Comparison between simultaneous and sequential updating in 2n+1−1 cellular automata

Properties of cellular automata with Packard-Wolfram code 2n+1-1 have been studied without and with noise on square (n = 4) and triangular (n = 6) lattices. The outer totalistic rule used assign a central spin opposite to the majority spin of its neighbourhood and flips it in the case of equality. Starting from random configurations with a variable proportion r of up spins, different relaxation behaviours and equilibrium structures are obtained: a ferromagnetic (F) structure and an antiferromagnetic (AF) structure for simultaneous and sequential updatings respectively on a square lattice. When the spins are processed simultaneously in the presence of noise, a first-order transition F → AF is observed for a square lattice and a second-order transition for a triangular lattice. The results show similarities with Ising models (domains) but also strong differences (magnetization as a function of r).

[1]  D. Stauffer,et al.  Test of three-dimensional Q2R Ising algorithm , 1987 .

[2]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[3]  Hans J. Herrmann,et al.  Fast algorithm for the simulation of Ising models , 1986 .

[4]  J. Morniroli,et al.  Crystalline defects in M7C3 carbides , 1983 .

[5]  S. Wolfram.  Statistical mechanics of cellular automata , 1983 .

[6]  Herrmann,et al.  Dynamics of spreading phenomena in two-dimensional Ising models. , 1987, Physical review letters.

[7]  M. H. Kalos,et al.  A new multispin coding algorithm for Monte Carlo simulation of the Ising model , 1984 .

[8]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

[9]  Scott Kirkpatrick,et al.  A very fast shift-register sequence random number generatorjournal of computational physics , 1981 .

[10]  B. Derrida,et al.  Effect of thermal noise and initial conditions in the dynamics of a mean-field ferromagnet , 1988 .

[11]  Universality two-dimensional Kauffman model for parallel and sequential updating , 1988 .

[12]  B. Derrida,et al.  Finite size scaling study of dynamical phase transitions in two dimensional models : Ferromagnet, symmetric and non symmetric spin glasses , 1988 .

[13]  G. Bhanot.  The Metropolis algorithm , 1988 .

[14]  G. Vichniac.  Simulating physics with cellular automata , 1984 .

[15]  E. Bauer-Grosse,et al.  Etude des défauts de structure dans le carbure defer métastable “Fe7C3” formé lors de la cristallisation d'alliages amorphes fer-carbone , 1981 .

[16]  S. Wolfram,et al.  Two-dimensional cellular automata , 1985 .