# Exercises for Chaos Under Control

## Chapter 9: Life, Sandpiles, and 1/f Noise

39. Consider the automaton sandpile model with five cells. The state of each
cell is the number, S(i), of grains of sand piled above that cell. Suppose S(5) = 0
always, and the rule is if S(i-1) - S(i) > 2 then one grain moves from cell i-1 to
cell i.

(a) Find the next two generations from the initial distribution

Answer
(b) Find the next five generations from the initial distribution

Answer
40. With the same set-up as in exercise 39, suppose the rule is if
S(i-1) - S(i) > 3 then one grain moves from cell i-1 to cell i.

(a) Find the next two generations from the initial distribution

Answer
(b) Find the next five generations from the initial distribution

Answer
41. Suppose the position of a particle is measured every tenth of a second,
yielding these values: 5, 4, 3, 4, 4, 5, 6, 5, 6, 5. Without computing a power
spectrum, does this look more like white noise or Brownian noise? Why?
Answer

42.[E] Get some dried peas. Place a flat dish inside a box (so that the peas
don't get everywhere) and pour peas onto it. Notice that, at first, the pile gets higher
and higher, but after awhile the shape of the pile on the dish doesn't change very much.
(That's the critical state.) Now, drop one pea at a time near the top of the (critical)
pile and observe the resulting avalanches. Do you see that most of the time very little
happens, but that occasionally a large avalanche springs loose? Repeat the procedure
using dried rice and dried lentils. In each case, observe the shape of the critical
pile. Can you draw a conclusion about the angle of repose of a pile and the character
of the individual grains from which it is made? Incidentally, rice is especially
interesting because, after many avalanches, the rice grains tend to get aligned.
In which direction is the alignment? Do you see a similar alignment with lentils?

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