Phase Mixing and Chaotic Mixing

A galaxy is a paradox of tempo. A large elliptical or a dark-matter halo contains so many stars that direct two-body encounters are utterly negligible over the age of the Universe — its collisional relaxation time exceeds the Hubble time by factors of millions. Yet these systems look thoroughly relaxed: smooth, near-equilibrium, having apparently forgotten how they were born, and they reached that state in a few crossing times, not a few relaxation times. Something smooths a collisionless system with no collisions at all — the smooth mean-field potential itself, through two mechanisms: phase mixing and chaotic mixing. Phase mixing is the gentle one. Whenever orbits carry a spread of frequencies — and any realistic anharmonic potential guarantees they do — a compact bundle of stars is sheared: each advances in phase at its own rate Ω(J), so the clump winds into an ever-thinner, ever-more-folded ribbon around the invariant tori. Chaotic mixing is the violent one: in the stochastic zones of non-integrable potentials — near a triaxial cusp or a central black hole — neighbouring orbits diverge not linearly but exponentially, at the largest Lyapunov exponent, filling their energy shell ergodically in a handful of e-folding times. Both are collisionless, both run on the dynamical time, and both leave the fine-grained distribution function exactly conserved while smoothing everything we can measure — the microscopic engine of Lynden-Bell's violent relaxation, and the reason the Universe is full of smooth galaxies far younger than their own relaxation clocks.

Key takeaways

Phase mixing is algebraic smoothing from a spread in orbital frequencies: nearby stars drift apart in phase as Δθ = Ω′ΔJ·t, winding a patch into filaments whose width thins as w ∝ 1/t.
Chaotic mixing is exponential and far faster: in stochastic regions neighbouring orbits separate as e^(λt) at the largest Lyapunov exponent λ, filling an energy shell ergodically in a few e-folding times.
Both are collisionless — no two-body encounters — and act on the crossing time t_cross, smaller than the two-body relaxation time t_relax by a factor ~10⁸ for a galaxy.
Liouville's theorem pins the fine-grained f exactly (its maximum never rises), but the observable coarse-grained f̄ smooths toward a lower-peak, phase-independent equilibrium — a function of the integrals of motion, as Jeans' theorem demands.
These are the mechanisms behind violent relaxation, tidal-stream survival, the erosion of triaxiality near galactic centres, and the multi-stream phase-space structure of cold dark matter.
Key idea: a collisionless stellar system relaxes not by collisions but because a spread in orbital frequencies (phase mixing) or exponential orbit divergence (chaotic mixing) shreds fine phase-space structure into unresolvably thin filaments; the fine-grained f is exactly conserved (Liouville), while the observable coarse-grained f̄ smooths toward equilibrium.

Where this lives today

The most spectacular modern sighting caught the Milky Way in the act. In 2018 Gaia, mapping the six-dimensional phase space of millions of stars, revealed a striking spiral — the "Gaia snail" — in the vertical position-velocity plane $(z,v_z)$ . It is a textbook phase-mixing signature: the disc was perturbed a few hundred million years ago (probably by a satellite like the Sagittarius dwarf), the vertical frequency varies with amplitude, and the coherent disturbance has wound ever tighter since — counting its wraps dates the event and weighs the local disc, phase mixing turned into a clock. The same physics is the daily bread of galactic dynamics: N-body simulations resolve violent relaxation driving haloes to their near-universal NFW profiles; frequency-map analysis (after Laskar) diagnoses chaotic zones and how central black holes erode triaxiality; and streams like GD-1 and Sagittarius are modelled as phase-mixing action-angle ribbons whose gaps betray dark-matter subhaloes. Cosmologically, folding the cold-dark-matter phase-space sheet makes caustics and a multi-stream local distribution, while the Liouville bound — mixing can never raise $\max f$ — sets a floor that constrains warm and fuzzy dark-matter models.

