Self-interactions of strands and sheets

Abstract

Physical strands or sheets that can be modelled as curves or surfaces embedded in three dimensions are ubiquitous in nature, and are of fundamental importance in mathematics, physics, biology and engineering. Often the physical interpretation dictates that self-avoidance should be enforced in the continuum model, i.e. finite energy configurations should not self-intersect. Current continuum models with self-avoidance frequently employ pairwise repulsive potentials, which are of necessity singular. Moreover the potentials do not have an intrinsic length scale appropriate for modelling the finite thickness of the physical systems. Here we develop a framework for modelling self-avoiding strands and sheets which avoids singularities, and which provides a way to introduce a thickness length scale. In our approach pairwise interaction potentials are replaced by many-body potentials involving three or more points, and the radii of certain associated circles or spheres. Self-interaction energies based on these many-body potentials can be used to describe the statistical mechanics of self-interacting strands and sheets of finite thickness.