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

TL;DR

This paper develops a first-principles statistical and Hamiltonian framework for two-dimensional Laplacian growth, incorporating surface tension effects, regularizing cusp formation, and analyzing the asymptotic surface morphology.

Contribution

It introduces a Hamiltonian formalism for Laplacian growth that includes surface tension and noise, providing new insights into surface regularization and morphology prediction.

Findings

Surface tension can be incorporated as a surface energy in the Hamiltonian.

The formalism prevents cusp singularities at finite times.

The spatial distribution of quasi-particles predicts asymptotic surface shapes.

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.