Kadanoff on Phase Transitions

Kadanoff has been doing work on renormalization group and nonequilibrium statistical mechanics from the beginning:

This paper Looks at the early theory of phase transitions. It considers a group of related concepts derived from condensed matter and statistical physics. The key technical ideas here go under the names of "singularity", "order parameter", "mean field theory", and "variational method". In a less technical vein, the question here is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor, "steam", come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in a some statistical variable of the system. The discontinuous property is called the order parameter. Each phase transitions has its own order parameter that range over a tremendous variety of physical properties. These properties include the density of a liquid gas transition, the magnetization in a ferromagnet, the size of a connected cluster in a percolation transition, and a condensate wave function in a superfluid or superconductor. A continuous transition occurs when that jump approaches zero. This note is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of this phenomenon.

Via Physics and Physicists.

Kadanoff makes an interesting epistemological point here. From the mathematical fact that no phase transition can occur within a finite system, he argues that "we must conclude that phase transitions and the definitions of different thermodynamic phases are the result of a process of extrapolating the real behavior of a theory of large bodies, to its infinite conclusion." Thus, the "extrapolation and simplifying ... suggests that at least this part of theoretical physics is not a simple result of the direct examination of nature but rather of the human imagination applied to an extrapolation of that examination."

I don't necessarily disagree with this point. I have noted elsewhere that our theories form part of a feedback loop with the stream of stimulus to our brains and our actions within the universe here. I do want to point out, however, that there is another possibility, which remains (unfortunately) inaccessible to scientific inquiry -- namely, that every seeming finite system is woven together into a fabric that is infinite.

Economists should take note of how the form of Kadanoff's argument here is so similar to the form of justification offered by economists for model-making. It's not that we believe our models are reality; it's that our models help us to perceive things about reality that would otherwise remain inaccessible to our senses -- at least until we learn how to use them better.

Comments

[EMP]

The relations between physics and economics are fascinating. I wonder if this will lead us somewhere? I am trying to put together a repository on these topics, at http://economicmanhattan.blogspot.com/

If you have suggestions, just dump them in the comments!

Great blog, keep up the good work :-)