10.2 random walks. random walk a random walk refers to the apparently random motion of an entity....
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10.2 Random Walks
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Random Walk
• A random walk refers to the apparently random motion of an entity.
• This is often the best model of a physical process (Brownian motion) or financial index (stock price), etc.
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Aside: Fooled By Randomness
HAMLET: Do you see yonder cloud that's almost in the shape of a camel?
POLONIUS: By the mass, and 'tis like a camel, indeed.
HAMLET: Methinks it is like a weasel.
POLONIUS: It is backed like a weasel.
HAMLET: Or like a whale.
POLONIUS: Very like a whale.
Hamlet Act 2, Scene 2
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1973 2004
Aside: Fooled By Randomness
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What’s the Trend?
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Simulating Motion: Cellular Automata
• If all we have to work with is a grid of cells (spreadsheet), how can we simulate a random walk?
• Moving a value from one cell to an adjacent cell is simulated by changing the contents of the cells over time:
time t time t +1
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The Wave
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Aside: The Wave/Particle Duality
• We tend to think of “true” motion as movement of an object from one location to another, and the wave phenomenon as an illusion of motion.
• But at the most fundamental level, energy and matter behave like both a particle (object) and a wave….
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Aside: The Wave/Particle Duality
Double-slit experiment (Young 1801):Light must be a wave, because interference produces more than two bands on screen.
Photoelectric effect (Thomson 1899):Light must consist of particles, becauselight waves knock electrons out of ametal plate.
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Cellular Automata
• A cellular automaton (plural automata) is a dynamical model discrete in space, time, and state (possible values, typically just 0 and 1).
• CA has a grid of cells, and a set of transition rules describing what happens to a cell based on its own state and the state of its neighbors.
• Simplest CA are one-dimensional….
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1D Cellular Automata
Sample rule: Look at a cell and its left and right neighbors. Change the cell according to the following table:
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ExCellular Automata
• Each row is the state of the CA at a particular time step.
• First row has random values:= IF(RAND()>0.5, 1, 0)
• Subsequent rows have rule:=IF( OR (AND( A1=1,B1=1,C1=1), AND (A1=1,B1=0,C1=0), AND (A1=0,B1=0,C1=0)), 0, 1)
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ExCellular Automata
• Drag second row, second column rightward to next-to-last• Keep first and last columns (boundary conditions) constant• Drag second row down 60 or 70 rows• Use Conditional Formatting / Color Scales to color cells
based on values• Zoom out to view whole display• Hit F9 key to restart with new initial conditions• Experiment with other rules:
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Cellular Automata in Nature
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2D Cellular Automata
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Cellular Automata as the Basis for Life, the Universe,
and Everything
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Algorithm for a Random Walk, With Diagonal Steps:
seed random number generator
let x, x0, y, and y0 be 0
let n be the number of steps
let list be a list containing the origin
do the following n times:
let rand be a random 0 or 1
if rand is 0
increment x by 1
else
decrement x by 1
do the same for y
append point (x, y) onto list
create and display graphics of walk
report distance between first and last points,
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Animation
Tricky in Excel, but fortunately a download is available here.
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Average Distance Traveled
Average distance traveled increases (but not monotonically) with time (# of steps n):