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Waves and patterns Science & Computers PHY307/PHY607 HWK MC #2 Random walks: the summation of random changes. Bacteria moves “forward”, but buffeted randomly by surrounding molecules. The DIRECTION accumulates random changes (a random walk on the sphere of directions.)

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hwk mc 2
HWK MC #2
  • Random walks: the summation of random changes.
  • Bacteria moves “forward”, but buffeted randomly by surrounding molecules.
    • The DIRECTION accumulates random changes (a random walk on the sphere of directions.)
  • Stock prices: Nobel prize winning model (Black-Scholes) for estimating future values of stocks and options.
    • The PRICE accumulates random changes (multiplied by a random number.)

PHY307, Fall 2002

slide3

from visual.graph import *

from random import *

stockchart = gcurve()

price = 100.0

for week in range(50):

growth = 0.9 + 0.25 * random()

price *= growth

stockchart.plot(pos=(week,price))

print `Final price is`, price

lab mc 2
Lab MC #2
  • Average displacement grows as the square root of the time (since average of the square of the position grew proportional to time.) Characteristic feature of a simple random walk.
  • The density of a diffusion created aggregate decreases as the aggregate grows: fractal structure with a dimension less than 3.

PHY307, Fall 2002

few coordinate systems
Few coordinate systems
  • Projectiles.
    • Coordinates might be x, y, and z positions and x, y, and z velocities.
  • Planets.
    • Similar coordinates.
  • Get periodic orbits and chaos.

PHY307, Fall 2002

extended many coordinate systems
Extended (many-coordinate) systems
  • Can’t be described by a few coordinates
    • (Often can for some purposes. For example, Earth is extended, but treated as a point in the solar system.)
  • “Many degrees of freedom”
  • Can exhibit quite complex phenomena, including periodic waves, pattern formation, chaotic waves, and turbulence.

PHY307, Fall 2002

why waves
Why waves?
  • Exchange between neighboring positions of
    • energy
    • material
    • whatever else
  • Also, convert from one form to another
    • Example: Height of water to motion of water.

PHY307, Fall 2002

oscillation
Oscillation
  • Exchange of energy in a few coordinate system as an example (not a wave yet.)
  • In a spring, the energy of distortion, which depends on position, is exchanged with the energy of motion, which depends on velocity.
  • Python demo.
    • Change in position is velocity
    • Change in velocity (acceleration) is force/mass, with force = - k * (position – rest_position)

PHY307, Fall 2002

concepts and procedures for simulating waves
Concepts and Procedures for Simulating Waves
  • Approximate continuous space by a grid.
  • Need a list of the values of the quantities (positions, velocities, chemical concentrations.) This list gives the values of the quantities at the grid points.
  • Need rules for how quantities affect their values at neighboring grid points.

PHY307, Fall 2002

example
Example
  • Demo of slinkie.
  • How do we describe the motion and simulate it?
  • FOR CODE DEVELOPED DURING LECTURE, SEE Codes page
    • http://physics.syr.edu/courses/PHY307/Codes/WaveInClass.py

PHY307, Fall 2002