No posts for three weeks… you know, we’ve been revolting in Spain, and there are times in which one has to care for politics. But physics is a jealous lover… :)

So, we have published a paper on kinetic roughening. What does it mean? OK, imagine that, while your mind is roaming through some the intricacies of a physics problem, the corner of your napkin falls into your coffee cup. You see how the liquid climbs up, and the interface which separates the dry and wet parts of the napkin becomes rough. Other examples: surface gowth, biological growth (also tumors), ice growing on your window, a forest fire propagating… Rough interfaces appear in many different contexts.

We have developed a model for those phenomena, and simulated it on a computer. Basically, the interface at any point is a curve. It grows always in the normal direction, and the growth rate is random. The growth, also, is faster in the concavities, and slower in the convex regions. After a while, the interfaces develop fractal morphology. I will show you a couple of videos, one in which the interface starts out flat, and another one in which it starts as a circle. The first looks more like the flames of hell, the second more like a tumor.

The fractal properties of those interfaces are very interesting… but also a bit hard to explain, so I promise to come back to them in a (near) future.

The work has been done with Silvia Santalla and Rodolfo Cuerno, from Universidad Carlos III de Madrid. Silvia has presented it at FisEs’11, in Barcelona, a couple of hours ago, so I got permission at last to upload the videos… ;) The paper is published in JSTAT and the ArXiv (free to read).

Very nice pictures, Javi, specially the second one … does the area enclosed by the curve increase in the evolution? A mathematical doubt, the motion is along the normal to the curve, but if the curve is not differentiable at any point, how do you define the normal line? I guess in simulations you consider piecewise smooth curves, but in the mathematical model you have to deal with nowhere differentiable curves … is it possible to reach a smooth curve from a rough curve, or does the fractal dimension always increase in the evolution?

Thanks!! :) And your comment is very appropriate… the curve is fractal, so the “local” curvature and normal are ill defined. The model is defined in the continuum, but only the discrete simulations make “full sense”. My idea is that, although I don’t think anybody has worked on that, the theory has to be “renormalized”. I mean: you can define the local normal, or the local curvature, when you “regularize”, i.e.: impose a minimum scale, or discretization. If you want to go “deeper”, the local normal and the local curvature changes, so the associated coefficients have to change accordingly in order to represent the same physics. This is exactly the same as we have in quantum field theory.

Very nice pictures, Javi, specially the second one … does the area enclosed by the curve increase in the evolution? A mathematical doubt, the motion is along the normal to the curve, but if the curve is not differentiable at any point, how do you define the normal line? I guess in simulations you consider piecewise smooth curves, but in the mathematical model you have to deal with nowhere differentiable curves … is it possible to reach a smooth curve from a rough curve, or does the fractal dimension always increase in the evolution?

Thanks!! :) And your comment is very appropriate… the curve is fractal, so the “local” curvature and normal are ill defined. The model is defined in the continuum, but only the discrete simulations make “full sense”. My idea is that, although I don’t think anybody has worked on that, the theory has to be “renormalized”. I mean: you can define the local normal, or the local curvature, when you “regularize”, i.e.: impose a minimum scale, or discretization. If you want to go “deeper”, the local normal and the local curvature changes, so the associated coefficients have to change accordingly in order to represent the same physics. This is exactly the same as we have in quantum field theory.