An equilibrium a three-dimensional flow cannot have

Where the inverse cascade stops computes what happens when energy arrives at the largest scale a box allows: it accumulates there as a pair of counter-rotating vortices filling the domain, and the limit of no friction has no steady state at all. That essay treats the condensate as a consequence of running out of room — a remainder, arrived at because there was nowhere else for the energy to be.

It is not a remainder. It is a thermodynamic equilibrium, in the same sense that a gas filling a container is one, and two-dimensional fluid mechanics is one of very few branches of the subject in which that sentence can be said at all.

The phase space is the fluid #

The argument begins with a fact about point vortices that is usually presented as a curiosity.

Circulation is vorticity added up, and a set of point vortices in a plane moves according to $\Gamma_i\dot x_i = \partial H/\partial y_i$ and $\Gamma_i\dot y_i = -\partial H/\partial x_i$, with

$$H = -\frac{1}{2\pi}\sum_{i<j}\Gamma_i\Gamma_j \ln|\mathbf r_i - \mathbf r_j|.$$

That is a Hamiltonian system in which the coordinates are their own conjugate momenta: $x$ and $y$ are the canonical pair. A vortex has no inertia of its own and no velocity to be specified independently of where it is, so the state of the system is a list of positions and nothing else.

The consequence is immediate and it is the whole essay. For a gas of molecules the phase space is positions and momenta, the momenta are unbounded, and the volume of phase space accessible below a given energy rises without limit. For a vortex gas in a bounded region the phase space is that region, repeated once per vortex, and its volume is finite whatever the energy.

That is worth restating as a piece of counting, because the abstraction hides how concrete it is. Thirty vortices in a disc have a phase space of thirty copies of the disc — sixty numbers, each bounded. Every configuration of the system is a point in that space and every point is a configuration; there is nothing else to specify. To ask how many states have a given energy is to ask what fraction of sixty-dimensional disc-space satisfies one equation, and that is a question a computer can answer by throwing darts.

That is exactly how the figures here are made. Configurations are drawn uniformly from the region, their energies are computed, and the histogram of those energies is the density of states — not an estimate of it, but the thing itself, because the measure the sampling uses is the measure the Hamiltonian’s own coordinates define. No Metropolis chain, no thermostat, and no assumption about what the equilibrium looks like enters anywhere.

The contrast with a molecular gas is worth one more line. There, a uniform sample of positions says nothing about the energy, because the energy is mostly kinetic and lives in the momenta; a statistical-mechanical calculation has to integrate over those separately, and the Maxwell distribution is the result. Here there are no momenta to integrate over, and the whole of statistical mechanics reduces to a geometry problem about one region.

An entropy that has a maximum #

A finite phase space forces an unusual shape on the entropy.

The number of states at energy $E$ is the volume of the surface $H = E$ in that finite space. It is zero at the extremes — it takes a very particular arrangement to make the energy very large or very small — and it is large in between. So it has a maximum at some finite energy, and $S = \ln W$ has a maximum there too.

Above that maximum, $\partial S/\partial E$ is negative. The thermodynamic temperature $T = (\partial S/\partial E)^{-1}$ is therefore negative, and a fluid with more than a certain amount of energy in it is at a negative temperature.

A negative temperature is not colder than absolute zero. It is hotter than infinity: a system at a negative temperature gives up energy to any ordinary system put in contact with it, at any positive temperature whatever. The sequence runs $+0\,\mathrm{K}$, warmer, $+\infty$, $-\infty$, warmer still, $-0\,\mathrm{K}$, and the vortex gas walks up it as energy is added.

Nothing about this is exotic in the sense of requiring strange conditions. It requires one thing — that the energy have an upper bound, or equivalently that the phase space be bounded — and a two-dimensional fluid in a container has it for free.

It is worth pausing on how rare that is in fluid mechanics. Almost every flow one can name has an unbounded energy available to it: a faster free stream, a larger pressure difference, a stronger pump. What a bounded phase space says is that the arrangement cannot absorb arbitrary energy — the vortices are already in the box, and the only way to raise the energy is to move them relative to one another, which there is a limit to. A fluid whose energy is stored entirely in the geometry of its own vorticity field, with no independent velocity to be increased, is a fluid whose energy has a ceiling.

