A relaxation that stops before it is finished

An elliptical galaxy has a hundred billion stars, and in the age of the universe essentially none of them has passed close enough to another to change its orbit. Two-body relaxation in such a system takes far longer than the universe has existed. Yet elliptical galaxies are in equilibrium — their stars’ speeds balance their gravity in exactly the proportion the virial theorem requires — and their light follows smooth profiles that look like the product of a thorough mixing. Something mixed them without encounters — and the equilibrium is real, because the speeds of the stars weigh the galaxy the way the theorem says they should.

Donald Lynden-Bell identified the mechanism in 1967. In a system collapsing from a cold, out-of-equilibrium start, the gravitational potential changes rapidly, on the same timescale as the orbits. A star’s energy is conserved only in a static potential; in one that changes in time, its energy changes by the rate of change of the potential where the star happens to be. Different stars, at different places and phases, gain and lose different amounts, so the collapse redistributes energies among them in a few crossing times — violent relaxation, which brings a system to virial equilibrium without a single encounter.

Lynden-Bell went further. He argued that the process should carry the system to its most probable state — the one with maximum entropy, given the constraints that collisionless dynamics imposes — and derived what that state looks like. The prediction has never matched what simulations produce. Collapses relax, and they relax quickly, but they stop short of the most probable state, and where they stop depends on how they started. The figures here show why, in the simplest system in which the question can be answered exactly.

A system with exact forces #

The system is a slab of parallel, infinite, self-gravitating sheets. Each sheet attracts every other with a force that does not depend on their separation — the field of an infinite plane is uniform — so a sheet’s acceleration is simply proportional to the mass on one side of it minus the mass on the other. Between the instants when two sheets cross, every sheet moves on a parabola; the total potential energy is a sum of the separations, known exactly. There is no softening, no approximation to the force, and the energy is conserved to a part in a hundred over the whole run.

The sheets also have no encounters in any useful sense. Two sheets passing through each other exchange nothing but their order; the force on each changes by a fixed amount, the same whether they pass at a metre or a light year. Relaxation by encounters does happen in a slab — the finite number of sheets makes the potential slightly grainy — but on a timescale of about the number of sheets times the crossing time, which for eight hundred sheets is ten times longer than anything in the figures. Everything that happens here is collective.

The figure follows eight hundred sheets in phase space. They start as a cold rectangle — uniform in position across a slab of width two, with velocities giving a tenth of the kinetic energy that equilibrium would need. The slab falls in; within three time units it has overshot and is rebounding, and the rectangle has sheared, because sheets that started further out fall further and take longer. By eight it is a spiral: each sheet orbits the centre of the slab at a period set by its own energy, the periods differ, and the initial line is wound up into an arm that grows longer and thinner with every turn. By a hundred and twenty the arms are too finely wound to resolve, and what remains is a steady state — a dense core surrounded by a scattering of sheets that were thrown out in the first collapse.

An equilibrium reached in a few crossings #

The virial theorem for this potential, which grows linearly with separation, says that a steady state has $2K = W$, twice the kinetic energy equal to the potential energy.

The approach to equilibrium is fast. The first swing is enormous — at maximum compression the kinetic energy is six times what equilibrium requires — and the next few are progressively smaller, so that by twenty time units, a dozen or so crossing times, the ratio oscillates by a few tens of per cent around one. That is the result the virial theorem predicts and observations of galaxies confirm: equilibrium is reached in a few dynamical times, regardless of the absence of encounters.

The figure also shows why the equilibrium is not the end of the story. The oscillation of the virial ratio is the potential changing in time — the very thing that redistributes the sheets’ energies. As the system settles, the oscillation decays, and with it the mechanism. The mixing is driven by the departure from equilibrium and removes it, so the process is self-limiting: it runs fast while there is a lot of it to do and slows as it succeeds. What remains at the end is a residual breathing of about a fifth in amplitude, which persists as long as the run does, because nothing in a collisionless system damps an oscillation that has become regular.

The state the mixing should have reached #

Lynden-Bell’s argument goes like this. A collisionless system’s phase-space density is conserved along each orbit — the fine-grained distribution is incompressible, and the regions of phase space initially occupied at a given density keep that density, merely being stretched and folded. What changes is the coarse-grained distribution, the average over small cells. Violent relaxation folds the initially occupied regions finely through the available phase space, and the most probable coarse-grained state, subject to conserving the total energy and the amount of each initial density, is a specific distribution with a Fermi–Dirac form, which for a dilute system is very close to the isothermal — the Boltzmann distribution of a gas in thermal equilibrium.

