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
| 12 | 8 | 6 | 1 | 0 |
(b) Find the next five generations from the initial distribution
| 3 | 3 | 3 | 20 | 0 |
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
| 12 | 8 | 6 | 1 | 0 |
(b) Find the next five generations from the initial distribution
| 3 | 3 | 3 | 20 | 0 |
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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