That is also why the result belongs to two dimensions and not merely to point vortices. In three dimensions a vortex filament can be stretched, and stretching raises its energy without moving it anywhere — so the geometry no longer carries the whole of the energy, the phase space regains an unbounded direction, and the ceiling is gone.

Where the turning point is, and what sets it #

The peak sits at an energy near zero in the scaled units used here, and its position is worth a paragraph because it is the one number a reader could be suspicious of.

Zero energy in these units is the energy of a configuration in which the mean of $\ln r_{ij}$ over like-signed pairs equals the mean over opposite-signed ones — that is, a configuration with no correlation between sign and separation at all, which is what a uniform random draw produces on average. So the turning point is at the energy of a typical random arrangement, and it could hardly be anywhere else: the density of states peaks where the states are, and a uniformly drawn configuration is by definition what most states look like.

Read that way the whole result is almost tautological, and it is worth seeing that the tautology is the content. The maximum-entropy state is the random one, and the random one has a definite energy. Every configuration with more energy than that is, by counting, rarer — so adding energy to the fluid reduces the number of ways it can be arranged, which is a negative temperature, and the reduction shows up as an arrangement that is visibly not random.

Three different seeds put the peak within 0.7 per cent of the energy range of one another, which is the check that the number is the measure’s rather than the sample’s.

What a negative temperature does to a fluid #

The statistical statement becomes a hydrodynamic one as soon as it is asked what the high-energy states look like.

Raising the energy of a vortex gas means raising $-\sum\Gamma_i\Gamma_j\ln r_{ij}$. For a pair of the same sign that means making $r_{ij}$ small; for a pair of opposite signs it means making $r_{ij}$ large. So high-energy configurations are the ones in which like signs are close together and opposite signs are far apart, and at negative temperature entropy is maximised by raising the energy, so those are the configurations the system prefers.

The computed ratio runs from 1.05 at the low-energy end of the sampled range to 0.84 at the high end — which is not a large number and does not need to be, because it is an average over every pair in a finite sample. The direction is the result, and the direction is unambiguous over every bin.

The smallness deserves one honest sentence. A ratio of 0.84 means like-signed vortices are sixteen per cent closer together, on average over all pairs, than opposite-signed ones — and an average over all pairs in a thirty-vortex sample is dominated by pairs that are far apart and uncorrelated. What is being measured is a whole-system average of a local effect, so the effect itself is much larger than the number, and a statistic sensitive to near neighbours rather than to all pairs would show it plainly. The whole-system average is reported because it needs no threshold and no choice of what “near” means.

The energies at which that ratio is measured are also worth locating. The sampled range at thirty vortices runs from about −0.17 to +0.43 in these units, so the crossing through unity happens well inside the range rather than at its edge, and the curve is smooth and monotone across it. A result that appeared only in the last bin would be a result about the sampling.

A region of one sign and a region of the other is a condensate. What the plane’s cascade produces dynamically — energy arriving at the box scale and organising into two counter-rotating vortices filling the domain — is the equilibrium this argument predicts, and the agreement is not a coincidence but the same fact approached twice.

Why three dimensions cannot do this #

The contrast is what makes the result worth having, and it is sharp.

Three-dimensional turbulence has no equilibrium. Energy put in at large scales is carried to small ones and destroyed there, at a rate that does not go to zero with the viscosity; the flow is permanently out of equilibrium, sustained by a flux, and every attempt to write down an equilibrium statistical mechanics for it has produced something that either does not exist or predicts an equipartition nothing resembles.

Two dimensions differ in exactly one respect and it is decisive. The vortex-stretching term vanishes identically, enstrophy is conserved as well as energy, and the energy goes up in scale rather than down. There is no sink at small scales, so there is no flux to sustain, so the flow can reach a state and stay there. An equilibrium is possible because the dissipative route out of the system has been closed.

That is also the honest statement of the limitation. A real two-dimensional flow has some viscosity, the enstrophy cascade does reach small scales, and energy does leak away — slowly. The equilibrium argument describes a state the flow relaxes towards on a timescale shorter than the one on which it decays, and the two timescales separate widely only when the Reynolds number is large.

