Lev Landau and his free energy
When my first paper on Statistical Physics of Design got published in some form, a friend asked me, in good faith: "If you are using the Landau free energy, why are you not citing Landau?" That is a good question which set off a long search for clues. But before we embark on a literature hunt, I need to clarify what the figure of Landau means and implies for physics in post-Soviet countries.
Lev Landau was not just a Soviet physicist, but the Soviet Physicist, the icon, the platonic ideal, and the highest aspiration. When I was starting my serious physics studies a dozen years ago, in late high school and early undergrad, the Landau and Lifshitz Theoretical Physics textbook (known as "Landafshitz") was considered the definitive bible. Landafshitz is a 10 volume mammoth that is intimidating in both size and the impenetrable style of prose (anyone not able to get through it is clearly not hardcore enough). Back when I was borrowing paper books from libraries, I borrowed the Quantum Theory volume, read ten pages, and returned it a semester later. I was taught quantum mechanics later through more pedagogically sound ways. Landafshitz is not a textbook from which you should learn stuff, but one where you look up all the answers, like a magic crystal ball. For a lot of Russian-speaking physics in the 20th century the figure of Landau and his ouevre was comparable to that of J.R.R. Tolkien in fantasy literature. The mysticism of Lev Landau among Russian-speaking students is comparable to that of A***** E******* and R****** F****** (we don't use those kinds of F-words in my household) among Western students. All three of these humans are deeply problematic by the standards of today's time - and also their time - but that's a discussion for another day.
With that kind of reputation, there must be no doubt that Landau was the one who invented the free energy named after him (if only by Stigler's law of eponymy). So I set out to scour the books. My Ph.D. advisor once remarked that those of his students who come from "somewhere east of Germany" inexplicably have pdfs of all the books. Coming from somewhere east of Germany, I of course knew how to get a hold of a bunch of electronic versions of the book, in different editions, in both Russian original and English translation to cross-check, but I was missing the crucial transition. The volume on Statistical Physics was actually the earliest to be published in 1938 and underwent many editions and revisions, in later versions becoming volume #5 of 10. Early editions (1938, 1940, 1951) talk about the phenomenological Landau theory, but don't list the formula for computing LFE from microscopic details. The formula appears later, in the 1976 edition as the Eqn. (147.2) for a fairly specific system. The source of this formula is listed as "L.D. Landau 1958" (note the lack of an exact journal reference). The 1976 Russian edition is the basis for the 1980 English translation and its later reprints that you can purchase today.
So can we just give Landau his due and cite any recent Landafshitz edition? Not quite. Apart from Landau and Lifshitz, a number of other authors were involved in the preparation of the books over the years, with complicated and bitter arguments over their contributions. In 1962 Landau got into a car accident and fell into coma, followed by a long period of recovery. While some argue he recovered sufficiently to return to physics several years later, he died following complications of a surgery in 1968. His overall contributions to physics, in both research and pedagogy, across the career are subject to much acclaim, but here I am only tracking a single question boiled down to a formula. The last edition of Statistical Physics that saw the light of day during Landau's lifetime is the one from 1964. We know that the formula was absent in the previous one (1951) and present in the next one (1976).
Unfortunately, that particular edition was missing from the sources I could get my hands on. I tapped an acquaintance to look up a specific page from the 5th volume of Landafshitz (1964) in Moscow State University library, but they could only get a hold of a recent edition used for current teaching, which doesn't help the historical question. After a desperate search, I discovered a paper copy - in original Russian - sitting on a shelf in the basement of Hayden library at MIT where I spent a lot of time studying in undergrad. While the book is long enough to include an equation numbered (147.2), it has nothing to do with free energies. Therefore, the editions of Landafshitz published during Landau's lifetime do not explain how to compute the LFE starting with a microscopic model. Such an explanation was included post mortem by his coauthors with a begrudging reference to something Landau might have said in passing in 1958, before his car accident. Following Landau's death, a full collection of his printed works was published - but he never wrote about the free energy, not in 1958 nor later.
So if Landau didn't come up with the formula, who did? The trail goes cold in Landafshitz editions, but the spirit of the formula was of course in the zeitgeist of 1960s and 70s. Since I learned about the difference between the phenomenological and the microscopic LFE from the 1992 textbook by Nigel Goldenfeld, I shot him an email about the origins. Nigel very kindly responded with the following:
I am sure Landau did not write it down, because he would consider his derivation from phenomenology is superior. [...]
I have known this formula all my life it seems, and I am surprised that other books do not discuss it. I am sure that I learned it from Jim Langer's papers when I was a graduate student working on phase transition kinetics. See the attached paper where some version of the formula appears and a description of its physical interpretation. Jim almost certainly was thinking about it in this way because of Leo Kadanoff's block spin ideas, but Leo's 1966 paper does not have this formula.
(note: this exchange happened long before the pandemic and Nigel's remarkable work on Covid)
The mentioned paper by Langer is from 1974, two years before the formula appears in Landafshitz. The language used by Langer is at least subjectively more uplifting and less system-specific than that in Landafshitz. I haven't found an earlier verifiable mention of the formula, and clearly the modern significance was not attached to it.
Langer wrote the formula out of desire for completeness, without an intention to evaluate the bulky sum. Now that we can actually compute the sum, the result needs to bear some name. Just like the Boltzmann machine was not invented by Ludwig Boltzmann but relies on his ideas, it makes sense to name the intermediate scale free energy it in honor of Lev Landau, even though he did not come up with it.