The critical point of water and the critical point of an iron magnet. Nothing in common microscopically ── and the numbers that come out agree exactly
Heat and pressurise water and at 374 °C and 218 atmospheres you arrive at a strange place called the critical point. The distinction between liquid and gas disappears, droplets of every conceivable size boil up at once, and clear water turns milky (critical opalescence). Separately, heat an iron magnet and its magnetism vanishes at 770 °C. That is a critical point too. One is about how molecules are packed, the other about the direction of electron spins ── microscopically unrelated. Yet measure the numbers that describe behaviour near the critical point (the critical exponents) and ── they agree to three decimal places. This is no coincidence. Repeat coarse-graining and the two theories flow to the same destination, discarding along the way the information about what they originally were.
| System | What it is microscopically | Critical exponent β | ν |
|---|---|---|---|
| Water (liquid–gas critical point) | density of H₂O molecules | ≈ 0.326 | ≈ 0.630 |
| A uniaxial ferromagnet | direction of electron spins | ≈ 0.326 | ≈ 0.630 |
| Phase separation in a binary liquid | the mixing ratio of two molecular species | ≈ 0.326 | ≈ 0.630 |
| A ±1 arrow model on a lattice (3D Ising model) | nothing at all ── a model invented on paper | ≈ 0.326 | ≈ 0.630 |
The last row is the punchline. A made-up toy model that does not exist anywhere gets the measured value for real water right ── with no molecular shapes, no details of the forces, not even quantum mechanics in it. This family is called the 3D Ising universality class.
If coarse-graining merely dropped information this could not happen. Merely dropping would blur the difference and cost accuracy. What actually happens is the opposite: coarse-grain and the difference disappears and the answers agree. It's exactly what happened in Episode 1 when A and B became indistinguishable.
Kadanoff's idea goes like this. You have ±1 arrows on a lattice. Group them into 2×2 blocks and replace each block by one arrow, by majority vote. The lattice is now half as fine and there are a quarter as many arrows. Same operation as Episode 1's block averaging.
Here is the new bit. What you end up with is a model of the same form ── ±1 arrows on a lattice. What has changed is the strength of the coupling between arrows. In other words, coarse-graining is a map from model to model. So you can do it again, and again.
The coupling \(K\) moves with each coarse-graining. The mathematics of following that motion is the renormalization group. "Group" sounds forbidding but it just means the obvious composition rule: coarse-graining twice equals coarse-graining once by a factor of four. Physical theories flow along an axis called resolution ── the theme of this series, and the punchline of Episode 7.
Wherever there is a flow there are points where the flow stops (fixed points) ── models that map to themselves under coarse-graining, i.e. that look the same at every magnification.
And around a fixed point, deviations split into two kinds. This is the key to everything.
| Direction | What coarse-graining does to it | What it means physically |
|---|---|---|
| relevant | the deviation grows and carries you away from the fixed point | very few of them. Temperature, magnetic field ── the knobs you turn in the lab |
| irrelevant | the deviation shrinks and disappears | overwhelmingly many. Molecular shape, force details, lattice type ── microscopic origin |
And there is the reveal. Every difference between water and a magnet lives in the irrelevant directions. So repeated coarse-graining shrinks those differences to nothing and both are drawn to the same fixed point. Since critical exponents depend only on how fast the flow expands around the fixed point, they do not depend on origin at all. The number 0.326 is not water's number and not iron's number ── it is the fixed point's number.
Particle physics says the same thing this way. Effects from the details at a high energy \(\Lambda\) (microscopic) are suppressed at low energy \(E\) (macroscopic) by $$\left(\frac{E}{\Lambda}\right)^{n}\qquad (n>0).$$ For atomic physics (\(E\sim\) eV) against the Planck scale (\(\Lambda\sim10^{28}\) eV) the ratio is \(10^{-28}\). That is why you can do chemistry without knowing quantum gravity. The rigorous statement is the decoupling theorem (Appelquist–Carazzone, 1975): heavy particles decouple from low-energy physics.
The figure below runs ±1 arrows on a lattice (the Ising model) toward thermal equilibrium live in your browser. Leftmost is the raw configuration (64×64). To the right are successive 2×2 majority-vote coarse-grainings (32²→16²→8²) ── literally the renormalization group flow.
Try the temperature slider in three regimes.
The pattern being scale-invariant at the critical point means there is no characteristic size there. Ordinary matter has landmark scales ── molecular size, the reach of correlations ── but at the critical point the correlation length diverges to infinity and the landmarks vanish. With no landmark length, microscopic lengths lose their meaning ── and so origin is forgotten. That vanishing of a characteristic length is what universality really is.
Episode 1 said "throw it away, same answer"; Episode 3 said "bigger discards faster." Episode 4 goes further ── discarding is what determines the answer.
The exponent 0.326 is not a property of water or of iron but a property of the discarding operation itself. That is why a made-up model gets it right. A physical quantity belonging to an operation rather than to a substance ── a startling thing, and indeed Wilson received the 1982 Nobel Prize in Physics for this understanding.
You can coarse-grain (Ep. 1) → you can repeat it (it's a map) → the flow has fixed points → only the fixed point's properties survive → and therefore the world can be described without knowing its microscopics.
Those five lines are very nearly the whole reason the enterprise of physics is possible.
Established: universality in critical phenomena (that the liquid–gas critical point, uniaxial ferromagnets, phase separation in binary fluids and the 3D Ising model share critical exponents); the values (\(\beta\approx0.326\), \(\nu\approx0.630\); nowadays determined to high precision by the conformal bootstrap); Kadanoff's block-spin picture; Wilson's renormalization group with fixed points and the relevant/irrelevant classification (1982 Nobel Prize in Physics); and \((E/\Lambda)^n\) suppression in effective field theory together with the decoupling theorem (Appelquist–Carazzone 1975) ── all established physics.
Caveats: (1) The figure is the two-dimensional Ising model (\(T_c=2/\ln(1+\sqrt2)\approx2.269\), \(\beta=1/8\)), shown because it can be seen. The numbers in the table are the three-dimensional values; these are different universality classes and the numbers do not match. (2) The lattice is finite (64×64) and the run time is finite, so critical behaviour is only approximately reproduced (finite-size effects). (3) Universality classes are set by spatial dimension, the symmetry of the order parameter, the range of the interaction and so on ── it is not that "everything comes out the same"; different classes have different exponents. (4) Irrelevant operators do reappear as corrections once you move away from the critical point (correction-to-scaling exponents). "Origin vanishes completely" holds only sufficiently near the fixed point.
The critical point of water, an iron magnet, a binary liquid, and a toy model on paper ── four things with nothing microscopically in common ── all produce the same critical exponents \(\beta\approx0.326,\ \nu\approx0.630\). The reason is that coarse-graining is a map from model to model, and repeating it makes theories flow to a fixed point. Around the fixed point, deviations split into relevant (grow and survive; very few) and irrelevant (shrink and vanish; overwhelmingly many) ── and all microscopic origin sits on the irrelevant side. So it is forgotten.
The pattern looks scale-invariant at the critical point because the correlation length diverges and the characteristic length disappears. With no landmark length, microscopic lengths lose meaning ── that is universality. In particle-physics language it becomes \((E/\Lambda)^n\) suppression and the decoupling theorem, turning "a chemist need not know about quarks" into a theorem. A physical quantity belonging to the operation of coarse-graining rather than to a substance ── that is the heart of this series.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the temperature slider changes the configuration, and each panel to the right is a further coarse-graining. The thing to look for is that at the critical point (2.269) every panel looks the same. "See the answer" opens each solution.