The physical picture

Return to the coffee cup. Stirred only by the differential rotation of the fluid, a blob of cream is drawn into thinner and thinner spiral streaks until the cup looks uniform — never destroyed, merely divided too finely to see. That is phase mixing: the shearing of a phase-space patch by a gradient of orbital frequency, conserving the fine-grained density while smoothing the coarse-grained view. Stir vigorously to turbulence and two nearby specks fly apart exponentially, going uniform almost instantly — that is chaotic mixing, the Lyapunov exponent as stretching rate. The clean way to see phase mixing is on the action-angle torus, where a regular orbit is an angle θ winding at constant rate Ω(J). Because Ω depends on J, a small square patch shears into a long diagonal ribbon that wraps the torus many times, thinning transversely to conserve area.

Figure

Phase mixing on the action-angle torus: an initial square patch in green shears because Ω(J) varies with action, so Δθ = Ω′(J)·ΔJ·t and it winds into about 4 turns after 8 crossing times while its width thins as w ∝ 1/t at fixed area

The difference between the two mixing modes is entirely one of rate. In a regular (integrable) region two nearby orbits separate only linearly — slow, algebraic phase mixing; in a chaotic region they separate exponentially, finishing the same job dozens of times faster. The second figure puts a number on it: with a Lyapunov rate of half a crossing time, ten crossing times of chaos separate a pair by e⁵ ≈ 148, while phase mixing over the same interval spreads them only about ten-fold.

Figure

Two ways nearby orbits separate: phase mixing spreads them only linearly, δ ∝ t, while chaotic mixing grows them exponentially, δ ∝ e^(λt) with rate λ = 0.5 per crossing time — so after 10 crossing times the chaotic pair has diverged e^5 ≈ 148-fold versus about 10-fold

Whichever mode acts, the galaxy we observe is always a coarse-grained view: no telescope, and no finite-resolution N-body snapshot, resolves the micron-thin filaments mixing produces. What we measure — density, velocity dispersion, the Jeans moments — reflects the smoothed f̄, which relaxes while the fine-grained f is only folded, never erased.

The core equations

The bedrock is that a collisionless phase-space fluid is incompressible: the fine-grained distribution function is constant along every orbit. This collisionless Boltzmann equation — Liouville's theorem for the one-particle DF — is precisely why "relaxation" here can never mean a rise in the fine-grained density.

\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 (1)

where: $f(\mathbf{x},\mathbf{v},t)$ — fine-grained phase-space distribution function; $\dot{\mathbf{x}}=\mathbf{v}$ and $\dot{\mathbf{v}}=-\nabla\Phi$ — motion in the smooth mean-field potential $\Phi$ . Reading: $f$ is merely transported, never created or destroyed, so its maximum is an exact invariant. For a regular orbit in an integrable potential the natural coordinates are action-angle variables, in which $H=H(\mathbf J)$ : the actions are frozen and the angles wind uniformly, so the evolution of $f$ is a rigid shear in the angle direction.

f(\mathbf{J},\boldsymbol{\theta},t)=f\big(\mathbf{J},\;\boldsymbol{\theta}-\boldsymbol{\Omega}(\mathbf{J})\,t\big) \qquad (2)

where: $\mathbf{J}$ — actions, conserved for regular orbits; $\boldsymbol{\theta}$ — angle variables, $0\le\theta_i<2\pi$ ; $\boldsymbol{\Omega}(\mathbf{J})=\partial H/\partial\mathbf{J}$ — orbital frequency vector. Reading: each torus winds at its own rate, dragging the pattern of $f$ around torus by torus. That variation is the whole story of phase mixing: two orbits differing by a small action $\Delta J$ drift apart in angle at the frequency-gradient rate, so their separation grows linearly — algebraically — in time.

