Complexity Explorer Santa Fe Institute

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Introduction to Renormalization

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3.2 Israeli and Goldenfeld; projection and commuting diagrams » Quiz Solution

1. In the cellular automata we consider in this unit, how many states do you have to keep track of at time t to know the state of any particular point at time t+2?​

A. 3

B. 5

C. 9

D. infinite

 

Answer: (B). One step back you only need to know the three points just above, and to the left and right; two steps back requires you to keep track of the neighbors of each of those three points. Each point depends on three things in the past, but there are overlaps, so in fact there are only five grid cells in total you need to track.

 

This, of course, causes problems for the time-based coarse graining. We had a rule f(x,y,z); but now it seems like we need a new rule altogether, f(a,b,c,d,e), when we coarse-grain in time. And (of course) that will get even worse as we go to longer and longer jumps. In contrast to the Markov Chain case, we don't stay within the same model class (space of lookup tables).

 

2. In the Israeli and Goldenfeld picture, how many grid cells are in a supercell when you coarse-grain in time, to go from t to t+2?

A. 1

B. 2

C. 3

D. arbitrary

 

Answer: (B). 

 

3. What does it mean for the coarse-graining and evolution to "commute"?

A. if you evolve forward with f, and then coarse-grain with P, it's the same as coarse-graining with P and then evolving forward with g.

B. coarse-graining is independent of scale.

C. the evolution operator is at a fixed point, and does not change with coarse graining.

D. the coarse-graining operator leaves the system unchanged (fractal).

 

Answer: (A). This is the basic trick of Israeli and Goldenfeld -- given a function f, can you find an g and P that makes (A) true. "Commute" is a bit of a technical term in mathematics: if two operations commute, then you can do them in either order. Addition commutes with subtraction (if you add one and then subtract five, it's the same as subtracting one and adding five), but does not commute with multiplication (if you add one and double, it's different from doubling and adding one). Remember this when you tip: it's one thing to tip based on the bill, it's another thing to tip based on the added tax.

 

(D) is the wrong answer because (spoiler alert for the 1980s) the answer is never "a fractal". Sorry.