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The complicated nature of DLA


Diffusion limit aggregates
Benoît Mandelbrot Mathematician
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Another phenomenon which I spent an immense amount of time on is diffusion limit aggregates, and this one started quite outside of my group of colleagues, but would not, I think, have arisen otherwise. For a long time, a very long time, in fact in the '20s, many scientists dealing with disorderly matters, often people working on oil fields, were finding branching structures. So for example, if you push water into a rock- for example, take an old oil field; oil that is under pressure has been collected, but there's plenty of oil left, so you want to push water in it hoping the water will push the oil and get more oil, - secondary recovery. Now the hope is that the water you pump in will move like a kind of wave. The fear is that more like a finger, it will just find some easy way out and you just put dirty water in and get dirtier water out and no oil. The reality was it moves more like a hand, with fingers spread out. In the '20s they had no idea how to analyse it. As a matter of fact, they said, "Let's cut these fingers off because we want to study them, and let's look at the shape of the blood inside." After fractal geometry arose, two physicists, Whitney and Sandor, studied an example in which such branching structures occur, an example they could control completely, they knew thoroughly, they knew inside out, and they obtained very strange shapes. They did not know about fractals, and walked down the corridor to show their colleagues, "Look what we've got. Do you know what to make out of that?" And one of their colleagues said "But that's a fractal." "How do you know?" "Well, it looks like one." "So what do you do?" "Well, make this and that and that measurement." They did. Indeed it was a fractal. That is, the way in which this system branches makes it into a fractal. How is the system defined? You can express in childish biology as follows. Imagine that there are little cells floating around, and a cell floats around, and here is a living, multi-cellular organism. As soon as this wandering cell touches that organism it gets stuck to it and the organism becomes bigger by one cell. And if another cell comes again it makes something grow. So it grows by corners and so on, and if you added a hundred, a thousand, ten thousand, a million cells, they don't add up to a wave going out as you might have thought, but they add up to a very, very loose and fingering structure, which was called DLA, diffusion limit aggregate. And the fractality was its most striking characteristic.

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, 22 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008