Laplacian Growth Models.

A class of non-equilibrium growth problems are studied, where the local velocity is given as a function of the Laplacian field surrounding the object. Here, the local velocity depends on all the other points on the object (a non-local property). In the first part, I present a detailed study of a typical Laplacian growth model--diffusion limited aggregation (DLA). Using both an exact enumeration approach and direct numerical simulation, it is found that the minimum growth probability does not scale. I also find a form for the tail part of the distribution. In order to interpret this result, a family of structural models for DLA are proposed, which consist of self-similar "voids" connected by narrow "channels." An analytic solution for the growth probability distribution is presented where the form of the distribution is the same as the one obtained by numerical simulation. Also, the form is the same for the whole family, suggesting that the form is a consequence of the "channels" and self-similar "voids," and is independent of further details of the model. In the second part of the thesis, an approach is made to develop and apply a position space renormalization group (PSRG) method for growth models. I applied the PSRG method to the problem of viscous fingering in the absence of surface tension, with an arbitrary viscosity ratio between the injected and displaced fluid. It is found that there are only two fixed points, the Eden and the DLA points. The Eden point is stable in all directions, while the DLA fixed point is a saddle point. Also studied are the evolution of patterns formed by injecting a reactive fluid with viscosity mu into a two-dimensional porous medium filled with a non-reactive fluid of unit viscosity. Formulated is a three-parameter PSRG, and two crossovers are found: (1) From the first DLA to the Eden point--due to finite viscosity, and (2) From the Eden to the second DLA--due to chemical dissolution.