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 matured in the 1950s and 60s and culminated with Kenneth Wilson’s development of the renormalization group in the first few years of the 1970s—a theoretical method to calculate critical exponents that also explains why there are universality classes at all. 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.

One thing that happened between 1976 and 1996 is that emergence became a hot topic in interdisciplinary physics through articles like PW Anderson’s More is different and the establishment of Santa Fe Institute and similar centers. 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, surely details do matter. Still, some conflation of the phenomena of emergence and universality might have led to overhyping the latter (my hypothesis).

A final puzzling thing about the Bak school of invoking universality 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?


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:

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