The point at which the two become one

Seal carbon dioxide in a thick glass tube at the right density and warm it. Below 31 °C there is liquid at the bottom and vapour above, with a meniscus between them that is easy to see. Warm it further and the liquid does not boil away and the vapour does not condense. The meniscus stays where it is and becomes harder to see, because the liquid is getting less dense and the vapour is getting more dense, and at 31.0 °C the two are the same substance and there is no longer anything for a surface to separate.

The quantity plotted there is the order parameter: something that is non-zero in the ordered phase, zero in the disordered one, and goes continuously to zero at the transition. For a fluid it is the density difference between the two phases. For a magnet it is the magnetisation. For a binary alloy it is the excess of one species on one sublattice. The three systems have nothing physical in common and the exponent with which the order parameter vanishes is the same in all of them, to the accuracy of the measurements. That a magnet and a fluid should share a number is the sort of coincidence which, in this collection, usually turns out to be one equation wearing two coats — and here it is, though the equation took a century to find and is not an equation about either magnets or fluids.

Where the two curves meet #

The dome under which two phases coexist is drawn by the equal-area construction at each temperature, and it closes at a single point.

Drawn as isotherms instead, the same dome is a family of curves with a flat portion in the middle, and the flat portion is the tie line between the two coexisting volumes. As the temperature rises that flat shortens, because the two volumes are converging, and on the critical isotherm it has contracted to a single point. What is left there is an inflection with a horizontal tangent: the first and second derivatives of pressure with respect to volume both vanish. That is the standard definition of the critical point, and every exponent in this essay comes out of those two simultaneous vanishings.

Two derivatives vanishing at once is a strong condition and it has an immediate consequence. If the pressure is an analytic function of the density near the critical point — which any equation written down in closed form necessarily makes it — then with the first and second derivatives gone, the leading behaviour is cubic.

That is the cleanest statement of what goes wrong. The measured exponent is not merely different from 3; it is not an integer, and a function with a power-series expansion cannot have a non-integer leading exponent. The failure is not in van der Waals’ particular equation. It is in the assumption that there is an equation.

A fluid that stops resisting #

Approach the critical point from above, along the density at which the dome closes, and the isothermal compressibility diverges.

A diverging compressibility is a statement about fluctuations as much as about mechanics, and the identity connecting them is exact:

$$\langle (\Delta N)^2\rangle = \frac{N kT}{V}\,\kappa_T .$$

The mean square fluctuation in the number of molecules in a region is proportional to the compressibility. A fluid that offers no resistance to being compressed is a fluid that compresses and rarefies itself, spontaneously, and near the critical point it does so on every scale at once.

That softness has a practical face that has nothing to do with fundamental physics. Supercritical carbon dioxide — the state above the critical point, where there is no distinction to make — is used industrially as a solvent precisely because its density can be swung over a wide range by a small change of pressure, so its solvent power is tunable in a way no liquid’s is. Decaffeinating coffee runs on the flatness of that critical isotherm.

The length that grows without limit #

What is actually happening near the critical point is that the fluid is becoming correlated over longer and longer distances. Two molecules a nanometre apart in an ordinary liquid know about each other; two a micrometre apart do not. Near the critical point the distance over which they do grows without bound.

Twenty millikelvin is the number worth carrying. A degree away the fluid is as clear as water; a hundredth of a degree away it is opaque. Nothing about the effect is gradual on the scale of an ordinary thermometer, which is why critical opalescence is described as switching on and why Andrews, who first saw it in 1869, described it as a “flickering” rather than as a cloud.

The sequence of the discoveries is worth recording because it is the reverse of the usual one. Andrews had the phenomenon in 1869 and no theory. Van der Waals produced an equation in 1873 that reproduced the whole topology — the dome, the tie line, the metastable region, the point at the top — and was rewarded with a Nobel Prize for it. The exponents were measured accurately enough to contradict him in the 1940s and 1950s, and it took until 1971 for anybody to explain why: Wilson’s renormalisation group, which does not solve the fluid at all but instead asks what happens to a description of it as the scale of the description is changed, and finds that the answer approaches a fixed point which knows nothing about molecules. Eighty years separated a correct picture from a correct number, and the correct number required abandoning the idea that the fluid had an equation.

Ordinary Rayleigh scattering from molecules is weak because the scatterers are small and independent, so their amplitudes add with random phases and the intensity goes as the number of them. Near the critical point the scatterers are correlated regions containing many molecules, the amplitudes within a region add coherently, and the intensity goes as the square of the number involved. Nothing has been added to the fluid and no new interaction has appeared; the same molecules have merely stopped being independent. That is the whole difference between a blue sky and a milky one.

The scattered light also carries the correlation length in its angular distribution, which is how the number is actually measured. A region of size ξ scatters coherently only into angles smaller than about λ/ξ, so the scattering is forward-peaked, and the width of that peak — measured with a laser and a rotating detector — gives ξ directly. That measurement is called light scattering off critical fluctuations and it is where the exponent 0.630 comes from, rather than from any thermodynamic quantity.

