# Universality: Stronger than logic

I was re-reading some old universality papers. You know, universality in the stat-mech sense—the critical exponents that characterize phase transitions are insensitive to details of your model or crystal structure of your material. This insight, allegedly flowing around in the statmech community in the 1960s, was first articulated by Robert Griffiths in 1970. It got a more solid theoretical ground with Kenneth Wilson’s development of the renormalization group a few years later. The renormalization group is a theoretical method to calculate critical exponents that also explain why some features (like if you have a square or hexagonal grid in your spin model) don’t affect the exponents. Leo Kadanoff—who many think should have shared Wilson’s Nobel prize for providing one of the key insights behind the renormalization group (the block-spin argument)—wrote in 1976:

In its simplest terms, the universality hypothesis is the statement that all critical problems may be divided into classes differentiated by:
(a) The dimensionality of the system;
(b) The symmetry group of the order parameter; and
(c) Perhaps other criteria.
Within each class, the critical properties are supposed to be identical or, at worst, to be a continuous function of a very few parameters.

LP Kadanoff, “Scaling, universality and operator algebras”, in Domb & Green, Phase Transitions and Critical Phenomena, vol. 5a (1976).

Before I continue, let me confess that I am a big fan of Per Bak. The way he was brazenly going for the big questions and the audacity of his ideas . . during the heydays of self-organized criticality, it must have felt (to everyone around him) like they had the key to the secrets of the universe in their hands. If I relived my life, I wish I could hang out with his crowd back then . . it must have been an experience on par with assembling computers in Steve Jobs’ garage or rapping with The Sugar Hill Gang.

Anyhow, reading about universality in Per Bak’s How Nature Works, it is remarkable how the tone changed from the physics literature 20 years earlier. In the 2nd chapter, it is now called “the universality principle” and is a “theorist’s dream” because it justifies studying simple models in general. There is no explicit mention of critical phenomena and examples (like optimizing milk production) that most likely aren’t.

I don’t think any radically new evidence elevated the universality hypothesis to a principle. Did it just become a convenient truism and battle call for physicists wanting to invade other disciplines (tacitly assuming readers wouldn’t question physical “principles”)? I don’t know, but it’s nevertheless bizarre.

It’s also worth recalling that the quest to validate Bak, Tang, and Wiesenfeld’s sandpile model, inadvertently led to a beautiful illustration that sometimes details do matter. When a research team at Oslo University set out to measure avalanches in sandpiles, they discovered that sand didn’t really produce power-law avalanches, but unboiled rice could. Well, not any rice though, but a particular type from a specific Asian import store.

One thing that happened between 1976 and 1996 is that emergence became a hot topic in interdisciplinary physics. Critical phenomena are one class of physical emergent phenomena, but there are much more mundane examples—like how different forms of matter get their physical properties away from the critical points—and for such phenomena, indeed, details do matter. Still, some conflation of the phenomena of emergence and universality might have led to overhyping the latter (my hypothesis).

There are other logical glitches in how the self-organized criticality literature invokes universality. One is that discovering “universal properties” (typically power-law exponents) is a highly valued result. But if the “universality principle” is so fundamental that it can even sweepingly justify simple toy models, then wouldn’t the discovery of similar exponents be close to trivial?

Another conundrum is that the “details don’t matter” mantra has not only been used to motivate stylized, minimal models. It can also be an excuse that model output doesn’t fit reality. This shows the scientifically biggest issue with Bak et al.’s (ab)use of the “universality principle.” It is dangerously close to a plea against critical thinking. How can we falsify anything if it’s OK when the pictures don’t look like reality because “we are painting with broad brush strokes here.” Where are we on the science vs. religion scale, really?

PS 1. The title comes from a chess video on the Agadmator channel that stuck to my mind and seemed somewhat appropriate.

PS 2. The idea for writing this post came from this tweet: