Applied Mathematics Colloquium
Princeton University
November 3, 2003

Title:    Transport-limited aggregation in two dimensions

Speaker:  Martin Z. Bazant (Department of Mathematics, MIT)

Abstract:

Over the past two decades, Diffusion-Limited Aggregation (DLA) has become
the canonical model of fractal growth controlled by bulk transport (as
opposed to interfacial kinetics). A key feature of DLA, also arising in
related phenomena of Laplacian growth such as viscous fingering, is the
assumption of steady diffusion, governed by a harmonic concentration
field. As first described by Hastings and Levitov (1998), this allows DLA
in the plane to be recast in terms of a stochastic conformal map with
``bumps'' chosen according to the harmonic measure.  Here, we apply
conformal mapping to certain systems of transport equations [1] to
generalize the Hastings-Levitov formalism to a new class of (discrete and
continuous) non-Laplacian growth phenomena limited by nonlinear diffusion,
advection-diffusion in a potential flow, and/or electrochemical transport
[2]. Motivated by the viscous-fingering analysis of Entov and Etingov
(1991), we also consider curved two-dimensional manifolds, including DLA
on a sphere or pseudo-sphere [3]. Another interesting example is
Advection-Diffusion-Limited Aggregation in a potential flow, which
exhibits a universal crossover from DLA to a new advection-dominated
regime, controlled by a time-dependent Peclet number. Remarkably, the
fractal dimension is not affected by spatial curvature or advection, in
spite of dramatic changes in anisotropy and growth rate.

[1] M. Z. Bazant, to appear in Proc. Roy. Soc. A (2003).
http://arXiv.org/abs/physics/0302086

[2] M. Z. Bazant, J. Choi, and B. Davidovitch, Phys. Rev. Lett. 91,
045503 (2003). http://arXiv.org/abs/cond-mat/0303234

[3] J. Choi, D. Crowdy, and M. Z. Bazant, in preparation.