A distribution function chosen by Jeans' theorem and closed by Poisson weighs the galaxy inside 10 kpc at two hundred billion solar masses — with no orbit integration at all.

Physics of the portrait

The collisionless Boltzmann equation is the meeting point of three lineages — the phase-space mechanics of Liouville, the kinetic theory of Boltzmann, and the stellar dynamics of Jeans — welded together and finally understood as a statement about mixing by Lynden-Bell.

Origins · 1 mincollisionless Boltzmann equation and how this was discovered

collisionless Boltzmann equation

df/dt = 0 in the mean-field potential · kinetic theory of stellar systems

The equation carries Boltzmann's name but is older than his transport equation in spirit. Joseph Liouville proved in 1838 that a Hamiltonian flow preserves phase-space volume — the entire content of the collisionless case. Ludwig Boltzmann, building the kinetic theory of gases in 1872, wrote the full transport equation with its collision integral on the right; setting that integral to zero, where encounters are too rare to matter, gives the collisionless form. The decisive application to the stars was James Jeans', who in 1915 took velocity moments to derive what are now the Jeans equations and proved the theorem that steady states depend only on the integrals of motion — the twin results that still frame every equilibrium galaxy model. Subrahmanyan Chandrasekhar made the approximation quantitative in his 1942 Principles of Stellar Dynamics, computing the two-body relaxation time and its Coulomb logarithm and so fixing exactly when the collision term may be dropped. The conceptual capstone came from Donald Lynden-Bell in 1967: his theory of violent relaxation explained how a collisionless system, with f rigorously conserved along orbits, nonetheless reaches a smooth coarse-grained equilibrium through phase mixing in a few dynamical times. The modern relativistic descendant of the same equation governs dark matter and free-streaming neutrinos in cosmology.

History

1838

— Liouville proves that Hamiltonian phase-space volume is conserved

1872

— Boltzmann publishes his transport equation with the collision integral

1915

— Jeans takes velocity moments and states the theorem for steady stellar systems

1942

— Chandrasekhar quantifies the two-body relaxation time and the collisionless limit

1967

— Lynden-Bell explains violent relaxation and phase mixing in collisionless systems

Going further

Texts. Binney and Tremaine, Galactic Dynamics, 2nd ed., §4.1–4.2 (the CBE and Jeans' theorems) and §4.8 (the Jeans equations) — the definitive treatment · Mo, van den Bosch and White, Galaxy Formation and Evolution, §5.4 (collisionless dynamics) · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 24–25 · Dodelson, Modern Cosmology, Ch. 3–4 (the relativistic Boltzmann equation for cosmological species). Key results in this article: the Cartesian CBE ∂ₜf + 𝐯·∇ₓf − ∇Φ·∇ᵥf = 0 follows from the vanishing six-dimensional divergence of the phase flow 𝐰 = (𝐯, −∇Φ); the first Jeans equation comes from the vⱼ-moment with the force term integrated by parts; the relaxation-time formula t_relax ≈ (0.1 N/ln N) t_cross gives ≈ 1.9×10¹⁶ yr for the galaxy and ≈ 2.5×10⁸ yr for the globular cluster; and the singular isothermal sphere ρ = σ²/(2πG r²), M(r) = 2σ²r/G yields 1.9×10¹¹ M_⊙ within 10 kpc for σ = 200 km/s. Only G and the factor 4π enter Poisson's equation as fixed constants.

Prerequisites. Distribution Function and Phase Space · Gravitational N-Body Problem · Two-Body Relaxation Time. Leads to. The Jeans Equations · Jeans' Theorem · The Virial Theorem for Stellar Systems · Action–Angle Variables.

Ask the tutor: Ask me why a galaxy's relaxation time beats the age of the Universe by a million, or to build an f(E) model and turn it into a rotation curve.

Is the collisionless Boltzmann equation the same as Liouville's equation?