For the slab, the isothermal state is known exactly: a density falling as $\mathrm{sech}^2(x/z_0)$, with a scale height set by the velocity dispersion. It has an extended exponential tail; it has no memory of where anything started; it is the state a slab would reach if its sheets could exchange energy freely for long enough.

The relaxed slab is not isothermal. It has a core of about the right density, a shoulder between two and four scale heights holding more than twice the isothermal share of the mass, and then nothing. The shoulder is the sheets thrown out in the first collapse, which gained energy as the potential deepened beneath them and were left on wide orbits when it stopped changing. The sharp edge is where the most energetic of them turn round: in a slab, unlike a sphere, every orbit is bound, and the outermost sheets mark the energy the most favoured of them happened to reach.

A thermal system would have smoothed the shoulder into its tail by continuing to exchange energy between the sheets. The collisionless slab cannot, because once the potential stops fluctuating, nothing changes a sheet’s energy at all.

Energies that remember their order #

The incompleteness shows even more directly in the energies of individual sheets.

The most probable state would carry no information about the initial one. A sheet’s final energy would be drawn from the equilibrium distribution regardless of where it started. What the collapse leaves is a weak but unmistakable memory: sheets that started near the edge of the slab, with the most potential energy, tend to end least bound, and sheets that started near the centre tend to end most bound. The correlation is not large — the collapse shuffled the energies thoroughly — but it is seven times larger than chance could produce, and it is there in every run.

This is the statement Lynden-Bell’s theory cannot accommodate. The theory assumes the mixing continues until the system has explored every configuration consistent with its constraints. The memory says it stopped while the configurations were still partly sorted.

A warmer start ends somewhere else #

If the end state were the most probable one, it would depend only on the conserved quantities — the total energy and the phase-space densities — and not on the details of the start. Starting the same slab warmer tests that.

The warm slab ends with a different shape: a flatter core and less mass in the outskirts, where the cold one had more. Both differ from the isothermal sheet, in opposite directions in the outer parts. The difference follows from the violence of the collapse. A cold slab falls through itself with great speed and the potential at maximum compression is deep and brief, so the energy changes it imposes are large and concentrated — enough to throw out a shoulder. A warm slab barely collapses, the potential changes gently, and the sheets keep their initial energies more nearly.

The memory is correspondingly stronger: a rank correlation of 0.57 rather than 0.25. The two runs reach virial equilibrium equally well; they reach it in different states, each remembering its start to a degree set by how violent the collapse was. The equilibrium is unique in the sense the virial theorem uses — the total energies balance — and not unique in any finer sense.

Why the theory overshoots #

The failure of the most-probable-state prediction is now well understood, and the figures show its ingredients. The mixing requires the potential to fluctuate on the timescale of the orbits. The fluctuations are produced by the system being out of equilibrium. As the system approaches equilibrium the fluctuations decay, and the mixing slows. Relaxation is therefore exponential in character at best, and in practice it stalls: the last stages of mixing, which would be needed to erase the memory and fill in the tails, are driven by fluctuations too weak to accomplish them before they die.

There is a second, subtler reason. Lynden-Bell’s statistical argument treats phase-space elements as freely exchangeable, but the folding of phase space is not random. It proceeds by the winding of the spiral seen in the first figure — phase mixing — which stretches each element into a thin filament along an orbit of fixed energy. Phase mixing makes the coarse-grained distribution look smooth without exchanging energy between orbits at all. Only the time-dependent potential exchanges energy, and it does so efficiently only while the spiral is still coarse enough to make the potential lumpy. Once the spiral is fine, the potential is smooth, and what looks like a mixed system is merely a well-wound one.

The winding is the same arithmetic that keeps a spiral arm from being made of a fixed set of stars: orbits with different periods turn any initial line into a tightening spiral, in a galaxy’s disc over a few rotations and in the slab’s phase space over a few crossings. In the disc it is why arms must be waves; in phase space it is why a mixed-looking system need not be mixed.

The two processes — phase mixing, which is free and complete, and violent relaxation, which costs a fluctuating potential and stalls — are easily confused in a picture, because both turn a cold rectangle into a smooth blob. They differ in what they do to energies, and the memory figure is what tells them apart.

