×

zbMATH — the first resource for mathematics

Equivalence of cellular automata to Ising models and directed percolation. (English) Zbl 1174.82315
Summary: Time development of cellular automata in \(d\) dimensions is mapped onto equilibrium statistical mechanics of Ising models in \(d+1\) dimensions. Directed percolation is equivalent to a cellular automaton, and thus to an Ising model. For a particular case of directed percolation we find \(\nu_{\parallel}=2\), \(\nu_{\perp}=1\), \(\eta_{\perp}=0\).

MSC:
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C32 Neural nets applied to problems in time-dependent statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Wolfram, Rev. Mod. Phys. 55 pp 601– (1983) · Zbl 1174.82319 · doi:10.1103/RevModPhys.55.601
[2] S. Wolfram, Physica (Utrecht) 10D pp 1– (1984)
[3] J. von Neumann, in: Theory of Self Reproducing Automata (1966)
[4] P. Grassberger, Physica (Utrecht) 10D pp 52– (1984)
[5] E. Ott, Rev. Mod. Phys. 53 pp 655– (1981) · Zbl 1114.37303 · doi:10.1103/RevModPhys.53.655
[6] H. Haken, in: Chaos and Order in Nature (1981) · Zbl 0462.00013 · doi:10.1007/978-3-642-68304-6
[7] T. R. Welberry, J. Appl. Crystallogr. 6 pp 87– (1973) · doi:10.1107/S0021889873008216
[8] T. R. Welberry, Acta Crystallogr., Sec. A 34 pp 120– (1978) · doi:10.1107/S0567739478000212
[9] G. Nicolis, in: Self Organization in Nonequilibrium Systems (1977) · Zbl 0363.93005
[10] S. Ulam, Annu. Rev. Biochem. 255 (1974)
[11] B. A. Huberman, Phys. Rev. Lett. 52 pp 1048– (1984) · doi:10.1103/PhysRevLett.52.1048
[12] A. M. W. Verhagen, J. Stat. Phys. 15 pp 213– (1976)
[13] I. G. Enting, J. Phys. C 10 pp 1379– (1977) · doi:10.1088/0022-3719/10/9/011
[14] I. G. Enting, J. Phys. A 11 pp 555– (1978) · doi:10.1088/0305-4470/11/3/015
[15] W. Kinzel, in: Percolation Structures and Processes, Annals of the Israel Physical Society, Vol. 5 (1983)
[16] D. Dhar, in: Stochastic Processes, Formalism and Applications (1983)
[17] J. Stephenson, J. Math. Phys. 5 pp 1009– (1964) · doi:10.1063/1.1704202
[18] J. Stephenson, J. Math. Phys. 11 pp 413– (1970) · doi:10.1063/1.1665154
[19] J. Stephenson, Phys. Rev. B 1 pp 4405– (1970) · doi:10.1103/PhysRevB.1.4405
[20] W. Kinzel, J. Phys. A 14 pp L163– (1981) · doi:10.1088/0305-4470/14/5/012
[21] R. B. Griffiths, J. Math. Phys. 8 pp 484– (1967) · doi:10.1063/1.1705220
[22] J. G. Mauldon, in: Proceedings of the Fourth Berkeley Symposium on Mathmatics, Statistics and Probability (1961)
[23] W. Feller, in: An Introduction to Probability Theory and its Applications (1968) · Zbl 0155.23101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.