\Delta\theta_i(t)=\frac{\partial \Omega_i}{\partial J_j}\,\Delta J_j\,t \qquad (3)

where: $\Delta\theta_i$ — angular separation at time $t$ ; $\Delta J_j$ — initial action separation; $\partial\Omega_i/\partial J_j$ — frequency gradient (anharmonicity), which must be non-zero. Reading: no frequency gradient, no phase mixing — a harmonic potential never shears. By Liouville area conservation the swept-out patch area is fixed, so stretching in angle forces the transverse filament width to collapse as $w\propto1/t$ (worked out in full in the first derivation below) — phase mixing thins structure only algebraically. Chaotic mixing replaces that algebraic law with an exponential one: linearising the motion about a reference orbit, a deviation vector grows in a stochastic region at the largest Lyapunov exponent, whose inverse is the chaotic mixing time.

\lambda=\lim_{t\to\infty}\frac{1}{t}\ln\frac{|\delta\mathbf{w}(t)|}{|\delta\mathbf{w}(0)|},\qquad |\delta\mathbf{w}(t)|\sim|\delta\mathbf{w}(0)|\,e^{\lambda t} \qquad (4)

where: $\lambda$ — largest Lyapunov exponent ( $\lambda>0$ chaos, $\lambda=0$ regular); $\delta\mathbf{w}$ — phase-space separation of two close orbits; $1/\lambda$ — e-folding (Lyapunov) time. Reading: symplectic exponents come in $\pm$ pairs summing to zero, so stretching one way is matched by compression another — volume is preserved as the blob shreds. What "relaxes" is not $f$ but its coarse-grained average over a small but finite cell — the only thing any measurement or simulation resolves.

\bar f(\mathbf{w},t)=\frac{1}{V}\int_{V(\mathbf{w})} f(\mathbf{w}',t)\,\mathrm{d}^6\mathbf{w}' \qquad (5)

where: $\bar f$ — coarse-grained DF over a cell of volume $V$ ; $V(\mathbf{w})$ — averaging cell at phase point $\mathbf{w}$ ; $f$ — fine-grained DF conserved by (1). Reading: once filaments thin below the cell it holds many interleaved strands, so as high- $f$ filaments blend with low- $f$ voids the coarse-grained peak can only fall, $\max\bar f(t)\le\max\bar f(0)$ , ending on a stationary function of the integrals of motion — the collisionless arrow of time and the microscopic justification of Jeans' theorem (third derivation below). Finally, why any of this matters: the mixing clock is the crossing (dynamical) time, smaller than the two-body relaxation time by a factor scaling with the enormous star count $N$ .

t_{\mathrm{cross}}\sim\frac{R}{v}\sim\frac{1}{\sqrt{G\bar\rho}}\;\ll\; t_{\mathrm{relax}}\sim\frac{0.1\,N}{\ln N}\,t_{\mathrm{cross}} \qquad (6)

where: $t_{\mathrm{cross}}$ — crossing/dynamical time (the mixing clock); $t_{\mathrm{relax}}$ — two-body (collisional) relaxation time; $N$ — number of stars; $\bar\rho$ — mean mass density. Reading: for a galaxy with $N\sim10^{11}$ the two clocks differ by $\sim10^8$ , so mixing wins overwhelmingly.

Derivation APhase mixing shears and thins a phase-space patch

The goal is to start from free streaming in angle variables and show, step by step, that a compact phase-space patch winds into a filament whose resolvable width decays as $1/t$ , so the coarse-grained distribution function smooths toward a phase-independent equilibrium while the fine-grained one is conserved exactly — leaning only on the angle advance and Liouville area conservation, no collisions and no chaos, just anharmonicity.

Step 1. On an invariant torus the actions are frozen and the angle advances uniformly, $\theta(t)=\theta_0+\Omega(J)\,t$ — Hamilton's equations in action-angle form, $\dot J=0$ and $\dot\theta=\Omega(J)$ , when $H=H(J)$ . Why: it isolates the one ingredient that drives phase mixing — a frequency that depends on which torus you are on.

Step 2. Two orbits differing by a small action $\Delta J$ separate as $\Delta\theta(t)=\Delta\theta_0+\Omega'(J)\,\Delta J\,t$ , growing linearly because their frequencies differ by $\Omega'\Delta J$ . Why: this linear-in- $t$ growth is the signature of integrable phase mixing, in contrast to the exponential chaotic case.

