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This page presents numerical results related to the article Asynchronism Induces Second Order Phase Transitions in Elementary Cellular Automata

Summary of the results

Tcode ECA alpha_c delta Tmin Tmax
BFGH 6 0.2825 (1) 0.160 (4) 1* 10 3 2.5 * 10 5
BCEFGH 18 0.71385 (5) 0.157 (3) 5* 10 1 2 * 10 4
BCEGH 26 0.47485 (5) 0.159 (1) 1* 10 3 2 * 10 5
BDFG 38 0.04085 (5) 0.159 (2) 2* 10 4 3.5 * 10 5
BCDEFGH 50 0.6282 (1) 0.159 (8) 1* 10 3 1 * 10 5
BCDEGH 58 0.3398 (1) 0.161 (4) 2* 10 4 2 * 10 6
BDEH 106 0.8146 (1) 0.157 (2) 5* 10 1 1 * 10 5
BFG 134 0.0821 (2) 0.162 (6) 2* 10 4 2 * 10 6
BCEFG 146 0.67505 (5) 0.158 (3) 5* 10 1 1 * 10 5
BCDEFGH 178 0.410 (1) 0.286 (4) 1* 10 2 2 * 10 5

Detailed Numerical Results for the Delta exponent(text file)
The table shows for each ECA, for different values of alpha, for different time intervals :

  • the measure of Delta (DD)
  • the relative difference (in %) with Delta_DP (Drel)
  • the root mean square error (last column, dimensionless units).

Detailed Numerical Results for the Beta exponent (text file)
The table shows for each ECA, for different values of alpha_C, for different delta_alpha intervals :

  • the measure of Beta
  • the root mean square error (RMS)

Plots

For the nine DP-ECA, the following plots present the three experiments descirbed in the article:
  • Stationary density as a function of synchrony rate: d_sta=f(alpha)
  • Evolution of the density with time near the critical point: d =f(t) in log-log scale
  • Divergence of asymptotic density near teh critical point: d_sta=f(|alpha-alpha_c|) in log-log scale
Ring sizes: First column n = 10 000, second column n = 40 000, he third column for n= 20 000. Click on images to display them in full size.
Tcode ECA d_sta=f(alpha) d=f(t) d_sta=f(|alpha-alpha_c|)
BFGH 6 _CLICKIMG2_(../Robustness/DensityExp/ECA6DensAlpha-small.png,../Robustness/DensityExp/ECA6DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA6.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA6.png)
BCEFGH 18 _CLICKIMG2_(../Robustness/DensityExp/ECA18DensAlpha-small.png,../Robustness/DensityExp/ECA18DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA18.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA18.png)
BCEGH 26 _CLICKIMG2_(../Robustness/DensityExp/ECA26DensAlpha-small.png,../Robustness/DensityExp/ECA26DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA26.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA26.png)
BDFG 38 _CLICKIMG2_(../Robustness/DensityExp/ECA38DensAlpha-small.png,../Robustness/DensityExp/ECA38DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA38.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA38.png)
BCDEFGH 50 _CLICKIMG2_(../Robustness/DensityExp/ECA50DensAlpha-small.png,../Robustness/DensityExp/ECA50DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA50.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA50.png)
BCDEGH 58 _CLICKIMG2_(../Robustness/DensityExp/ECA58DensAlpha-small.png,../Robustness/DensityExp/ECA58DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA58.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA58.png)
BDEH 106 _CLICKIMG2_(../Robustness/DensityExp/ECA106DensAlpha-small.png,../Robustness/DensityExp/ECA106DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA106.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA106.png)
BFG 134 _CLICKIMG2_(../Robustness/DensityExp/ECA134DensAlpha-small.png,../Robustness/DensityExp/ECA134DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA134.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA134.png)
BCEFG 146 _CLICKIMG2_(../Robustness/DensityExp/ECA146DensAlpha-small.png,../Robustness/DensityExp/ECA146DensAlpha.png ) _CLICKIMG_W_(200,ExpF4/DensTimeECA146.png) _CLICKIMG_W_(200,BetaF2/BetaFitECA146.png)

The case of ECA 178

The first and second plots were obtained for n = 10 000, the thord plot for n = 80 000.
d_as=f(alpha) d-kinks=f(alpha) d-kinks=f(t)
_CLICKIMG2_(../Robustness/DensityExp/ECA178DensAlpha-small.png,../Robustness/DensityExp/ECA178DensAlpha.png ) _CLICKIMG2_(../Robustness/KinksExp/ECA178KinksAlpha-small.png,../Robustness/KinksExp/ECA178KinksAlpha.png ) _CLICKIMG_W_(200,DP2exp/DensTimeECA178.png)

Last Update : December 2008