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.

An electromagnet is a wire coil in which the magnetic field is produced by the flow of an

electric 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).

Iron filings that have oriented in the magnetic field produced by a bar

magnet & a coil

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

Complex Systems Research Project: Predicting the

growth of ramified networks

Alfred Hübler

Center for Complex Systems ResearchUniversity of Illinois at Urbana-Champaign

Research supported in part by the National Science Foundation

(DMS-03725939 ITR)

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.

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-wiresM. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

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

Phenomenology Overview

12 cm

t=0s 10s 5m 13s 14m 7s

14m 14s 14m 41s 15m 28s 77m 27s

stage I:strand

formation

stage II:boundary

connection stage III: geometric expansion

stationary state

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.

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)

The number of trees is not an emergent property

J. Jun, A. Hubler, PNAS 102, 536 (2005)

Can we predict the structure of the emerging transportation network?

?

Predicting the Fractal Transporatation Network

Left: Initial condition, Right: Emergent transporation network

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

Random Growth Model: Randomly select two neighboring particles

Typical connection structure from RAN algorithm.

Distribution of termini produced from 105 permutations run on a single

experiment.

Number of termini produced for all experiments, plotted as a function of

N.

Minimum Spanning Tree Model: Randomly select pair of very close neighbors

Typical connection structure from MST algorithm.

Distribution of termini produced from 105 permutations run on a single

experiment.

Number of termini produced for all experiments, plotted as a function of

N.

Propagation Front Model: Randomly select connected pair of neighbors

Typical connection structure from PFM algorithm.

Distribution of termini produced from 105 permutations run on a single

experiment.

Number of termini produced for all experiments, plotted as a function of

N.

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

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 flowApplications: Hardware implementation of neural nets, absorbers, batteriesM. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the

Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

random initial distribution compact initial distribution