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White noise and fixed points


Development of work with turbulence and multifractals
Benoît Mandelbrot Mathematician
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Now this idea embodies in itself the basic thought of similarity, self-similarity, and it was taken up around 1940 by Kolmogorov and others but in a very analytic fashion. They produced this marvellous work, K-5/3 spectrum of turbulence that played such a big role in science and also in my life. But for me, in a certain sense, the K-5/3 was an over simplification. It was already analytic and too crude, because it assumed that turbulence was homogenous, that even though there was eddy structure hidden under it, the turbulence was the same at every point. However as it was pointed out very soon to Kolmogorov and to others, turbulence is anything but homogenous. If you think of a wind, the wind comes in gusts, and you think of a gust, a gust is just gust but it has separate sub-gusts, it splits into pieces. Turbulence has not only a self-similar scaling analytic structure, it has a self-similar scaling physical structure, and the question was how to illustrate how to represent it. Now for this goal I introduced the idea of scaling going well beyond the analytics to the geometry, and this was the foundation of the theory of multifractals that I mentioned, not because of their mathematical interest. The mathematical interest came second. It was very great, but the most important idea was that these hierarchies were meant to represent the structure of turbulence. Many techniques were developed: I hasten to say these techniques were developed completely on the basis of traditional mathematics. I was not aware of any work in physics proceeding in the same fashion. For the rivers, say the River Nile, or any of the other rivers with one exception which is the Rhine, the structure is very different. What I did was to generalise Brownian motion by introducing fractional Brownian motion. I add again, as I often do, that later on I discovered that fractional Brownian motion with that name was present as esoterica in one or two papers before my work. Those papers had attracted no attention because, as a defined formula, it was not a topic of exciting content. As I defined it, it was simply a translation into mathematical terms of the idea that seven lean years, seven fat years, seventeen of each, seventy of each, seven hundred of each, and the development of this approach was very different from multifractals. If I had been only, how should I say, led by analytic similarity, by the fact that the 1/f noise occurs in both cases, I don't know where I would have ended. But because one of the principal lessons I learned from my studies is that generality is - well, generality is for the birds! It is not offered to mathematicians when it is interesting, but in general, if I dare say, generality itself takes us away from what you can see and compare and have on our fingers and study in detail. So I certainly studied 1/f noises as one phenomenon, or several kinds of them, including variation of rivers, and multifractals were a separate thing even though I had gone from one to the other for the purpose of keeping the 1/f character and having something richer. Today I just finished putting together a book which puts together my papers of the '60s on this topic, and their diversity and their unity impress themselves upon me again. They're all the same phenomenon, if you look by this narrow window of Fourier spectrum, harmonic analysis; they are very different phenomena from any other viewpoints. It is a case also of where the mathematical haste of saying this is almost independent is absolutely terminal. One gets nothing because one must proceed very, very carefully.

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, 45 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008