Phase transitions near the "game of Life"

Nordfalk, Jacob; Alstrøm, Preben

doi:10.1103/physreve.54.r1025

Cited by 21 publications

(10 citation statements)

References 9 publications

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“…1 . This density i s b e t w een the previous estimates of three per cent 1 { 4 and 2.6 per cent [5] and agrees excellently with the 2.85 0.05 per cent of Bagnoli et al 11 Indeed Fig. 2 shows very small nite-size eects; from thousand lattices with L = 6 4 w e found a density 0.0294 (our errors are of the order of 10 4 ).…”

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confidence: 82%

Search for Asymptotic Death in Game of Life

Gibbs

Stauffer

1997

Int. J. Mod. Phys. C

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Simulations with more than 4 10 10 sites conrm that the density of active sites in Game of Life cellular automata approaches a positive limit, of about 2.87 per cent.Conway's \Game of Life" cellular automata have been discussed for some time in connection with self-organized criticality. 1 { 5 F or this purpose one introduces a perturbation to a stationary distribution and investigates the avalanche produced by it. These simulations were made for rather small lattice sizes, typically 10 3 10 3 , and gave stationary densities of only about three per cent for the fraction of active sites.Such concentrations of about three per cent h a v e been observed in several bootstrap percolation models 6 { 9 as threshold values but turned out to be artifacts of the nite system size L of the L L square lattice: the threshold concentration varies, for example, as 1/log L when a site remains empty if and only if at least three of its four nearest neighbors are empty (otherwise it is lled permanently). If similar eects would happen in the more complicated Game of Life, the whole question of critical behavior would be obsolete since in an empty lattice no avalanches can be produced. There is also evidence for extreme nite-size eects in critical phenomena at similar transitions, 10 where even L = 20032 turned out too small. Thus we present here simulations of much larger lattices than before 1 { 5 to check for the size-dependence of the equilibrium density. Since Ref. 11 checked the extent t o which the nal density depends on the initial density, w e always start with half of the sites occupied. 601 Int. J. Mod. Phys. C 1997.08:601-604. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/20/15. For personal use only.

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confidence: 82%

Search for Asymptotic Death in Game of Life

Gibbs

Stauffer

1997

Int. J. Mod. Phys. C

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“…The frequency-area distribution of avalanches is a power law and satisfies (1.1) with α ≈ 1.4. A number of articles discuss in detail the relationship of the Game of Life to self-organized criticality, including , Bak (1992), Garcia et al (1993), Sales (1993), Alstrom and Leao (1994), Nordfalk and Alstrom (1996), Blok and Bergersen (1997), Newman et al (1997) and Ninagawa et al (1998). Bak and Sneppen (1993) introduced a cellular-automata model for evolution.…”

Section: The Game Of Lifementioning

confidence: 99%

Self-organized criticality

1999

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The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box accumulates four particles they are redistributed to the four adjacent boxes or lost off the edge of the grid. Redistributions can lead to further instabilities with the possibility of more particles being lost from the grid, contributing to the size of each 'avalanche'. These model 'avalanches' satisfied a power-law frequency-area distribution with a slope near unity. Other cellular-automata models, including the slider-block and forest-fire models, are also said to exhibit self-organized critical behaviour. It has been argued that earthquakes, landslides, forest fires, and species extinctions are examples of self-organized criticality in nature. In addition, wars and stock market crashes have been associated with this behaviour. The forest-fire model is particularly interesting in terms of its relation to the critical-point behaviour of the sitepercolation model. In the basic forest-fire model, trees are randomly planted on a grid of points. Periodically in time, sparks are randomly dropped on the grid. If a spark drops on a tree, that tree and adjacent trees burn in a model fire. The fires are the 'avalanches' and they are found to satisfy power-law frequency-area distributions with slopes near unity. This forest-fire model is closely related to the site-percolation model, that exhibits critical behaviour. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi steady-state cascade gives a power-law frequency-area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees.

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“…Asymptotic density and time evolution in GOL have been analyzed in detail using earlier theoretical approaches [9,10]. Generalizations of GOL include deterministic systems with variations in the rules [11] or expansion in the neighborhood [12] and stochastic systems involving probabilistic evolution [13,14] or asynchronous updating [15]. These generalizations are accompanied by additional tunable parameters, and they lead to the conclusion that GOL is a subcritical system dominated by quiescent states, but the same framework is still associated with critical transitions in its dynamics.…”

Section: Introductionmentioning

confidence: 99%

Deterministic phase transitions and self-organization in logistic cellular automata

2019

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We present a simple extension in which a single parameter tunes the dynamics of cellular automata (CA) by consequently expanding their discrete state space into a Cantor set. Such an implementation serves as a potent platform for further investigation of several emergent phenomena, including deterministic phase transitions, pattern formation, autocatalysis, and self-organization. We first apply this approach to Conway's Game of Life and observe sudden changes in the asymptotic dynamics of the system accompanied by the emergence of complex propagators. Incorporation of the new state space with system features is used to explain the transitions and formulate the tuning parameter range where the propagators adaptively survive by investigating their autocatalytic local interactions. Similar behavior is present when the same recipe is applied to Rule 90, an outer totalistic elementary one-dimensional cellular automaton. In addition, the latter case shows that deterministic transitions between classes of CA can be achieved by tuning a single parameter continuously.

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Phase transitions near the "game of Life"

Cited by 21 publications

References 9 publications

Search for Asymptotic Death in Game of Life

Search for Asymptotic Death in Game of Life

Self-organized criticality

Deterministic phase transitions and self-organization in logistic cellular automata

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