Monday, August 18, 2014

The Mandelbrot Approach To Statistical Mechanics

B Mandelbrot (gasp, color!)
The mathematician Benoit Mandelbrot is most famous for coining the term "fractal" and being one of the first to treat them as anything more than theoretical curiosities. He was widely known to be in the first rank from a young age, working some of the best scientific institutions in France and the United States. Fans of mathematical genealogy will be pleased to learn that he was sponsored at the Institute of Advanced Study by John von Neumann himself! He spent most of his mathematical life working at IBM, where he used and developed both the tools (computers, visualization, statistics) he would use for the rest of his life and found the problems that interested him the most. Mandelbrot was a "visual thinker", more so than most, and his most famous book The Fractal Geometry of Nature was a best seller at least partly because he was able to expound his mathematical view of the world without dense equations. I'd recommend it, but it has been quite some time since I've even so much as leafed through it.

Strangely enough, this post isn't a place for going into Mandelbrot's most famous views and discoveries, the things that will make him immortal. Instead, I want to highlight some really interesting applied mathematics work he did at IBM. Last post, I sort of mentioned a few approaches to statistical mechanics, two of which got big time names behind them the Boltzmann (single set-up, large number of particles) and Gibbs (ensemble, ergodic system) approaches. I mentioned that there are two other approaches, both avoiding interpretation as much as possible in different ways. The last approach mentioned was to do straight up dynamical systems. This approach is very rigorous, very slow, and open to complaints that we no longer believe in dynamics smooth in the very small (in other words, it's still classical but nature is quantum). Good work continues to be done in this field, such as this description of a deterministic random number generator by Fields Medalist Artur Avila. The other method was to avoid dynamics as much as possible, postulate the existence of free energy functions and derive what you need from there. This method is highly visual and very fruitful, but has certain scientific limitations because, as mentioned before, free energy functions are not always analytic. What I completely failed to mention is that Mandelbrot made some huge contributions to this field

Wrong kind of field

The first contribution is this paper, which is very excellent. It concerns a statistical interpretation of the free energy function approach, making it more clear that this really is a kind of statistical mechanics. This is in many ways a precursor to Jaynes's method, but doesn't have the same subjective implications because the probabilities aren't conceived as coming from a subjectively defined ensemble. In fact, it is very important to Mandelbrot that the estimators will converge on the truth, which he calls "self-consistency". The paper shows that many thermodynamic variables of interest (such as temperature) are sufficient statistics of whatever it is that underlies them. That is, if you know the volume, temperature, mass, et cetera of a system, you can't learn more by tracking the particles or otherwise zooming in. One remarkable about this paper is its inversion of our usual values. We usually think of the randomness and chaos of statistical mechanics as being the interesting part, the entropy, the second law. In this approach, it is the measurability (the zeroth and first laws) of these systems that has the deep implications and powerful theorems!

This work is extended in this paper and its sequels, in which several more classical concepts are introduced. This paper is, in many ways, better than the first, being more clearly written, but the first is more focused and impressive. It contains an interesting "uncertainty relationship": the better energy is estimated, the worst (inverse) temperature is estimated. This is a classical uncertainty relationship, a consequence of the linearity of Laplace transforms. This paper also pushes a more philosophical line, which is interesting in itself. Mandelbrot was obviously impressed by the positivists, and isn't afraid to let a little subjectivity get into his arguments as a result. I will suspend judgment about this.

I think this part of Mandelbrot's work is of the very highest quality, but I will remain agnostic about judging it against the other methods, especially due to problems about phase transitions, etc. I think his contributions as being good enough that I will call this approach The Mandelbrot Approach, placing him along side Gibbs and Boltzmann.

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