Step 3. After time $t$ the patch spans $\Delta\theta=\Omega'\Delta J\,t$ radians, wrapping the torus $n(t)=\Omega'\Delta J\,t/2\pi$ times — in the periodic angle coordinate, a set of $\sim n$ parallel diagonal strands. Why: the winding number sets how thoroughly a coarse-grained cell is criss-crossed, hence how smooth $\bar f$ becomes.

Step 4. Liouville fixes the patch area, $A=\Delta\theta\cdot w=\mathrm{const}$ , so as $\Delta\theta$ grows the width must collapse, $w(t)=A/(\Omega'\Delta J\,t)\propto t^{-1}$ . Why: here fine-grained conservation and coarse-grained smoothing reconcile — the density is preserved by being spread too thin to resolve.

Step 5. Once $w$ drops below the cell, the coarse-grained value becomes the angle-average, $\bar f\to\langle f\rangle_\theta$ — smooth and phase-independent, a function of the actions as Jeans' theorem requires. Why: it closes the loop, delivering equilibrium $\bar f$ from a conserved, non-equilibrium $f$ .

Go deeper · Derivation B · 1 minExponential divergence and the Lyapunov exponent

Derivation BExponential divergence and the Lyapunov exponent

The goal is to linearise the equations of motion about a reference trajectory and show that in a chaotic region a deviation vector grows exponentially, defining the largest Lyapunov exponent and the mixing timescale $1/\lambda$ , while a symplectic flow forces the exponents into cancelling $\pm$ pairs so exponential stretching coexists with exact volume conservation — the divergence that makes chaotic mixing far faster and more thorough than phase mixing.

Step 1. Write the flow as $\dot{\mathbf w}=\mathbf F(\mathbf w)$ with $\mathbf w=(\mathbf x,\mathbf v)$ and $\mathbf F=(\mathbf v,-\nabla\Phi)$ — Hamiltonian motion in six-dimensional phase space. Why: it puts positions and velocities on one footing so a single deviation vector tracks the full separation of two orbits.

Step 2. For a small separation $\delta\mathbf w$ , differentiating gives $\dot{\delta\mathbf w}=(\partial\mathbf F/\partial\mathbf w)|_{\mathbf w(t)}\,\delta\mathbf w\equiv\mathbb J(t)\,\delta\mathbf w$ , with $\mathbb J$ the Jacobian along the reference orbit. Why: near-neighbour behaviour is set entirely by the local linearisation, whose eigenstructure decides regular versus chaotic.

Step 3. Integrating, $|\delta\mathbf w(t)|=|\delta\mathbf w(0)|\exp\!\int_0^t\sigma\,\mathrm dt'$ , with $\sigma$ the instantaneous expansion rate. In a regular region stretching and squeezing average out (algebraic growth); in a stochastic region the average is net positive. Why: the sign of that time-averaged rate is the boundary between quasiperiodic and chaotic motion.

Step 4. The largest Lyapunov exponent is the long-time average $\lambda=\lim_{t\to\infty}\tfrac1t\ln[|\delta\mathbf w(t)|/|\delta\mathbf w(0)|]$ : $\lambda>0$ is chaos with $|\delta\mathbf w|\sim e^{\lambda t}$ , and near a triaxial cusp $1/\lambda$ is only a few crossing times. Why: $1/\lambda$ is the physical clock of chaotic mixing, directly comparable to the crossing time.

Step 5. Because the flow is symplectic, the spectrum comes in pairs $\{+\lambda_i,-\lambda_i\}$ summing to zero: a blob stretched one way is compressed another, six-volume exactly preserved, while it folds ergodically over its energy shell. Why: this is how exponential mixing and Liouville conservation coexist, and why the fine-grained peak still cannot rise.

Go deeper · Derivation C · 1 minThe coarse-grained H-function and the arrow of time

Derivation CThe coarse-grained H-function and the arrow of time

The goal is to make the irreversibility of mixing precise: to build a functional of the coarse-grained distribution function that increases monotonically as mixing proceeds — even though every functional of the fine-grained one is exactly conserved — thereby proving that the coarse-grained peak $\max\bar f$ can only fall. The tool is convexity, via Jensen's inequality applied to the gap between cell-averaging and integrating over all phase space: the collisionless analogue of the second law.

