10.2 Random Walks

1. 10.2 Random Walks

2. 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.

3. 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

4. Aside: Fooled By Randomness 1973 2004

5. What’s the Trend?

7. 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 +1 time t

8. The Wave

9. 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….

10. Aside: The Wave/Particle Duality Photoelectric effect (Thomson 1899): Light must consist of particles, because light waves knock electrons out of a metal plate. Double-slit experiment (Young 1801): Light must be a wave, because interference produces more than two bands on screen.

11. 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….

12. 1D Cellular Automata Sample rule: Look at a cell and its left and right neighbors. Change the cell according to the following table:

13. 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)

14. 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:

15. Cellular Automata in Nature

16. 2D Cellular Automata

17. Cellular Automata as the Basis for Life, the Universe, and Everything

18. 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,

19. Animation Tricky in Excel, but fortunately a download is available here.

20. Average Distance Traveled Average distance traveled increases (but not monotonically) with time (# of steps n):