What it means for haloes and galaxies #

The same physics operates in three dimensions, with complications. Simulated dark-matter haloes, formed by the collapse and merging of cold structures, converge on nearly the same density profile regardless of their initial conditions — the NFW profile, with a cusp at the centre and a steeper fall-off outside — and that convergence has been called evidence that violent relaxation does reach a universal end state. But the universal profile is not the isothermal sphere Lynden-Bell’s theory predicts, and haloes formed from unusual initial conditions — very hot, or very smooth — do not converge on it.

The present understanding is that the universality comes from the hierarchical way haloes form, through many mergers each of which partially relaxes the result of the last, rather than from any one collapse reaching its most probable state. The innermost profile is still argued about, and part of that argument is about how much memory of their formation haloes retain. The slab says that some memory is to be expected — and that how much depends on how violent the formation was.

For elliptical galaxies the memory has an observable form. A galaxy held up by disorder has its random motions shared unequally between directions, and stars in the outer parts of ellipticals have radially elongated orbits more often than an isothermal distribution would give, and metallicity gradients — stars formed in the dense centre being richer in heavy elements — survive in many ellipticals despite their having been through violent relaxation. A complete mixing would have erased the gradients; an incomplete one, of the kind the slab shows, would preserve them partially, in proportion to how gently the galaxy was assembled.

Half a century of sheets #

The slab is not a new laboratory. Sheet models were among the first self-gravitating systems integrated on computers, in the 1960s, precisely because their forces are exact and their crossings can be followed event by event without any approximation. Frank Hohl and colleagues ran waterbag collapses of the kind drawn here within a few years of Lynden-Bell’s paper, and found what the figures show: rapid approach to virial equilibrium, a core–halo end state, and no convergence on the predicted distribution. The experiments have been repeated many times since with millions of sheets, and the disagreement has hardened rather than softened.

What the sheets offer that three-dimensional simulations do not is certainty about the cause. In a three-dimensional simulation of a collapsing halo, the force on each particle is softened, the number of particles is a compromise with the computing budget, and a disagreement with theory can always be blamed on the numerics. In a slab there is no softening to blame and the energy is conserved to a part in a hundred over the whole run, so a disagreement with the most-probable-state prediction is a disagreement with the theory. The sheets established, before any halo was simulated, that collisionless relaxation is real, fast and incomplete — and the three-dimensional work has since found the same three properties in systems that matter more.

Where the slab stops #

The sheets are infinite planes. Real systems are three-dimensional, where orbits have angular momentum, the potential is not linear in separation, and the outermost particles can escape rather than turning round. The slab’s sharp edge has no exact counterpart in a sphere, where the energetic tail can extend indefinitely.

The number of sheets is finite. Encounters do eventually matter: given long enough, the graininess of the potential drives a slow, genuinely collisional relaxation toward the isothermal state — the process that, in three dimensions, makes a star cluster boil away its outer members and heat up as it loses energy. In the slab, eight hundred sheets are enough to make the collective behaviour clear and few enough that the graininess of the potential could matter on long timescales; it does not over the hundred and twenty time units drawn, because the two-body time is about ten times longer.

The isothermal comparison stands in for Lynden-Bell’s distribution. For the low phase-space densities reached here the two are nearly the same; the exact Lynden-Bell state for a waterbag start is a Fermi–Dirac distribution with a degeneracy that makes the core slightly flatter, which does not bring it closer to the relaxed slab’s shoulder and edge.

Each run is one realisation. Different random placements of the sheets give the same picture within a few per cent; the correlations in the memory figures are many times their own statistical uncertainty.

Still open: what decides the end state #

What the slab shows is that the end state of violent relaxation is not the most probable state and does depend on the start. What it does not supply is a theory of where the relaxation stops. Several proposals exist — that it stops when the fluctuation spectrum has decayed below some threshold, that it reaches a “core–halo” state whose core is maximally mixed and whose halo is frozen, that a different kind of entropy is maximised — and each fits some simulations and not others.

The question is not academic. Dark-matter haloes and elliptical galaxies are the two most common products of violent relaxation in the universe, and the profiles of both are measured with growing precision and compared with models that assume one end state or another. A theory that predicted, from the initial conditions of a collapse, how far toward the most probable state it would get, would turn the shapes of haloes and galaxies into a record of how violently they formed. The slab, where the forces are exact and the run takes a second, is where such a theory would first be tested.