Step 1. For any $C$ , the fine-grained integral $\int C(f)\,\mathrm d^6w$ is constant, because Liouville flow only rearranges the values of $f$ without changing the measure of any level set. Why: the fine-grained DF carries no arrow of time on its own — reversibility is exact microscopically.

Step 2. Take $C$ convex and coarse-grain: inside a cell, $C(\bar f)=C(\langle f\rangle_V)\le\langle C(f)\rangle_V$ by Jensen's inequality, with equality only if $f$ is uniform there. Why: convexity is the one fact that breaks the symmetry between fine and coarse graining and injects an inequality.

Step 3. Integrate over all cells: $\int C(\bar f)\,\mathrm d^6w\le\int C(f)\,\mathrm d^6w=\mathrm{const}$ . With $H[\bar f]=-\int C(\bar f)\,\mathrm d^6w$ this reads $H[\bar f(t)]\ge H[\bar f(0)]$ , strict whenever sub-cell structure develops. Why: it turns the local inequality into a global, monotone functional — an H-theorem with no collisions at all.

Step 4. Mixing is exactly what generates that sub-cell structure, so the inequality is strict and $H[\bar f]$ climbs toward its extremum, where $\bar f$ depends only on the integrals of motion and $\max\bar f$ has fallen. Why: it names mixing as the driver of the collisionless arrow of time and fixes the smoother, lower-peak end state we observe.

Limiting cases

How mixing behaves in the regimes that bracket the physics:

Case	Result	Meaning
Harmonic potential (no frequency spread)	$\partial\Omega/\partial J=0 \Rightarrow$ no phase mixing	All orbits share one frequency; a clump orbits rigidly and never shears. Mixing requires anharmonicity.
Integrable regular orbits	$\lvert\Delta\theta(t)\rvert\sim\lvert\partial\Omega/\partial J\rvert\,\Delta J\,t\propto t$	Phase mixing: linear separation, filament width thins as $1/t$ . Slow and cold — streams survive.
Chaotic (stochastic) orbits	$\lvert\Delta\mathbf w(t)\rvert\sim\lvert\Delta\mathbf w_0\rvert\,e^{\lambda t},\ \lambda>0$	Chaotic mixing: exponential divergence at rate $\lambda$ , completing in a few $1/\lambda$ , ergodically filling the shell.
Collisional limit (small $N$ )	$t_{\mathrm{relax}}\sim(0.1\,N/\ln N)\,t_{\mathrm{cross}}$	For small $N$ (globulars) two-body relaxation eventually dominates the late evolution mixing cannot reach.

Common trapPhase mixing erases the information in the distribution function

It does not. The fine-grained distribution function is conserved exactly by Liouville and its maximum never changes — the information is preserved microscopically, merely folded too thin to resolve; only the coarse-grained density evolves, because averaging over a finite cell blends high- $f$ and low- $f$ strands. Two further traps: do not confuse this collisionless smoothing with two-body relaxation (utterly different clocks, $t_{\mathrm{cross}}$ versus $t_{\mathrm{relax}}$ , a factor $\sim10^8$ apart); and phase mixing needs a non-zero frequency gradient $\mathrm d\Omega/\mathrm dJ\neq0$ — a perfect harmonic potential does not phase mix at all, a clump just orbits rigidly forever.

Worked examples

1 · Why a galaxy is collisionless but a globular cluster is not (fully worked)

Compare the two-body relaxation time to the crossing time for a Milky-Way-scale galaxy ( $N\approx10^{11}$ , $t_{\mathrm{cross}}\approx2\times10^{8}$ yr) and for a globular cluster ( $N\approx10^{6}$ , $t_{\mathrm{cross}}\approx10^{6}$ yr). Which relaxes collisionlessly, and which eventually relaxes by collisions?