Why the exponents are the same for everything #

Once the correlation length is much larger than a molecule, the fluid near its critical point has no idea what it is made of. Its behaviour is decided by fluctuations on scales of hundreds of molecules, and at that scale the details of the intermolecular potential have been averaged into invisibility. What is left is the dimension of space, the number of components in the order parameter, and the range of the interaction.

That is the argument for universality, and it is the reason the same four exponents describe a liquid–gas critical point, a uniaxial magnet at its Curie temperature, a binary alloy ordering, and the separation of a mixture of two liquids. Three-dimensional systems with a one-component order parameter and short-range forces are all in one class, and the measured exponents are 0.326, 4.8, 1.24 and 0.630 for every one of them.

There is a way of seeing why the fluctuations must matter that needs no machinery. A mean-field theory computes the force on one molecule from the average density around it, and that is a good approximation when the number of neighbours contributing is large — because then the actual density is close to the average. Near the critical point the correlation length is enormous, so the region a molecule is effectively coupled to is enormous, and one might expect the averaging to become better. It becomes worse, because the fluctuations grow faster than the region does: the compressibility diverges, so the variance of the density in any region diverges too, and the average stops being representative of anything. The condition for mean-field theory to work near a critical point is called the Ginzburg criterion, and evaluating it for a three-dimensional fluid gives the answer that it fails within a few per cent of the critical temperature — which is exactly where the interesting measurements are.

Mean-field theories — van der Waals for a fluid, Weiss for a magnet, Bragg–Williams for an alloy — get the same wrong answers for the same reason: each replaces a molecule’s neighbours by an average, which is precisely the step that deletes the fluctuations. They are not bad theories. They are exactly right above four spatial dimensions, where there are enough neighbours for the average to be a good one, and the fact that they fail in three is a statement about how much room there is.

Surface tension exists because a molecule at a surface has fewer neighbours than one inside — so it costs energy to make surface, and a drop minimises its area. At the critical point there is no surface, because there is nothing on either side of it to differ. Surface tension therefore vanishes there too, with an exponent of its own of about 1.26, and a fluid approaching its critical point wets everything it touches, foams uncontrollably, and has no meniscus left to measure. The last of those is the standard laboratory sign that the critical point has been reached: the boundary between the two phases does not move, it fades.

Two exponents, not four #

Four exponents have been quoted — 0.326 for the order parameter, 4.8 for the critical isotherm, 1.24 for the compressibility, 0.630 for the correlation length — and they are not four independent numbers.

They satisfy relations. The first was found by Rushbrooke:

$$\alpha + 2\beta + \gamma = 2,$$

with $\alpha$ the exponent of the specific heat’s own divergence, measured at about 0.110. Putting the measured values in gives $0.110 + 0.652 + 1.237 = 1.999$. Widom’s relation says $\gamma = \beta(\delta - 1)$, which gives $0.326 \times 3.79 = 1.235$ against a measured 1.237. And the hyperscaling relation ties the thermodynamic exponents to the correlation length through the dimension of space, $2 - \alpha = d\nu$, which gives $1.890$ on both sides in three dimensions.

Three relations among five exponents leaves two independent, and that is the strongest quantitative statement universality makes. It is not merely that unlike systems share exponents; it is that any one system’s exponents are so constrained that measuring two of them determines the rest.

The relations were first derived as inequalities from thermodynamic stability arguments — Rushbrooke’s proof gives $\ge 2$ — and only later shown to be equalities by the scaling hypothesis. That history is worth noting because it is the reverse of the usual order: thermodynamics could bound the exponents before anybody could compute them, and the bounds turned out to be saturated.

What makes them equalities is the assumption that near the critical point there is only one length in the problem, the correlation length, and that every quantity scales as some power of it. Given that, all the exponents are ratios of powers of one thing and the relations follow by counting. The assumption is justified by the renormalisation group and it is the whole content of the word scaling.

Where the failure begins #

The Ginzburg criterion was named above and it is worth putting a number to, because it says how close is close enough for the interesting physics to appear.

Mean-field theory is self-consistent when the fluctuations in the order parameter within a correlation volume are small compared with the order parameter itself. Both sides of that comparison can be computed from the mean-field theory, and requiring the inequality gives a reduced temperature below which the theory contradicts itself.

Evaluating it for a fluid, with a molecular size for the microscopic length, gives a window of a per cent or so: closer to the critical temperature than about $10^{-2}$ in reduced units, mean-field theory is wrong, and further away it is right. Which is precisely why van der Waals looked so good for eighty years — almost every measurement made before 1950 was outside the window.

The same criterion evaluated for a superconductor gives a window of $10^{-14}$, which is why the mean-field theory of superconductivity works essentially perfectly and its critical fluctuations have never been an experimental issue. The difference is the coherence length: a superconductor’s is thousands of atomic spacings even far from its transition, so the number of neighbours contributing to the average is enormous.

And the criterion has a dimension in it, which is where the four comes from. Carrying the same calculation in $d$ dimensions, the window shrinks to nothing as $d$ approaches four from below and mean-field theory becomes correct arbitrarily close to the transition at four and above. Four is therefore not a numerical accident but the dimension at which the fluctuation term stops competing, and the modern calculations of the exponents work by expanding in the difference between four and three.

