Towards a theory of growing surfaces: Mapping two-dimensional Laplacian growth onto Hamiltonian dynamics and statistics

Abstract

I show that the evolution of a two dimensional surface in a Laplacian field can be described by Hamiltonian dynamics. First the growing region is mapped conformally to the interior of the unit circle, creating in the process a set of mathematical zeros and poles that evolve dynamically as the surface grows. Then the dynamics of these quasi-particles is analysed. A class of arbitrary initial conditions is discussed explicitly, where the surface-tension-free Laplacian growth process is integrable. This formulation holds only as long as the singularities of the map are confined to within the unit circle. But the Hamiltonian structure further allows for surface tension to be introduced as an energetic term that effects repulsion between the quasi-particles and the surface. These results are used to formulate a first-principles statistical theory of pattern formation in stochastic growth, where noise is a key player.