From equilibrium spin models to probabilistic cellular automata
Tóm tắt
The general equivalence betweenD-dimensional probabilistic cellular automata (PCA) and (D+1)-dimensional equilibrium spin models satisfying a “disorder condition” is first described in a pedagogical way and then used to analyze the phase diagrams, the critical behavior, and the universality classes of some automata. Diagrammatic representations of time-dependent correlation functions of PCA are introduced. Two important classes of PCA are singled out for which these correlation functions simplify: (1) “Quasi-Hamiltonian” automata, which have a current-carrying steady state, and for which some correlation functions are those of aD-dimensional static model. PCA satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes. (2) “Linear” (and more generally “affine”) PCA for which the diagrammatics reduces to a random walk problem closely related to (D+1)-dimensional directed SAWs: both problems display a critical behavior with mean-field exponents in any dimension. The correlation length and effective velocity of propagation of excitations can be calculated for affine PCA, as is shown on an explicitD=1 example. We conclude with some remarks on nonlinear PCA, for which the diagrammatics is related to reaction-diffusion processes, and which belong in some cases to the universality class of Reggeon field theory.
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Tài liệu tham khảo
W. Kinzel,Z. Phys. B 58:29 (1985).
S. Wolfram,Theory and Applications of Cellular Automata (World Scientific, Singapore, 1986).
S. Wolfram,Rev. Mod. Phys. 55:601 (1983);Physica 10D:1 (1984).
J. Stephenson,J. Math. Phys. 11:420 (1970);Phys. Rev. B 1:4405 (1970);Can. J. Phys. 47:2621 (1969).
T. R. Welberry and G. H. Miller,Acta Cryst. A 34:120 (1978).
T. R. Welberry and R. Galbraith,J. Appl. Cryst. 6:87 (1973);8:636 (1975); T. R. Welberry,J. Appl. Cryst. 10:344 (1977).
I. G. Enting,J. Phys. C, Solid State Phys. 10:1379 (1977).
P. Rujan,J. Stat. Phys. 29:231, 247 (1982);34:615 (1984).
C. Bennet and G. Grinstein,Phys. Rev. Lett. 55:657 (1985).
E. Domany and W. Kinzel,Phys. Rev. Lett. 53:311 (1984).
E. Domany,Phys. Rev. Lett. 52:871 (1984).
P. Rujan, Cellular automata and statistical mechanical models, IFK der KFA Jülich,J. Stat. Phys. 49:139 (1987).
E. Domany and J. Gubernatis,Phys. Rev. B 32:3354 (1985).
M. T. Jaekel and J. M. Maillard,J. Phys. A Math. Gen. 18:1229 (1985).
A. M. W. Verhagen,J. Stat. Phys. 15:219 (1976).
N. C. Chao and F. Y. Wu,J. Phys. A Math. Gen. 18:L603 (1985).
I. G. Enting,J. Phys. C Solid State Phys. 6:3457 (1973); T. Garel and J. M. Maillard,J. Phys. C 19:L505 (1986).
J. M. Maillard, Lecture, Meeting on Random Schrödinger Equations, University of Arizona (February 1987); P. Rujan, inNon-linear Equations in Classical and Quantum Field Theory, N. Sanchez, ed. (Springer-Verlag, 1985), p. 286.