Why it took a change of subject #

The eighty-year gap between the right picture and the right numbers deserves one paragraph on what was actually missing, because it was not a technique.

Every attempt before 1971 tried to compute the fluid — to find a better equation of state, a better approximation to the partition function, a better treatment of the interactions. All of them failed for the same reason, which is that the answer does not depend on the fluid, so a better description of the fluid cannot produce it.

What worked was to change the object of study from the system to the description. Take the fluid, average over the smallest fluctuations, and ask what equation describes what is left. That gives a new description at a coarser scale, with different parameters. Repeat. The parameters move, and near a critical point they move toward a fixed point at which further coarse-graining changes nothing.

The exponents are properties of that fixed point — of how the parameters flow near it — and the fixed point is reached from a whole basin of starting descriptions, which is universality stated as a geometry. Two fluids, a magnet and an alloy have four different starting points and one destination.

The step that made it work was therefore not a better calculation but the recognition that the quantity to study is what happens when the scale of the description is changed, and that a critical point is precisely the state in which the answer is that nothing happens. Everything about the subject that is strange — non-integer exponents, systems with nothing in common sharing numbers, the failure of every closed-form equation — follows from that one sentence.

What the whole business looks like from further away #

Step back to the ordinary pressure–temperature diagram and the peculiarity becomes topological rather than numerical. On it the critical point is a single dot, and what is remarkable about the dot is that a line ends there. The liquid–vapour line ends. The solid–liquid line does not, and as far as anybody knows cannot: a solid and a liquid differ by a symmetry — one has long-range order and the other does not — and a symmetry is either present or absent, with no continuous route between.

A liquid and a gas differ only in degree, so their line is allowed to stop, and once it has stopped there is a route around the end of it by which one becomes the other with nothing discontinuous happening anywhere along the way. That is what “supercritical” means, and it is why the question of whether a supercritical fluid is a liquid or a gas has no answer.

The latent heat says the same thing in energy rather than in geometry. Crossing a first-order boundary costs a fixed amount of heat per unit mass, paid to rearrange the molecules from one arrangement into the other, and along the liquid–vapour line that cost falls as the critical point is approached and reaches zero exactly there. It has to: the two arrangements have converged, so there is no rearrangement left to pay for. A first-order transition whose latent heat has vanished has become a continuous one, and the critical point is the single temperature at which that happens.

Where the model stops #

Gravity ruins the measurement. A fluid with a diverging compressibility settles under its own weight, so a centimetre-tall sample near its critical point has a density gradient large enough to smear out everything being measured. The best terrestrial measurements use samples a millimetre deep, and the exponents were pinned down to their third decimal place only in microgravity.

The approach has to be along the right path. Coming in along the critical isochore, along the coexistence curve and along the critical isotherm give three different exponents of the same quantity, and quoting one without saying which path was taken is meaningless.

Not every transition has a critical point. A first-order line can end in one only if the two phases differ by a continuous quantity rather than by a symmetry, which is why liquid and gas have one and solid and liquid do not.

Impurities change the class. A second component turns a one-component order parameter into something with more structure, so a fluid mixture near its consolute point is in a different universality class from a pure fluid near its critical point, and a trace impurity in a supposedly pure sample shifts the exponents rather than merely shifting the temperature. Mixing is never entirely reversible and it is not entirely innocent here either.

And the classical description of the fluctuations fails at the shortest scales. Everything above treats the fluid as a continuum with a correlation length; when that length is a few molecules, which is where the power laws are entered, the description is being used at the edge of its validity and the corrections to scaling are large.

What the pictures cannot show #

Every figure here is a curve on logarithmic axes, and the thing the curves describe is a fluid that looks like nothing in particular: transparent, homogeneous, and — over the last twenty millikelvin — visibly milky. A photograph of the phenomenon would show a tube that has gone cloudy, which conveys none of the arithmetic.

Nor can any static figure show a fluctuation. What is happening near the critical point is that regions of every size, up to the correlation length, are appearing and dissolving on every timescale up to a matching one, and a drawing of the density field at an instant would be a snapshot of noise. The self-similarity across scales — the thing that makes the exponents non-integer — is exactly the property a single frame cannot carry.

Where this ladder goes next #

The rungs below built the phase boundary as a line where two phases have equal free energy, showed that a boiling point is a pressure rather than a temperature, and then found the part of the isotherm no fluid follows. This rung is the end of that line, and it is where the machinery of the previous three stops working — not through an approximation degrading but through a qualitative change in what kind of function is required.

The habit worth carrying away is a diagnostic. A non-integer exponent is a statement that no closed-form equation of state exists. Whenever a measurement gives a power law whose index is not a simple fraction of small integers, the thing being measured is not the leading term of an expansion; it is a property of fluctuations on every scale at once, and the correct tools for it are the ones that treat the change of scale itself as the object of study.

What is left on this ladder is the transition that has no critical point — melting, where the symmetry argument forbids one — and the ways a first-order transition can nonetheless be approached continuously, which is the subject of nucleation and belongs with the metastable region rather than with the point at its top.