Abstract
This paper investigates the consequences for our understanding of physical theories as a result of the development of the renormalization group. Kadanoff’s assessment of these consequences is discussed. What he called the “extended singularity theorem” (that phase transitons only can occur in infinite systems) poses serious difficulties for philosophical interpretation of theories. Several responses are discussed. The resolution demands a philosophical rethinking of the role of mathematics in physical theorizing.
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Notes
In fact, we most often don’t recognize the theories as phenomenological until long after they become well-established.
Much of interest, philosophically, concerns the nature of these analogical derivations. Gibbs, for one, was very cautious about claiming to have found identities that would allow one to claim that the thermodynamic relations have been reduced to relations of statistical mechanics. See [4] for a discussion.
Roughly, this involves a claim to the effect that what there is in the world, or perhaps what we can know about the world, is a product or function of human thought.
Actually, I’m not completely sure this was his view. In private conversation, the first time I met him, Leo told me he was a Platonist. This reflects a kind of idealism, but not one that particularly fits well with a simple reading of this sentence. And, as we will see below, he later argued for what seems to me, a very different understanding of the consequences of the extended singularity theorem.
A prominent proponent of this viewpoint is Jeremy Butterfield [7, 8] (see also, [21]) . On his view, the use of infinite limits receives a “straightforward justification”:
The use of the infinite limit ...is justified, despite N being actually finite, by its being mathematically convenient and empirically correct (up to the required accuracy). [7, p. 1077]
This is a hallmark of Kuhn’s characterization of “normal science.”
Kadanoff notes that similarities between different systems were well known to early investigators [14, p. 103]. The idea that the “details of the system undergoing the phase transition might be irrelevant and that many different phase transitions problems might be essentially identical” was already noted in Landau and Lifshitz in the context of mean field theory. I believe that this mean field understanding of universality also contributed to that theory’s entrenchment in the scientific community.
One needs this for first order transitions as well, but the singularities are weaker.
It was in part Gibbs’ recognition that his ensembles failed to provide an account of phase transitions that he exercised extreme caution in claiming to have identified statistical mechanical analogs of thermodynamic properties like temperature and entropy. See [4] for a discussion of Gibbs’ caution and his awareness that phase transitions would pose theoretical problems for the statistical mechanical theory he developed. See [22, Chap. 3] for a discussion of Gibbs’ thermodynamic analogies.
As an interesting aside, Clifford Truesdell, in a lecture entitled “The Ergodic Problem in Classical Statistical Mechanics,” discusses Khinchin’s use of asymptotics to try to justify the identification of infinite time averages of certain functions with phase (or ensemble) averages. The last paragraph reads as follows:
I should like to be able to say that statistics is unnecessary, that the name “statistical mechanics” is a misnomer for “asymptotic mechanics”, as far as equilibrium is concerned. This is almost true, but not quite so. [23, p. 82]
On my understanding, Truesdell is stressing the importance of appealing \(N\rightarrow \infty \) limits in justifying certain claims about finite systems whose observable properties are represented by sum-functions. Actually, I believe it is possible to tell an RG story to justify the use of the microcanonical probability distribution for calculating equilibrium values for functions of this form. See [4] and [18, pp. 139–142] for details.
In his 1970 lecture [14] Kadanoff says the following:
The physical origin of this hypothesis [of universality] is the observation that critical phenomena are dominated by fluctuations in M [the order parameter] and H [the energy] which appear in a special scale which is very long compared to the force range. Then these fluctuations cannot probe all the details of the interatomic potential. Rather they only see certain gross features of the potential .... [14, p. 104]
It is also important to recognize that at the critical point the fluid exhibits fractal-like, self-similarity. It is the recognition of this fact that allowed Kadanoff to introduce the blocking operations and Wilson to think about finding fixed points of the RG transformation.
A referee objects to this way of putting things because according to the point of view in Sect. 2.2 the use of infinities is merely a matter of mathematical convenience and so, the claim I’m making here is completely in line with the claims of those supporting the idea that real phase transitions aren’t sharp. In response, it seems to me that if one is going to hold that the use of the infinite limits is a convenience, then one should be able to say how (even if inconveniently) one might go about finding a fixed point of the RG transformation without infinite iterations. I have not seen any sketch of how this is to be done. The point is that the fixed point, as just noted, determines the behavior of the flow in its neighborhood. If we want to explain the universal behavior of finite but large systems using the RG, then we need to find a fixed point and, to my knowledge, this requires an infinite system.
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Thanks to Michael Miller for helpful comments.
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This article is dedicated to the memory of Leo Kadanoff.
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Batterman, R.W. Philosophical Implications of Kadanoff’s Work on the Renormalization Group. J Stat Phys 167, 559–574 (2017). https://doi.org/10.1007/s10955-016-1659-9
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DOI: https://doi.org/10.1007/s10955-016-1659-9