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Critical opalescence, Onsager and work in physics
Benoît Mandelbrot Mathematician
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Now we are coming close to a period of the early '70s which is logical in this context, because it is in '71 or '72 that I became aware through Herbert Curran of the fact that at the same time as I, totally separately, totally independently of me, a number of physicists had been studying a phenomenon called critical fluctuations. And they were using tools, which for some extraordinary coincidence were very much the same as those I'd been using. Now, what is critical opalescence? It is rather important and easy to say: it is a tube with liquid and gas above it; as you heat them at one point the boundary between liquid and gas vanishes and the liquid becomes quite opaque, which is why it's called opalescence; we heat even further, it is neither a gas nor liquid, it is something else, and again it is transparent. It is a phenomenon that struck the imagination in the nineteenth century and was very much discussed in books of science for young people or science for amateurs that I was reading as a young man, as a child. Critical opalescence was discovered therefore at that time. Some time later Pierre Curie, a very great scientist, discovered that if you heat a magnet enough above a certain temperature it ceases to be a magnet. That temperature is called the critical temperature or the Curie temperature. And more generally, many phenomena in physics have different phases and under certain critical conditions their character changes. A breakthrough in that context was in 1944. An extraordinary calculation performed by Lars Onsager, nominally a chemist, but in the book of many of my friends, one of the great mathematicians of this century, who succeeded in solving the problem of the Ising model in two dimensions. That is: you imagine that a big magnet is made of a gigantic collection of small magnets which can move up and down; that all the neighbours interact but not beyond neighbours - interaction is very much short range; that these magnets are being hit by an environment which is at some temperature, so the higher the temperature, the stronger the inversions of these magnets forced by the environment; and the question arises, is it possible for these small magnets acting together to create gross visible microscopic magnetism? And Onsager solved this problem in two dimensions. In three dimensions it's gigantically more complicated. Now, Onsager did that in '44. From what I understand for the next twenty years his ideas were very, very slowly digested and around '65 or shortly before, the study of critical phenomena became again extremely active and produced several great scientists who established a very very great chapter of physics around them, based again upon scaling, upon normalisation, upon fixed points, and all these phenomena which I'd been studying. Now, I became acquainted with those people very rapidly and in a short time established a very close working relation with several of them, so my work in physics can be described in two parts. When I was working alone, solo, on these questions that are a part of physics, you could say it was physics but not physics proper; in a sense. It is physics of a kind that was carried out by people who did not call themselves physicists; some were called fluid mechanics experts, others were electronics experts or hydrologists. Each dealt with a physical phenomenon, but physicists didn't have an interest in them, whereas critical phenomena were very much part of physics proper.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 4 minutes, 29 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008