Ostwald ripening and separation of scales

The physics group at IUB is quite small so the number of exciting seminars is relatively limited. But the mathematicians have a weekly colloquium that is usually worth attending. I should have started earlier reporting on this, then I would have been talking about generalized theta functions in number and string theory amongst other things but this was the past and today we had Barbara Niethammer who talked about Ostwald ripening.

I have to admit, I had no idea what this is but I learned that this is a theory of crystal formation in a solution. You first start out with many small crystals but eventually the bigger ones will grow on the expense of the smaller ones. The big question is whether there is self similar behaviour.

The model is quite simple: You start with a number of spherical crystals at random fixed positions and with random sizes. Then (on very short time scales) there is diffusion in the solution. This can be described by a local chemical potential u. The diffusion makes sure u is harmonic in the bulk and at the surface of the spherical crystals it is given by 1/radius of the crystal. You can easily solve this Dirichlet problem. But now you turn on the growth/shrinking of the crystals by imposing the time derivative of the crystal volume to be given by the integral over the flux of gradient(u) over the boundary of the shere. This induces a long range interaction between the crystals and it turns out that the sum of the crystal volumes is constant in time.

Now you assume that the crystals are sparse and you can make a mean field approximation: Each crystal just sees u with boundary conditions given by the average u which turns out to be 1/average radius. Effectively, this is now a system of coupled ODE's, with the radii of the crystals and the average u as time dependant variables.

But there is a second step of idealization in which you only consider the average u and the number density of crystals of radius r: f(r,t). The evolution becomes now a transport equation for this f and you can study its late time behaviour starting from some initial data.

Of course r has to be positive and crystals of radius 0 drop out of the system. Studying the transport equation you find that it is basically given by a stretching in r then imposing the cut-off and then renormalizing (as the total volume was conserved). And indeed, this approaches static distributions (after scaling out a trivial t^1/3 growth). At least for nice inital data. Today, only initial data with compact support was discussed and it turned out that the form of the asymptoptic solution only depended on the form of the initial data at the upper end of the support of the initial data: This is can easily be understood from the strech-cut off-renormalize form of the transport equation. For example, if the distribution ends with a delta function peak, that is there is at least one largest crystal, all the volume eventually ends up in these larges crystals. If the inital distribution goes to zero with some power at its upper end, the static solution is characteristic of that power. And finally if there is no largest power (for example because it is given by a power series in 1/(r-r_max) ) then there is no asymptotic solution. There were also pictures which looked quite interesting.

The reason why I talk about this is, that I think that this is a nice example of a non-renormalizable system. It's late time behaviour depends on the infinitely small scales (at the upper end of the distribution function): So in the long run (late times) the behaviour is completely determined by the UV. (Well, one could say, at late times the second approximation of describing a large number of crystals in terms of a distribution function breaks down. But one could forget about the crystals and consider the distribution function theory as the microscopic one).

If the real theory of everything had such a behaviour, we would just have to wait long enough and find out how the world looks at infinitely small scales. Maybe waiting for larger and larger accelerators is a similar endeavour.