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Return to France and disillusionment with mathematics in France


Work on thermodynamics in Geneva
Benoît Mandelbrot Mathematician
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Let me again go briefly through the second part of my work during this period. I also worked on the foundations of thermodynamics, and there too, what could a person bring to a topic, which has been so often gone over by so many brilliant people? The topic that actually interested me was the definition of temperature, because in non-statistical thermodynamics there's well defined temperature. If you go to statistical thermodynamics, then temperature is well defined for canonical systems. It is the parameter of distribution of energy. No problem with that, it's straight probability theory. Now everybody wants to define the temperature of a micro-canonical system, an isolated system. There is no parameter, just a number. The energy is given; what's the temperature? That has been a field of extraordinary fights between great minds. At one point Landau and Lifschitz had a definition of temperature for closed systems, and Kittel wrote them an angry letter, and in the later edition they didn't have a definition of it. Then Kittel had an angry letter against somebody else, in Physics Today, much later, and I responded to him by saying that indeed, the idea as defined by Landau and Lifshitz was ill-defined, but one could give a meaning, by defining the temperature of a closed system as being the estimate of the temperature of a canonical system of which that would be a sample. In other words, adding to statistical thermodynamics, proper statistics and estimation of parameters, which is something, which was totally unknown in physics, in fact in mathematics, a very esoteric notion in one community that did not communicate enough. Well, it turned out that part of my work on thermodynamics had been scooped by many years previously by Szilard, but only part, and that in a way I had revived an approach by Szilard, which nobody knew about, I thought, and added to it this fundamental idea of thermometry, which Szilard did not tackle. I would like just to add a few words here. The reason why I became aware of Szilard is because of a mysterious note, a footnote, in Von Neumann's Foundations of Quantum Mechanics. The footnote says, "When I think of thermodynamics of course I think in terms as described by Szilard." The 'of course' looked to me ridiculous because nobody I knew had read that paper, but having read it after having redone it, I understood it. Another great man, Gilbert Lewis, in thermodynamics had also discovered that part of Szilard, but none had gone all the way to this other aspect. But again, I'm not saying this was a practical breakthrough; it was hair-splitting in a perfect system.

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: Bernard Sapoval Daniel Zajdenweber

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.

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

Duration: 3 minutes, 6 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008