Formulating a first-principles statistical theory of growing surfaces in two-dimensional Laplacian fields

Abstract

A statistical theory of two-dimensional Laplacian growths is formulated from first-principles. First the area enclosed by the growing surface is mapped conformally to the interior of the unit circle, generating a set of dynamically evolving quasi-particles. Then it is shown that the evolution of a surface-tension-free growing surface is Hamiltonian. The Hamiltonian formulation allows a natural extension of the formalism to growths with either isotropic or anisotropic surface tension. It is shown that the curvature term can be included as a surface energy in the Hamiltonian that gives rise to repulsion between the quasi-particles and the surface. This repulsion prevents cusp singularities from forming along the surface at any finite time and regularizes the growth. An explicit example is computed to demonstrate the regularizing effect. Noise is then introduced as in traditional statistical mechanical formalism and a measure is defined that allows analysis of the spatial distribution of the quasi-particles. Finally, a relation is derived between this distribution and the growth probability along the growing surface. Since the spatial distribution of quasi-particles flows to a stable limiting form, this immediately translates into predictability of the asymptotic morphology of the surface. An exactly-solvable class of arbitrary initial conditions is analysed explicitly.