P. Bak,Rep. Prog. Phys. 45:587 (1982); I. Peschel and V. J. Emery,Z. Phys. B 43:241 (1981).
F. Y. Wu,Rev. Mod. Phys. 54:235 (1982).
M. T. Jaekel and J. M. Maillard,J. Phys. A Math. Gen. 17:2079 (1984);18:2271 (1985).
M. Y. Choi and B. A. Huberman,J. Phys. A Math. Gen. 17:L765 (1984).
I. Peschel and T. T. Truong,J. Stat. Phys. 45:233 (1986).
M. T. Jaekel and J. M. Maillard,J. Phys. A Math. Gen. 18:641 (1985).
A. Georges, D. Hansel, P. Le Doussal, and J. M. Maillard,J. Phys. A Math. Gen. 20:5299 (1987).
G. Grinstein, C. Jayaprakash, and Y. He,Phys. Rev. Lett. 55:2527 (1985).
I. G. Enting,J. Phys. A Math. Gen. 11:2001 (1978).
D. Dhar and J. M. Maillard,J. Phys. A Math. Gen. 18:L383 (1985).
A. Georges, D. Hansel, P. Le Doussal, and J. M. Maillard,J. Phys. A Math. Gen. 19:1001 (1986).
A. Georges, D. Hansel, P. Le Doussal, and J. M. Maillard,J. Phys. A Math. Gen. 19:L329 (1986).
W. Kinzel, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Israel Physical Society, 1983).
B. K. Chakrabarti and S. S. Manna,J. Phys. A Math. Gen. 16:L113 (1983).
J. Cardy,J. Phys. A Math. Gen. 16:L355 (1983).
P. C. Hohenberg and B. I. Halperin,Rev. Mod. Phys. 49:435 (1977).
S. P. Obukhov,Physica 101A:145 (1980).
J. L. Cardy and R. L. Sugar,J. Phys. A Math. Gen. 13:L423 (1980).
E. Domany and W. Kinzel,Phys. Rev. Lett. 47:5 (1981).
S. W. J. Blöte and H. J. Hilhorst,J. Phys. A Math. Gen. 15:L631 (1982).
S. M. Bhattacharjee,Phys. Rev. Lett. 53:1161 (1984).
E. W. Montroll, inApplied Combinatorial Mathematics, E. F. Beckenbach, ed. (Wiley, New York, 1964).
S. M. Bhattacharjee, J. F. Nagle, D. A. Huse, and M. E. Fisher,J. Stat. Phys. 32:361 (1983).
J. des Cloiseaux,J. Phys. 36:281 (1975).
W. Feller,An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971), Vol. 1, p. 352.
A. Aharony, E. Domany, R. M. Hornreich, T. Schneider, and M. Zannetti,Phys. Rev. B 32:3358 (1985).
G. Vichniac,Physica 10D:96 (1984).
D. S. Zimmerman, C. Kalln, and A. J. Berlinsky,Phys. Rev. Lett. 58:1881 (1987).
R. M. Hornreich, M. Luban, and S. Shtrikman,Phys. Rev. Lett. 35:1678 (1975); D. Auerbach, Master Thesis, Weizmann Institute (1986).
G. Parisi and N. Sourlas,Nucl. Phys. B 206:321 (1982).
J. Vinals and J. D. Gunton,J. Phys. A Math. Gen. 19:L933 (1986).
H. Abarbanel, J. Bronzan, and A. Whote,Phys. Rep. 21:119 (1975); M. Moshe,Phys. Rep. C 37:255 (1978).
R. C. Brower, M. A. Furman, and H. Moshe,Phys. Lett. 76B:213 (1978).
W. Klein and W. Kinzel,J. Phys. A Math. Gen. 14:L405 (1981).
F. Schlögl,Z. Phys. 253:147 (1972).
P. Grassberger and K. Sundermeyer,Phys. Lett. 77B:220 (1978).
P. Grassberger and A. De La Torre,Ann. Phys. 122:373 (1979).
R. C. Brower, M. A. Furman, and K. Subbarao,Phys. Rev. D 15:1756 (1977).
J. L. Cardy and R. L. Sugar,Phys. Rev. D 12:2514 (1975).
D. Hansel and J. M. Maillard, Preprint LPTHE 87/29, and references therein.
G. Vichniac, P. Tamayo, and H. Hartman,J. Stat. Phys. 45:875 (1986).
B. Chopard and M. Droz, University of Geneva Preprint (1987).
Z. Racz,Phys. Rev. Lett. 55:1707 (1985); J. Marro, J. L. Lebowitz, H. Spohn, and M. H. Kallos,J. Stat. Phys. 38:725 (1985).
S. Katz, J. M. Lebowitz, and H. Spohn,J. Stat. Phys. 34:497 (1984); J. S. Wang and J. L. Lebowitz, Phase transitions and universality in nonequilibrium steady states of stochastic Ising models, Preprint.
Liggett,Interacting Particle Systems.
T. Schneider, M. Zannetti, and R. Badii,Phys. Rev. B 31:2941 (1985); T. Schneider and M. Schwatz,Phys. Rev. B 31:7484 (1985).