A103 ライフゲームの過渡状態におけるカスケード過程(形態形成関連)

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The "Game of Life" is one of the two-dimensional cellular automata. It has significant characteristics such as universal computation, self-organized criticality, and 1/f fluctuation and these are caused by a variety of different initial configurations. The implementation of a universal computer in the Game of Life needs carefully adjusted initial configurations. Self-organized criticality emerges at stable configurations and 1/f fluctuation is observed in the evolution from random initial configurations. The array size needed for the universal computation is supposed to be enormous and it has not been implemented yet. The connection between array size and transient behavior caused by perturbing stable configurations has been made clear by the study of self-organized criticality. But the dependence of the transient behavior from random initial configuration upon array size has not been investigated yet. In this paper, we study the transient behavior of the Game of Life from random initial configurations by investigating numerically the average transient time steps with various array size. After sufficiently long time steps, any evolutions in the Game of Life with finite array size must enter periodic configurations. The structures which remain after sufficiently long time steps are almost always the ones with time period two and three. So mostly the period of the stable configurations is six, although higher period structures are exceptionally generated. The array consists of N×N cells and null (absorbing) and periodic boundary conditions are used. Each array is started from random initial configurations in which each site takes state 0 or state 1 randomly with independent equal probabilities. N is varied from 100 to 4000 with increments of 100 (with increments of 500 when N is more than 2000) and for each value of N, the evolutions from 150 distinct random initial configurations are sampled. The simulations show that the average transient time steps <T> increase logarithmically with square array size N×N, <T>=ln(N) in null and periodic boundary conditions although it has been supposed that <T> increase algebraically within about N×N. This result means that the average transient time steps with array size MN×MN is the sum of the ones with array size M×M and with array size N×N. It also suggest that the transient behavior can be divided into the one in the smaller arrays and the other in the larger arrays. We measure the evolution of the distribution of cluster size in square array to verify this expectation. The results show that the larger clusters disappear quickly and the smaller ones remain long. Moreover we delete deliberately the smaller cluster in the middle of the evolution. As a result, while the larger clusters disappear, the smaller ones are created These results reveal that the transition from the larger clusters to the smaller ones occurs in the evolutions. This phenomenon is reminiscent of a cascade process of turbulence in viscous fluid which is the transfer of the kinetic energy of a flow from the large scale eddies to the smaller ones.
2001-11-14

A103 ライフゲームの過渡状態におけるカスケード過程(形態形成関連)

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