Almost, and the distinction is worth getting right. Liouville's equation, ∂ₜf + [f,H] = 0, governs the full 6N-dimensional distribution for the entire system and is exact for any Hamiltonian flow. The CBE is its one-particle projection: f(𝐱,𝐯,t) lives in the six-dimensional phase space of a single star, and the equation looks identical only because, in the mean field, each star moves independently in the smooth potential. The projection is exact precisely when correlations between stars — the discrete encounters — are negligible. So the CBE is Liouville's theorem for one particle in a self-consistent mean field; restore encounters and you must add a collision term, leaving the strict Liouville form behind.

If f is conserved, how does a galaxy ever relax or reach equilibrium?

This is the paradox violent relaxation resolves. The fine-grained f is rigorously conserved along orbits, so in the strict sense a galaxy never relaxes at all — no information about the initial phase-space density is lost. What smooths out is the coarse-grained f̄, the density averaged over small but finite phase-space cells. Phase mixing winds f into filaments far thinner than any cell, so an observer averaging over a cell sees a steadily smoother distribution even though f itself is untouched. During a rapid collapse the potential Φ(t) fluctuates violently, scrambling energies and driving f̄ toward a smooth quasi-equilibrium in a few dynamical times. Equilibrium is thus a property of the coarse-grained field, not the conserved fine-grained one.

Why can we drop the collision term for a galaxy but not for a globular cluster?

It comes down to a single ratio, the number of orbital crossings a star makes before cumulative small-angle encounters redirect it: t_relax/t_cross ≈ 0.1 N/ln N. Because this grows almost linearly with N, a galaxy (N ≈ 10¹¹) needs about 4×10⁸ crossings to relax — roughly 10¹⁶ years, a million Hubble times — so encounters are irrelevant and the collision term vanishes. A globular cluster (N ≈ 10⁵) needs only about 900 crossings, a few hundred million years, shorter than its age. So the cluster genuinely relaxes: it mass-segregates, ejects stars, and can undergo core collapse — physics that requires the Fokker–Planck collision term the galaxy may ignore.

What is the difference between the distribution function f and the density ρ?

The distribution function f(𝐱,𝐯,t) is the full six-dimensional object: how many stars sit near a given position and a given velocity simultaneously. The mass density ρ(𝐱,t) is a shadow of it — the zeroth velocity moment, ρ = ∫f d³v, summing over all velocities and keeping only spatial dependence. You can always compute ρ from f, but not the reverse: many different velocity distributions give the same ρ. That is why the Jeans equations, involving only moments like ρ, streaming velocity, and dispersion, are underdetermined and need extra assumptions, whereas a distribution-function model built by Jeans' theorem specifies everything at once.

Does the collisionless Boltzmann equation apply to dark matter?

Yes — dark matter is the ideal collisionless fluid, and the CBE (in its general-relativistic form for cosmology) is its governing equation. Cold dark matter starts as an extremely thin sheet in phase space, the single-stream limit f = ρ δ³(𝐯 − 𝐮), and the CBE reduces to continuity plus pressureless Euler until the sheet folds over at shell crossing, forming caustics and multi-stream regions. Every cosmological N-body simulation of structure formation solves this by sampling the dark-matter sheet with particles. The main caveat: self-interacting dark-matter models deliberately add a collision term, which is how simulations test whether observed halo cores rule the standard collisionless picture in or out.

Is the phase-space density really constant, given that galaxies clearly evolve?

Both statements are true because they refer to different objects. The fine-grained phase-space density — the value of f riding along with a star's orbit — is exactly constant, a theorem, not an approximation. Yet the galaxy's observable structure evolves: bars grow, spiral arms wind up, mergers scramble orbits, and the coarse-grained density smooths out. There is no contradiction: winding f into ever-finer filaments changes what any finite-resolution measurement reports while leaving f itself untouched. A sharp consequence is the phase-space density bound — since the fine-grained maximum of f can only be diluted by coarse-graining, never raised, the central coarse-grained f of a relaxed system can never exceed its initial peak, a constraint used to bound neutrino and dark-matter masses.