Eric Weeks - personal pages - pictures

Cellular Automata Phase Diagram

weeks@physics.emory.edu

For this plot, D ranges from -1 to 1 (horizonal axis), A = C and range from -1 to 1 (vertical axis; -1 at top of the plot), and B=0.2 is fixed. In this plot, black is zero, white is 1/2. Given that the arithmetic of the rule is done mod 1, the plot is periodic in D.

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Here is a sequence of pictures with D=0, B=0.2. In these pictures A = C = 0.395, 0.40, 0.41, 0.42, 0.45. For values of C less than 0.4, the final state tends towards zero (black) as A+B+C is less than 1.0. For C=0.4, the final state is the average of all of the initial states, mostly. For C greater than 0.4, the CA never reaches an equilibrium. Instead, for A+B+C slightly greater than one, cells quickly reach the average value of their neighbors. However, the average state of cells keeps increasing, and eventually a cell "overflows" and gets set back to zero (since the math is done mod 1). In some cases, large amounts of neighboring cells all overflow simultaneously; however, in some cases, they do not overflow, causing the streaked appearance. This CA has been called a "dripping rail" analogous to water dripping off a horizonal rail; every once in a while the water piles enough to release a drop, in some cases releasing a local "avalanche" of drops.

Here is another sequence of pictures with D=0, B=0.2. In these pictures A = C = 0.50, 0.55, 0.60, 0.70. In these cases, A+B+C is larger than one, and larger than the above pictures. As A and C grow larger, the final state gets more and more disorganized (even more than shown).