Theory of diffusion controlled growth
Abstract
We present a new theoretical framework for Diffusion Limited Aggregation and associated Dielectric Breakdown Models (DBM) in two dimensions [R. C. Ball and E. Somfai; PRL 89, 135503 (2002); and cond-mat/0210598]. The first key step is to understand how these models interrelate when the ultra-violet cut-off strategy is changed, leading us to propose a new set of equivalence relations between generalisations of the original DBM. This enables us to approach these models through the dynamics of conformal maps with a fixed charge cut-off. Using logarithmic field variables then leads to coupled moment equations which appear to be renormalisable. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is discussed. We find angular resonances relatable to the cone angle theory, and we are led to predict a new Screening Transition in the DBM at large η.
- Publication:
-
APS March Meeting Abstracts
- Pub Date:
- March 2003
- Bibcode:
- 2003APS..MARK15006B