1. Materials Computation Center, University of Illinois
Duane Johnson and Richard Martin, NSF DMR The Growth of Fractal Transportation Networks
from co-PI: Alfred Hubler
Research: Understanding the dynamic processes that form fractals and fractal-like trees remains a key open problem in physical phenomena as diverse as dielectric breakdown, viscous fingering, fracturing, river formation and in biological phenomena like branching in plants, fungi, and blood vessels. There are two major difficulties posed by studying the dynamics of fractals by experiment: 1) time scales are either too fast (a bolt of lightning is an example of dielectric breakdown) or too slow (as in river formation) and 2) control of experimental parameters are often out of the researchers’ hands. We use graph theoretical models to predict the growth of fractal transportation networks.
The geometry of the initial state can affect the topology of the network. Because the binding of particles to the boundary is strong, the network does not reconfigure easily; therefore, the way the particles are distributed during stage II largely determines how the particles connect to one another. This constraint sets the relative number of trunks, branches, and termini that the network forms. This figure shows how different initial states lead to different final networks.
Joseph K. Jun and Alfred W. Hubler, Formation and structure of ramified charge transportation networks in an electromechanical system, PNAS 102, 536–540 (2005).
Alfred W. Hubler, Predicting Complex Systems with a Holistic Approach, Complexity 10, (2005)