Step 1. Use equation (6), $t_{\mathrm{relax}}\approx(0.1\,N/\ln N)\,t_{\mathrm{cross}}$ . For the galaxy, $\ln N=\ln(10^{11})=25.3$ , so the ratio $0.1N/\ln N=0.1\times10^{11}/25.3=3.9\times10^{8}$ .

Step 2. Multiply by the crossing time: $t_{\mathrm{relax}}=3.9\times10^{8}\times2\times10^{8}\ \mathrm{yr}=\mathbf{7.9\times10^{16}\ yr}$ — about six million Hubble times ( $t_H\approx1.4\times10^{10}$ yr). Two-body encounters are irrelevant, yet mixing smooths the galaxy on $t_{\mathrm{cross}}\sim2\times10^{8}$ yr, a factor $\sim4\times10^{8}$ faster. The galaxy is collisionless.

Step 3. Now the globular cluster: $\ln N=\ln(10^{6})=13.8$ , so $0.1N/\ln N=0.1\times10^{6}/13.8=7.2\times10^{3}$ , and $t_{\mathrm{relax}}=7.2\times10^{3}\times10^{6}\ \mathrm{yr}=\mathbf{7.2\times10^{9}\ yr}$ .

Step 4. That is a fraction of the Universe's age, so an old globular has time to two-body relax — hence the collisional signatures (mass segregation, equipartition, core collapse) that galaxies never show. The single parameter $N$ decides it: mixing smooths collisionlessly on $t_{\mathrm{cross}}$ ; only for small $N$ does the slow $t_{\mathrm{relax}}$ clock catch up within a Hubble time.

2 · How fast does chaos fill a triaxial centre? (fully worked)

A box orbit passing near a triaxial galactic centre is chaotic with Lyapunov time $1/\lambda=3\,t_{\mathrm{cross}}$ . Two orbits start a fraction $\delta_0/L=10^{-3}$ of the system size apart. How many crossing times until they diverge across the whole system, and how does that compare with phase mixing?

Step 1. Exponential divergence, equation (4): $\delta(t)=\delta_0\,e^{\lambda t}$ . The pair fills the system when $\delta/\delta_0=10^{3}$ , i.e. $e^{\lambda t}=10^{3}$ , so $\lambda t=\ln(10^{3})=6.9$ .

Step 2. Convert to crossing times using $1/\lambda=3\,t_{\mathrm{cross}}$ : $t=6.9/\lambda=6.9\times3\,t_{\mathrm{cross}}=\mathbf{21\ t_{cross}}$ . Chaotic mixing completes in a couple of dozen crossing times.

Step 3. Contrast phase mixing, where separation grows only linearly, $\delta\propto t$ . To reach the same factor $10^{3}$ takes of order $10^{3}\,t_{\mathrm{cross}}$ — roughly 50 times longer than the chaotic route.

Step 4. This is why the centres of triaxial and barred galaxies relax first: box orbits supporting the triaxial figure pass through the chaotic zone near the cusp or black hole, redistribute over their energy shells in a few Lyapunov times, and the triaxiality erodes toward axisymmetry. Chaos does in ~21 crossing times what phase mixing needs ~1000 to do.

3 · Why tidal streams survive: the winding of a stream (fully worked)

A stream from a disrupted dwarf orbits with period $T\approx250$ Myr and a fractional frequency spread $\Delta\Omega/\Omega\approx0.05$ . After $t=8$ Gyr, how far has it wound, and what does that say about survival?

Step 1. In $8$ Gyr the stream completes $t/T=8000/250=32$ orbits, so it winds $n=(\Delta\Omega/\Omega)\times32=0.05\times32=\mathbf{1.6}$ turns — an angular stretch $\Delta\theta=\Delta\Omega\,t=1.6\times2\pi\approx\mathbf{10\ rad}$ .

