Local structure approximation as a predictor of second-order phase transitions in asynchronous cellular automata

Abstract

The mathematical analysis of the second-order phase transitions that occur in α-asynchronous cellular automata field is a highly challenging task. From the experimental side, these phenomena appear as a qualitative change of behaviour which separates a behaviour with an active phase, where the system evolves in a stationary state with fluctuations, from a passive state, where the system is absorbed in a homogeneous fixed state. The transition between the two phases is abrupt: we ask how to analyse this change and how to predict the critical value of the synchrony rate α. We show that an extension of the mean-field approximation, called the local structure theory, can be used to predict the existence of second-order phase transitions belonging to the directed percolation university class. The change of behaviour is related to the existence of a transcritical bifurcation in the local structure maps. We show that for a proper setting of the approximation, the form of the transition is predicted correctly and, more importantly, an increase in the level of local structure approximation allows one to gain precision on the value of the critical synchrony rate which separates the two phases.

Appendix

Definitions of polynomials d _x , d _y , d _z and d _v for local structure equations for rule 6:

$$\begin{aligned} d_x&= ( x+y ) ( v+y )\left( 3 {v}^{2}+vx-zv-v+8 yv+yx-y\right.\\&\quad\left. -\,2 zy+5 {y}^{2} \right) {\alpha }^{3}- ( v+y-z) \left( 3 {v}^{2}x+3 y{v}^{2}-vxz +3 {y}^{2}v\right. \\&\quad\left. -\,zvy+4 yvx+x{y}^{2}\right) {\alpha }^{2}- ( v+y ) y \left( 2 yv+vx+2 {y}^{2}\right. \\&\quad\left. -\,2 z y+yx-2 xz \right) \alpha -x (v+y)^{2} ( x+y ) \\ d_y&= ( x+y ) ( v+y ) \left( 3 {v}^{2}+vx-zv\right. -v+8 yv+yx-y-2 zy\\&\left.\quad +\,5{y}^{2} \right) {\alpha }^{3}+ \left( 6 zvxy-17 vx{y}^{2} \right. -16 {v}^{2}xy+2 yvx+4 {v}^{2}xz\\&\quad+\,4z{v}^{2}y+2 zx{y}^{2}-2 v{x}^{2}y+5 {y}^{2}zv-{z}^{2}vx-yv{z}^{2} -13 {y}^{3}v\\&\quad-\,14 {v}^{2}{y}^{2}-6 x{y}^{3}-{v}^{2}{x}^{2}+{y}^{3}-{x}^{2}{y}^{2}-4 {y}^{4}-5 y{v}^{3 }+y{v}^{2}\\&\quad+\,{v}^{2}x+{y}^{3}z+2 {y}^{2}v+x{y}^{2}\left. -5 {v}^{3}x \right) {\alpha }^{2}+ ( v+y )\left( 2 y{v}^{2}\right. \\&\quad+\,2 {v}^{2}x+2 yvx-2 vxz-2zvy\left. +{y}^{2}v-{y}^{3} \right) \alpha +y ( v+y )^{2} ( x+y ) \\ d_z&= ( 3 {v}^{2}+vx-zv-v+8 yv+yx-y-2 zy+5 {y}^{2} ){\alpha }^{3}- ( v+y )\\&\quad\;\; ( 5 v-3 z+x-1+6 y ) {\alpha }^{2}+ ( 2 {y}^{2}+2 {v}^{2}\\&\quad-\,3 zy-2 zv+4 yv ) \alpha +z ( v+y ) \\ d_v&= ( x+y ) ( v+y ) \left( 3 {v}^{2}+vx-zv\right. -v+8 yv+yx-y-2 zy\\&\left.\quad +\,5 {y}^{2} \right) {\alpha }^{3}+ \left( 7 zvxy-29 vx{y}^{2} \right. -25 {v}^{2}xy+4 yvx+4 {v}^{2}xz\\&\quad+\,4z{v}^{2}y+3 zx{y}^{2}-4 v{x}^{ 2}y+6 {y}^{2}zv-{z}^{2}vx-yv{z}^{2} \\&\quad-\,23 {y}^{3}v-22 {v}^{2}{y}^{2}- 11 x{y}^{3}-2 {v}^{2}{x}^{2}+2 {y}^{3}-2 {x}^{2}{y}^{2}-8 {y}^{4}\\&\quad-\,7 y{v}^{3}+2 y{v}^{2}+2 {v}^{2}x+2 {y}^{3}z\left. +4{y}^{2}v+2 x{y}^{2}-7 {v}^{3}x \right) {\alpha }^{2}\\&\quad+\,( v+y ) ( 5 y{v}^{2} +5 {v}^{2}x+v{x}^{2}-3 zvy+10 yvx\\&\quad+\,8{y}^{2}v-vx-yv-3 vxz-{y}^{2}- {y}^{2}z+3 {y}^{3}+5 x{y}^{2}\\&\quad+\,{x}^{2}y-zxy-yx )\alpha-v ( v+y )^{2} ( x+y ) \end{aligned}$$

Definitions of A ₁ and A ₂ for Eq. (30):

$$\begin{aligned} A_1&= (2 {\alpha }-3) (32 {\alpha }^{14}+1776 {\alpha }^{13}-32304 {\alpha }^{12} +248136 {\alpha }^{11}-1156158 {\alpha }^{10} +3746559 {\alpha }^9-9102790 {\alpha }^8+17374596 {\alpha }^7\\&\quad -26738472 {\alpha }^6 +33372704 {\alpha }^5-33402048 {\alpha }^4+26068992 {\alpha }^3-14802944 {\alpha }^2+5259264 {\alpha }-884736)\\&\quad +12 \sqrt{3\left( 8 {\alpha }^9+56 {\alpha }^8-426 {\alpha }^7+940 {\alpha }^6-821 {\alpha }^5-588 {\alpha }^4+2656 {\alpha }^3-3220 {\alpha }^2+1824 {\alpha }-400\right) }\\&\quad \;\left( 4 {\alpha }^4-20 {\alpha }^3+45 {\alpha }^2-16 {\alpha }-32\right) \left( 4 {\alpha }^6-32 {\alpha }^5+109 {\alpha }^4-214 {\alpha }^3+284 {\alpha }^2-256 {\alpha }+128\right) \\ A_2&=16 {\alpha }^{10}-576 {\alpha }^9+5048 {\alpha }^8-22816 {\alpha }^7+65969 {\alpha }^6-134476 {\alpha }^5\\&\quad +199844 {\alpha }^4-213632 {\alpha }^3+161536 {\alpha }^2-82944 {\alpha }+21504 \end{aligned}$$