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Diffusion limit aggregates


Benoît Mandelbrot Mathematician
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I would like to give two examples of my work in physics, perhaps more as I speak. One of them is the understanding of the phenomenon called percolation. I would like to do it first of all as physics or mathematics, and then perhaps as art. It started as an exercise in mathematics due to two Englishmen, Broadbent and Hammersley, who asked themselves what happens in the percolation of a coffee-making machine. To simplify they assumed, in my words, a very large floor - in the case of dimensional percolation - on which you would lay tiles that were either copper or rubber. If they're copper then they conduct electricity; if they're rubber, they don't. So you just put them at random, some with a proportion so much of copper and so much of rubber. If there is no copper to speak of, one or two percent, they are sort of lost in a sea of rubber and current cannot be conducted. If there is lots of copper and very little rubber, then the rubber is neutralised and the system conducts electricity over an endless distance. Then the discovery of Hammersley which was very important in mathematics and even more important in physics. That was that some particular probability of copper and rubber passed from a state in which electricity is not transmitted to one in which it was transmitted. In some cases for some shapes of tiles that's one half; for other shapes of tiles it is not one half. In a short time it was realised that this model, even though introduced as a mathematical curiosity, but not esoterica, was very much discussed around and was a model for many phenomena in the physics of disorder. The study of percolation clusters became a topic of very great interest to many individuals. The most striking aspect of percolation clusters is the following. If you take a criticality, which are these very magical probabilites of the two kinds of tiles, you take a copper tile and all the tiles that touch it. Together the (copper tiles) form a cluster, which is a fractal shape. It has the same properties at all scales. It has very, very subtle properties. It is very weakly attached to itself, that is, at some points if you remove one copper tile and replace by rubber, the cluster is disconnected; in other places if you move copper tiles out it doesn't disconnect it. So there are points that are 'privileged ' in a sense, in that they are the fundamental points where it can be disconnected. Those points also are a fractal. As a matter of fact, every property of this object is fractal. That is, if you don't look at the whole thing but only parts of it, they're all fractal. And then another aspect of it is that if one makes a list of all the interesting subsets of it, plus the fractal dimension of each, those properties together determine all the physics. It is a most remarkable situation because one of the models, the dream, the Holy Grail of much of physics is to prove that geometry can govern physics. The great example, of course, is Einstein's General Relativity, in which the geometry of space determines gravitation and everything else. It is a very beautiful situation. The geometry in the case of Einstein is a matter of curvature, it is a matter of classical properties in a very smooth structure; the geometry in the case of percolation clusters is a matter of wiggliness, of irregularity, of connectedness or disconnectedness. But the same words apply: for percolation cluster geometry determines structure. The study of percolation clusters became an extraordinarily important and central topic in physics, and much of it is fractal - not everything, but much of it is fractal. That is, if one understands the fractal structure of it and its parts, one understands the way it reacts in many different ways and those ways present a metaphor, sometimes more than a metaphor, to many natural phenomena. One of them is forest fires. There are many phenomena in physics, and also outside physics, in which these clusters are very important.

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: 5 minutes, 4 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008