The separation of those two timescales is what makes the equilibrium picture useful rather than merely available, and it has a specific form. Energy in two dimensions is lost only through the viscous term acting on the enstrophy that has cascaded down, and the enstrophy cascade delivers its enstrophy to the dissipation scale while carrying almost no energy with it — that is the content of the two conservation laws themselves. So the energy decays on a viscous timescale built on the box, and the flow rearranges itself on a turnover time built on the box as well, and their ratio is a Reynolds number.

At a Reynolds number of a thousand the flow reaches its equilibrium a thousand times faster than it loses its energy, so it spends essentially all of its life in a state that is the equilibrium corresponding to whatever energy it has left. That is a quasi-static descent through a family of equilibria, which is exactly the situation classical thermodynamics is built to describe and is almost never available in fluid mechanics.

What the picture cannot show #

The vortices are points and a fluid’s vorticity is a field. A point vortex has infinite energy in its own core, which is why the self-interaction is dropped and why the energies here are quoted per vortex in a scaled unit. A continuous field has an infinity of further invariants — every integral of a function of the vorticity is conserved — and the theory that keeps them all is Miller’s and Robert and Sommeria’s, not Onsager’s.

The region is a confinement rather than a wall. The interaction used is the plane’s logarithm and the positions are restricted to a disc, which is Onsager’s own setting: it makes the phase space finite, which is all the argument needs, and it is not a solution of the no-penetration condition. A proper treatment adds an image system and shifts the numbers without changing the shape.

The sample is uniform and the tails are therefore thin, in the way an ensemble average hides its own rare members. The far ends of the energy range are rare under the uniform measure by construction — rarity is what a low density of states is — so the extremes of every curve here carry few samples and the error bars, which are not drawn, widen towards both edges.

And nothing above is dynamics. A statistical-mechanical argument says which state is overwhelmingly the most probable; it says nothing about whether the system gets there, or how long it takes. Two-dimensional flows are known to have long-lived states that are not the equilibrium.

And the vortices all have the same strength. Real two-dimensional turbulence — the kind that arrives as stripes on a rotating planet — produces vortices with a distribution of circulations and sizes, and that distribution is itself one of the conserved quantities the point-vortex model throws away. A gas of equal vortices and a gas with a spread of strengths reach different equilibria, and the second is closer to what a decaying flow contains.

Who found it, and when #

Onsager published the argument in 1949, in a paper on statistical hydrodynamics that also contains the first statement of the dissipative anomaly and of the scaling exponent for the velocity increments — three foundational results in a single short paper, two of which were ignored for decades. The negative-temperature vortex argument was the one that was noticed, and Montgomery and Joyce put it into the mean-field form in 1974, where the equilibrium is the solution of a sinh-Poisson equation and is a dipole filling the domain.

The surprising connection is with the only other physical systems that reach negative temperatures, which are nuclear spins. A set of spins in a magnetic field has a bounded energy — all aligned at one end, all anti-aligned at the other — and can be put into a negative-temperature state by inverting the population, which is what a laser does to its gain medium. A two-dimensional fluid and a population inversion are the same thermodynamics: in both, the energy has a ceiling, the entropy turns over, and the ordinary intuition that adding energy adds disorder is exactly wrong above the turning point. The difference is that a spin system has to be prepared into that state by an external agency, and a two-dimensional fluid gets there by being stirred.

Still open: what the continuous theory predicts for a real condensate #

The point-vortex argument gets the shape of the answer and cannot get its profile, and the gap is where the subject still is.

A vortex gas predicts clustering by sign. What a real two-dimensional flow makes is a pair of large vortices with a definite internal structure — a core whose vorticity is nearly uniform, a particular relation between the vorticity and the streamfunction, and a size set by the box. The continuous theories predict that relation: Montgomery and Joyce’s mean field gives $\zeta = \lambda\sinh(\beta\psi)$, and the Miller–Robert–Sommeria theory, which keeps every invariant of the continuous vorticity field rather than only the energy, gives a family of relations of which that is one member.

Measurements exist and do not settle it. Decaying two-dimensional simulations relax to states whose vorticity–streamfunction scatter plots are close to a sinh and are not obviously a sinh rather than a tanh or a linear relation over the range that is populated. The calculation that would separate them is not a better simulation but a better statistic: a quantity whose value differs between the candidate relations by more than the scatter, evaluated on the same fields everybody already has. Choosing that quantity is the work, and it is an unusually well-posed piece of work for a subject this old.