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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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Waves and patterns science computers phy307 phy607 l.jpg

Waves and patternsScience & ComputersPHY307/PHY607

PHY307, Fall 2002


Hwk mc 2 l.jpg
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


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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


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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


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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


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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




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Waves II,

movie




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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


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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


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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


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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