1. Last week: Magnets & transformers A magnet is a material or object that produces a magnetic field. A method to detect a magnetic field is to scatter iron filings and observe their pattern. magnetic field An electromagnet is a wire coil in which the magnetic field is produced by the flow of an electric current. magnetic field current A transformer are two coils that transfer electrical energy from one circuit to another through magnetic coupling. A changing current in the first coil (the primary ) creates a changing magnetic field; in turn, this magnetic field induces a changing voltage in the second coil (the secondary).electrical energycircuitcurrentinduces Iron filings that have oriented in the magnetic field produced by a bar magnet & a coil

2. Complex systems: A system with a large throughput of - a fluid = turbulence, river networks - chemicals = flames & explosion - tension = fracture - electrical current = lightning, dielectric breakthrough - information = internet, social networks The throughput is large means “sudden appearance of a pattern or dynamics (self-organization)” This self-organization causes emergent properties. Today: Highlights from my research - Complex Systems

3. Complex Systems Research Project: Predicting the growth of ramified networks Alfred Hübler Center for Complex Systems Research University of Illinois at Urbana-Champaign Research supported in part by the National Science Foundation (DMS-03725939 ITR)

4. We study: Growth of networks in a reproducible lab experiment, here: the structure of materials with high-voltage currents; which quantities a reproducible? We find: Materials produced in a high-voltage current develop open-loop, fractal structures which maximize the conductivity for the applied current. These fractal structures can be predicted with graph- theoretical models.

5. Potential applications : -Predict and control the dynamics of networks, -i.e. use resonances to efficiently detect, grow, nourish, destabilize, disintegrate networks: o ramified chemical absorbers (better batteries, better sensors, better purifiers) o multi-agent mixed reality systems o the rise and fall of social networks -Non-equilibrium materials: maximum strength in a strong gradient -Atomic neural nets: integration & processing of information in ‘super-brains’ out of digital nano-wires M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

6. Experimental Study of Structural Changes in Materials due to High-voltage Currents: Growth of Fractal Transportation Networks 20 kV needle electrode sprays charge over oil surface air gap between needle electrode and oil surface approx. 5 cm ring electrode forms boundary of dish has a radius of 12 cm oil height is approximately 3 mm, enough to cover the particles castor oil is used: high viscosity, low ohmic heating, biodegradable particles are non-magnetic stainless steel, diameter D=1.6 mm particles sit on the bottom of the dish

7. Phenomenology Overview 12 cm t=0s10s5m 13s14m 7s 14m 14s14m 41s15m 28s77m 27s stage I: strand formation stage II: boundary connection stage III: geometric expansion stationary state

8. Adjacency defines topological species of each particle Termini = particles touching only one other particle Branching points = particles touching three or more other particles Trunks = particles touching only two other particles Particles become termini or three-fold branch points in stage III. In addition there are a few loners (less than 1%). Loners are not connected to any other particle. There are no closed loops in stage III.

9. Emergent property: Relative number of each species is robust Graphs show how the number of termini, T, and branching points, B, scale with the total number of particles in the tree. J. Jun, A. Hubler, PNAS 102, 536 (2005)

10. The number of trees is not an emergent property J. Jun, A. Hubler, PNAS 102, 536 (2005)

11. Can we predict the structure of the emerging transportation network? ?

12. Predicting the Fractal Transporatation Network Left: Initial condition, Right: Emergent transporation network

13. Predictions of structural changes in materials due to a high voltage current: Predicting fractal network growth Task: Digitize stage II structure and predict stage III transporation network. 1) Determine neighbors, since particles can only connect to their neighbors. All the links shown on the left are potential connections for the final tree. 2) Use a graph-theoretical algorithms to connect particles, until all available particles connect into a tree. Some particles will not connect to any others (loners). They commonly appear in experiments. We test three growth algorithms: 1) Random Growth: Randomly select two neighboring particles & connect them, unless a closed loop is formed (RAN) 2) Minimum Spanning Tree Model: Randomly select pair of very close neighbors & connect them, unless a closed loop is formed (MST) 3) Propagating Front Model: Randomly select pair of neighbors, where one of them is already connected & connect them, unless a closed loop is formed (PFM) loner

14. Random Growth Model: Randomly select two neighboring particles Typical connection structure from RAN algorithm. Distribution of termini produced from 10 5 permutations run on a single experiment. Number of termini produced for all experiments, plotted as a function of N.

15. Minimum Spanning Tree Model: Randomly select pair of very close neighbors Typical connection structure from MST algorithm. Distribution of termini produced from 10 5 permutations run on a single experiment. Number of termini produced for all experiments, plotted as a function of N.

16. Propagation Front Model: Randomly select connected pair of neighbors Typical connection structure from PFM algorithm. Distribution of termini produced from 10 5 permutations run on a single experiment. Number of termini produced for all experiments, plotted as a function of N.

17. Comparison of all models to experiments Main Result: The Minimum Spanning Tree (MST) growth model is the best predictor of the emerging fractal transportation network

18. Structural changes of materials in high voltage current Experiment: J. Jun, A. Hubler, PNAS 102, 536 (2005) 1) Three growth stages: strand formation, boundary connection, and geometric expansion; 2) Networks are open loop; 3) Statistically robust features: number of termini, number of branch points, resistance, initial condition matters somewhat; 4) Minimum spanning tree growth model predicts emerging pattern. 5) To do: random initial condition, predict other observables, control network growth, study fractal structures in systems with a large heat flow Applications: Hardware implementation of neural nets, absorbers, batteries M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999) random initial distributioncompact initial distribution