Step 2. So the stream spans $\sim10$ rad of phase but has wound only $\sim1.6$ times: long and thin, its velocity width thinned only algebraically ( $\propto1/t$ ). Had the orbit been chaotic with $\lambda\sim\Omega$ the separation would have grown like $e^{2\pi\times32}$ — dissolving the stream almost at once. Cold streams stay coherent for gigayears precisely because they phase mix algebraically in the regular part of the potential. That slow $1/t$ thinning makes streams like GD-1 and Sagittarius sharp probes of the Milky Way's mass and dark-matter substructure.

Physics of the portrait

Phase mixing and chaotic mixing were made rigorous by one 1967 insight: that a collisionless system relaxes by rearranging phase space, not by colliding. The portrait is Donald Lynden-Bell, who named violent relaxation and later helped prove the mixing theorem beneath it.

Origins · 1 minDonald Lynden-Bell and how this was discovered

Donald Lynden-Bell

1935–2018 · violent relaxation and the statistical mechanics of collisionless stellar systems

Donald Lynden-Bell was among the most inventive theoretical astrophysicists of the twentieth century, and he essentially defined the problem of collisionless relaxation. The puzzle was stark: elliptical galaxies look smooth and relaxed, yet their two-body relaxation times exceed the age of the Universe by factors of millions. In his landmark 1967 paper (MNRAS 136, 101) he introduced violent relaxation — during a rapid collapse the potential fluctuates strongly, energies are not conserved, frequencies are violently reshuffled, and the system phase- and chaotically mixes to near-equilibrium in a few crossing times. He went further, predicting the relaxed distribution function by counting phase-space microstates under the constraint that the fine-grained density is conserved, obtaining the Fermi-Dirac-like form now called Lynden-Bell statistics. The subtlety that mixing conserves the fine-grained f while smoothing the coarse-grained one became precise in 1986, when Tremaine, Hénon and Lynden-Bell proved coarse-grained H-theorems for any convex functional of the smoothed density. His reach was vast — the "Great Attractor" bulk flow, the prediction that quasars are powered by accretion onto supermassive black holes, the inaugural 2008 Kavli Prize — but violent relaxation and phase-space mixing remain his most quietly universal legacy.

History

1942

— Chandrasekhar sets out the two-body relaxation time, showing galaxies are effectively collisionless

1960

— dynamicists sharpen the paradox that galaxies look relaxed far faster than two-body times allow

1964

— Hénon and Heiles exhibit chaotic stellar orbits in a simple galactic potential

1967

— Lynden-Bell formulates violent relaxation and the statistical mechanics of collisionless systems

1986

— Tremaine, Hénon and Lynden-Bell prove the coarse-grained mixing H-theorem

Going further

Texts. Binney and Tremaine, Galactic Dynamics, 2nd ed., section 4.10 (phase mixing) and section 3.7 (chaotic orbits, Lyapunov exponents) · Carroll and Ostlie, An Introduction to Modern Astrophysics, Ch. 24–25 · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 5 · Lynden-Bell (1967), MNRAS 136, 101 (violent relaxation) · Tremaine, Hénon and Lynden-Bell (1986), MNRAS 219, 285 (H-function theorem) · Dodelson, Modern Cosmology. The key self-consistency checks are: Hamilton's equations in action-angle form give $\dot{\boldsymbol\theta}=\boldsymbol\Omega$ and $\dot{\mathbf J}=0$ ; Liouville area conservation forces the filament width $w\propto t^{-1}$ ; symplectic exponents come in $\pm$ pairs summing to zero; and Jensen's inequality gives $\int C(\bar f)\le\int C(f)$ for any convex $C$ . The worked numbers are: galaxy $t_{\mathrm{relax}}=7.9\times10^{16}$ yr, globular $t_{\mathrm{relax}}=7.2\times10^{9}$ yr, chaotic fill $\approx21\,t_{\mathrm{cross}}$ , and stream winding $n\approx1.6$ . Star counts and crossing times are standard order-of-magnitude values.

Prerequisites. Collisionless Boltzmann Equation · Distribution Function and Phase Space · Action-Angle Variables · Two-Body Relaxation Time. Leads to. Jeans' Theorem · Virial Theorem in Stellar Systems · Orbits in Axisymmetric and Triaxial Potentials.

