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title: "Why very different materials behave alike near a critical point"
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[How order survives, changes and gets frustrated](https://sandboxphysics.com/stories/thermal-complexity/#order)

Heat · Chance · Collective behaviour

# Why very different materials behave alike near a critical point

How did Wilson make sense of fluctuations at many different sizes?

Sandbox Physics · 2026-10-24

Nobel contribution: 1982-1

## Near the transition, patches stop acting independently

A fluid close to its liquid–vapour critical point can become cloudy. Large patches fluctuate between slightly denser and thinner states and scatter light. A magnet near the temperature where it loses its permanent magnetization has a related difficulty: many atomic magnets fluctuate together. The substances look very different, yet some of their measured changes follow the same mathematical pattern.

Far from a critical point, a small patch usually has a limited reach. Its neighbours are affected, but the influence becomes weak over a short distance. Near the transition, coordinated changes can extend across many particles. Physicists call the distance over which the changes remain related the correlation length. As it grows, fluctuations on many different sizes contribute to the response of the whole sample.

Sources: [1] [2]

## A local average hid the important part

An early approximation replaced a particle’s fluctuating surroundings by one average environment. This mean-field picture often gives a useful account of ordering. It can describe a tendency for neighbouring atomic magnets to line up, or for molecules to favour a dense liquid. Close enough to a critical point, however, large coordinated fluctuations defeat that replacement.

The problem is not simply that someone forgot a small correction. A fluctuation contains smaller fluctuations, and those contain still smaller ones. There is no obvious single size to choose as the typical one. Calculating the whole material by treating one molecule or spin at a time misses how these sizes interact.

The experimental clue was that the way a response grows or vanishes near the transition could be grouped across different substances. Materials with quite different microscopic forces sometimes shared the same critical exponents: the powers describing those changes. A successful theory had to explain which details survived in these powers and which faded away.

Sources: [1]

## Change the size of the picture and change the theory with it

Leo Kadanoff had proposed thinking in blocks: combine a small group of spins into one larger effective unit, then view the system from farther away. Kenneth Wilson developed this scale-changing idea into a calculation. The interaction rules between the larger units change when small-scale details are averaged over. Keeping track of those changes is the central task of the renormalization group.

Imagine a picture made from tiny light and dark squares. Replace each small block by a representative shade, then repeat. Most fine details disappear. Near a critical point, large patches remain at each step, so a new picture can resemble the old one after distances are rescaled. The physical calculation must update the interactions as well as the picture; a purely visual majority vote is only an illustration.

A fixed point is a set of effective rules that reproduces itself under this repeated change of scale. Different starting materials can approach the same fixed point. Their microscopic details still determine such things as the actual transition temperature, but their large-scale critical behaviour can fall into one universality class. Spatial dimension, the kind of symmetry being broken and the distance over which interactions act help determine which class applies. Long-range forces can change the universality class.

Wilson’s 1971 papers showed how scaling laws could follow from these changing interactions. With Michael Fisher, he also developed a systematic expansion for critical exponents near four spatial dimensions. The calculations connected the qualitative picture of scale after scale with numbers that could be compared with experiments.

Sources: [2] [3]

## A finite picture shows the approach to criticality

A simulation contains a finite number of sites. Its largest fluctuation can grow only as far as the system allows, so a sharp transition becomes rounded. Comparing several sizes reveals how the rounding and the location of response peaks change. That is why temperature and system size are both meaningful controls in a classical equilibrium Ising experiment.

The 1982 Physics Prize recognized Wilson’s theory of critical phenomena. Its lasting answer is that large-scale behaviour has to be studied through the way a description changes with scale. A fluid, a magnet and a simple statistical model can share a critical pattern when the relevant large-scale ingredients agree.

Sources: [4] [1]

Continue with the physics

## Try the related models

### [Watch thermal domains near a transition](https://sandboxphysics.com/labs/ising-critical-fluctuations/)

Compare temperatures below, near and above the transition, then change the lattice size.

The finite two-dimensional ferromagnetic Ising model samples thermal configurations.

## Sources and reading notes

1. [Kenneth G. Wilson, The Renormalization Group and Critical Phenomena (Nobel lecture, 1982)](https://www.nobelprize.org/uploads/2018/06/wilson-lecture-2.pdf)

   Use the discussion of fluctuations on many length scales to see why a uniform molecular environment is insufficient near criticality.

2. [Kenneth G. Wilson, Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture (1971)](https://doi.org/10.1103/PhysRevB.4.3174)

   Follow the relation between scale changes, changing effective interactions and fixed points.

3. [Kenneth G. Wilson and Michael E. Fisher, Critical Exponents in 3.99 Dimensions (1972)](https://doi.org/10.1103/PhysRevLett.28.240)

   Read the dimension expansion as a method for calculating critical exponents, separate from a literal claim of a laboratory in fractional dimension.

4. [The Nobel Prize in Physics 1982](https://www.nobelprize.org/prizes/physics/1982/summary/)

   Check the award’s scope: critical phenomena in connection with phase transitions.

[Heat, chance and collective behaviour](https://sandboxphysics.com/stories/thermal-complexity/) [A liquid can flow and still keep a direction](https://sandboxphysics.com/stories/thermal-complexity/de-gennes-liquid-crystals/) [A long molecule behaves like a wandering chain](https://sandboxphysics.com/stories/thermal-complexity/polymer-scaling/) [The magnets that cannot satisfy all their neighbours](https://sandboxphysics.com/stories/thermal-complexity/parisi-spin-glass/)