Soft Matter and Complex Systems Seminar

Laplacian growth is one of the fundamental mechanisms of pattern formation, driving such natural processes like viscous fingering or two-dimensional combustion in a Hele-Shaw geometry. The characteristic features of these processes include a strong competition between spontaneously formed dendrite-like structures, and tip-splitting effects when dendrites bifurcate into secondary branches. Many of these processes can be described in terms of a simple deterministic model in which growth takes place only at the tips of the fingers and the dynamics is expressed in terms of the Loewner equation [1].

In other systems, the dynamics is to a large extent noise-driven. Examples include diffusion-limited aggregation, dielectric breakdown or fracturing processes. The description of such processes can be provided by means of a simple model, in which growth is represented as a random sequence of elementary conformal maps [2], which is convenient for numerical treatment.

During the seminar I will discuss both the deterministic and the stochastic model and I will present the resulting structures and characteristic effects.

[1] L. Carleson and N. Makarov, J. Anal. Math. 87, 103 (2002).
[2] M.B. Hastings, L.S. Levitov, Physica D 116 (1998) 244-252