Waves and patterns Science & Computers PHY307/PHY607

1. Waves and patternsScience & ComputersPHY307/PHY607 • Thurs. Presentations: Have simulations available at computer, add handouts if you like. People will wander and ask questions. • Current grades: projects, labs, HWK. • 2:30, Friday, Dec. 13 is Final Exam. Room 106. • Start today with course surveys. PHY307, Fall 2002

2. From particles to waves • Particles (few coordinates): Get simple (fixed points or periodic orbits) and complex (chaos) behaviors, using simple rules [example: logistic map.] • Extended systems: The universe can be even more complicated, when considering extended systems, including possibility of pattern formation. PHY307, Fall 2002

3. Waves • Interactions between neighboring bits of space or material can lead to waves. • In lab so far, looked at case where mechanical forces act to bring distorted pieces of an elastic medium (string or surface) together. • Saw simple propagating waves. PHY307, Fall 2002

4. How can complexity arise? • Consider equilibrium vs. nonequilibrium. • Equilibrium: whole system is at same temperature – energy is evenly distributed. • Example:molecules inisolated box. PHY307, Fall 2002

5. Equilibrium (no energy source): white noise – order, patterns extremely improbable (in fact, maximum disorder/entropy)

6. Out of equilibrium • In a nonequilibrium, or driven, system, energy or material is constantly supplied and extracted: Source Flow through system PHY307, Fall 2002

7. Examples • A pot of boiling water. • A computer factory and the manufactured computers. • The surface of the Earth. • Biological systems. PHY307, Fall 2002

8. Patterns • How complicated does the nonequilibrium model need to be to generate complex behavior? • Relatively simple nonequilibrium systems can generate complexity: • stripes • spots • spatio-temporal chaos PHY307, Fall 2002

9. Pattern formation example

10. Pattern formation example

11. Extreme complexity • Nonequilibrium systems exhibit patterns that do not occur in equilibrium systems. • Don’t yet know all the details, but it is quite plausible that nonlinear nonequilibrium physical and chemical systems can generate arbitrary complexity, including weather and life and computers. • One of the challenges facing science is to more clearly define the bridges between pattern formation and natural selection. PHY307, Fall 2002

12. Turing, again (1951) • Same person of Turing machine fame. • Coined the term morphogen. [meaning?] • Showed that simple wave-type equations that include chemical reaction type terms (“nonlinear”) can generate complex patterns in space and time. PHY307, Fall 2002

13. Gray-Scott Equations • Suppose you have two chemicals, U and V, with the following properties: • U and V diffuse (spread out.) • A U molecule can react to form a V molecule, in the presence of 2 V molecules. • U and V decayat a given rate. • U is supplied to the system (continuously “sprinkled”.) PHY307, Fall 2002

14. Lab • VPython too slow for this purpose. • Use a web page that uses Java to simulate the G-S equation. • Look for distinct types of behavior, as the parameters k and F are varied. PHY307, Fall 2002