When did "scale invariance" started to be seen as an important concept in the theory of phase transition?

Phase transition and critical points started to be investigated in earnest in the middle of the 19th century, with the study of critical opalescence (Andrew/Faraday) : at the critical point of a liquid, the liquid start to get cloudy/diffuse, a phenomenon widely attributed to the scale invariance of the medium nowadays. I like to pinpoint when scale invariance was first observed, and when it started to be seen as something important in the theory of phase transition.

It seems that a lot of what was needed to introduce scale invariance in physics was already here at the start of the 19th century, with brownian motion and fractals being investigated. Marian Smoluchowski was the first (at the beginning of the 20th century) to guess that opalescence was due to large fluctuations in the fluid (but that's not "scale invariance").

Conformal invariance (a generalisation of scale invariance), started to get applied to phase transition in the 70's. So that's an upper boundary I guess. A google ngram seems to indicate that "scale invariance" only became a thing when conformal invariance started. But one of A. Migdal anecdote seems to indicate that scale invariance was already seen as "old news" in the beginning of the 70's (pdf, page 34-35).

Sorry if the "history of science" question is out of bounds for this stackexchange.


1 Answer 1


According to this paper by Michael Fisher, one of the pioneers in this area, the notion of scale invariance, in the context of the critical point, appeared in the mid-1960s. Let me quote two passages from the paper; much more information can be found the paper, of course.

From the introduction of the paper:

Whence came RGT? This is a good question to start with. I will try to respond, sketching the foundations of RGT in the critical exponent relations and crucial scaling concepts of Leo P. Kadanoff, Benjamin Widom and myself developed in 1963– 1966 — among, of course, other important workers, particularly Cyril Domb and his group at King’s College London, of which, originally, I was a member, George A. Baker, Jr., whose introduction of Padé approximant techniques proved so fruitful in gaining quantitative knowledge, and Valery L. Pokrovskii and A. Z. Patashinskii in the Soviet Union who were, perhaps, the first to bring field-theoretic perspectives to bear.

and from Section 3:

The 12 months from late-1965 through 1966 saw the clear formulation of scaling for the thermodynamic properties in the critical region and the fuller appreciation of scaling for the correlation functions. One may highlight Widom’s (1965) approach since it was the most direct and phenomenological — a bold, new thermodynamic hypothesis was advanced by generalizing a particular feature of the classical theories. But Domb and Hunter (1965) reached essentially the same conclusion for the thermodynamics based on analytic and series-expansion considerations, as did Patashinskii and Pokrovskii (1966) using a more microscopic formulation that brought out the relations to the full set of correlation functions (of all orders). Kadanoff (1966), however, derived scaling by introducing a completely new concept, namely, the mapping of a critical or near-critical system onto itself by a reduction in the effective number of degrees of freedom.

Obviously, scaling concepts in other settings can be found earlier (for example, in Kolmogorov's work on turbulence in the early 1940s).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.