THE SHELL MAP: The structure of froths through a dynamical map

TL;DR

This paper introduces the shell map, a simple dynamical system model that captures the structure of foams by combining geometrical tiling and disorder effects, using a radial logistic map approach.

Contribution

It presents a novel dynamical map framework for foam structures, integrating geometrical and disorder aspects into a unified mathematical model.

Findings

The shell map effectively models foam structure and disorder.

The radial map is described by the logistic map, capturing complexity.

The model demonstrates invariance properties and includes effects of space curvature.

Abstract

The shell map is a very simple representation of the structure of foams, combining the geometrical (random tiling) and dynamical (loss of information from an arbitrary cell out) aspects of disorder. The structure is built from the central cell outward like an ever expanding jigsaw puzzle without boundary. The radial map, from one spherical layer of cells to the next, is the logistic map, and the geometrical tiling is expressed mathematically as a dynamical system. The isotropy of the disordered structure is expressed locally by averaging over each layer. The over-all translational invariance is manifest in the independence of the structure and properties on the choice of the central cell. The radial map from one layer to the next includes both effects of disorder and of space curvature. We will illustrate it and give several examples, including a few arising from discussions in Cargese.