Laplacian networks: growth, local symmetry and shape optimization

Abstract

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.

References in corpus (6)

• 2D growth processes: SLE and Loewner chains
• A Guide to Stochastic Loewner Evolution and its Applications
• The Loewner equation: maps and shapes
• Exact Solutions for Loewner Evolutions
• Fingered growth in channel geometry: A Loewner equation approach
• Dimensions, Maximal Growth Sites and Optimization in the Dielectric Breakdown Model

Cited by in corpus (4)

• Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics
• Three-dimensional nanoprinting via charged aerosol focusing
• Geodesic Loewner paths with varying boundary conditions
• Tip growth in a strongly concentrated aggregation model follows local geodesics