Ask the tutor: Ask me to unwind the Gaia phase-space spiral, estimate a Lyapunov time for a box orbit, or show exactly how Liouville conservation and coarse-grained relaxation coexist.

Is phase mixing the same thing as violent relaxation?

Related but not identical. Phase mixing and chaotic mixing are the microscopic mechanisms — the shearing and stretching that smooth the coarse-grained density. Violent relaxation is the rapid episode in which they run at full tilt: while a system collapses or merges, the potential Φ(x, t) fluctuates strongly, energies are not conserved, and frequencies are violently reshuffled. That reshuffling maximises the frequency spread and generates widespread chaos, so both modes act together to reach quasi-equilibrium in a few crossing times. In a static potential you still get phase mixing, just more gently. Violent relaxation is simply the regime where mixing is fastest and most complete.

If the fine-grained distribution function is conserved, in what sense does the system relax?

Only the coarse-grained f̄ — the average over a small but finite cell — evolves. Mixing shreds the fine-grained f into ever-thinner filaments; once those fall below the cell size, any real measurement averages over many of them and sees a smooth, phase-independent function whose peak has dropped and which tends to a stationary function of the integrals of motion. That is a relaxed equilibrium. The microscopic information is all still there — Liouville guarantees it — but folded too fine to resolve, so it is inaccessible in practice. Relaxation here is about what we can observe, not what the dynamics forgets.

Why does phase mixing require a spread in orbital frequencies?

Because phase mixing is nothing but shearing by a frequency gradient. Two nearby orbits drift apart at Δθ = Ω′(J)·ΔJ·t, proportional to the anharmonicity Ω′ = ∂Ω/∂J. If every orbit shares one frequency — a harmonic potential, or the harmonic core of an isochrone — then Ω′ = 0 and a clump orbits rigidly, returning to its shape once per period forever: nothing shears, winds, or thins. Real galactic potentials are strongly anharmonic (frequencies fall with energy and depend on orbit shape), so their gradients are large and phase mixing is efficient. The harmonic case is the singular exception where it switches off entirely.

Why is chaotic mixing so much faster than phase mixing?

Linear versus exponential growth. In a regular region nearby orbits separate only linearly, δ ∝ t, so spreading a pair a thousand-fold takes of order a thousand crossing times. In a chaotic region the separation grows as e^(λt), so the same factor is reached when λt = ln(1000) ≈ 6.9 — a few e-folding times, a couple of dozen crossings. Exponential growth also means a chaotic orbit ergodically fills its whole energy shell rather than winding on one torus. So chaotic mixing finishes sooner and mixes more thoroughly — which is why the chaotic centres of triaxial galaxies relax first.

What sets the maximum central density of a dark-matter halo?

Ultimately, Liouville's theorem. Cold dark matter is born with a tiny velocity dispersion, occupying a thin phase-space sheet of large but finite fine-grained density f. Mixing during halo formation stretches and folds that sheet but can never raise the maximum of f — it only spreads it over finer structure — and coarse-graining lowers the observable peak. That bounded density caps the central phase-space density a halo can reach, which constrains models: warm dark matter, born with a larger dispersion, starts with a lower f and cannot build cores as dense as cold dark matter, so dwarf-galaxy centres constrain the particle.

How can mixing be irreversible if the underlying dynamics is time-reversible?

The same puzzle as in ordinary statistical mechanics. The fine-grained dynamics is perfectly reversible — run the collisionless Boltzmann equation backwards and the filaments unwind. The irreversibility lives entirely in the coarse-graining: averaging over a finite cell discards the sub-cell information about how the filaments are arranged, which is exactly what reversing the evolution would require. Formally, any convex functional of the coarse-grained density evolves one way only (the H-theorem), so the peak can only fall. No contradiction — the microscopic laws are reversible, but the macroscopic quantities we track are not, because measuring them throws away the fine structure. The arrow of time here is an arrow of observability.