Abstract
Discrete dynamical systems and mesoscopic lattice models are considered from the standpoint of their symmetry groups. Universal specific features of deterministic dynamical system behavior associated with nontrivial symmetries of these systems are specified. Group nature of soliton-like moving structures of the “spaceship” type in cellular automata is revealed. Study of lattice models is also considerably simplified when their symmetry groups are taken into account. A program in C for group analysis of systems of both types is developed. The program, in particular, constructs and investigates phase portraits of discrete dynamical systems modulo symmetry group and seeks dynamical systems possessing special features, such as, for example, reversibility. For mesoscopic lattice models, the program computes microcanonical distributions and looks for phase transitions. Some computational results and observations are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kornyak, V.V., On Compatibility of Discrete Relations, Computer Algebra in Scientific Computing, Ganzha, V.G., Mayr, E.W., and Vorozhtsov, E.V., Eds., Berlin-Heidelberg: Springer, 2005, pp. 272–284, http://arXiv.org/abs/math-ph/0504048.
Kornyak, V.V., Discrete Relations on Abstract Simplicial Complexes, Programmirovanie, 2006, no. 2, pp. 32–39 [Programming Comput. Software (Engl. Transl.), 2006, vol. 32, no. 2, pp. 84–89].
Conway, J.H. and Sloane, N.J.A., Sphere Packings, Lattices, and Groups, New York: Springer, 1988. Translated under the title Upakovki sharov, reshetki I gruppy, Moscow: Mir, 1990.
Baxter, R.J., Exactly Solved Models in Statistical Mechanics, Academic, 1982. Translated under the title Tochno reshaemye modeli v statisticheskoi mekhanike, Moscow: Mir, 1985.
Harari, F., Graph Theory, Reading, Mass.: Addison-Wesley, 1969. Translated under the title Teoriya grafov, Moscow: Mir, 1973.
McKay, B.D., Practical Graph Isomorphism, Congressus Numerantium, 1981, vol. 30, pp. 45–87, http://cs.anu.edu.au/:_bdm/nauty/PGI.
Gardner, M., On Cellular Automata Self-Reproduction, the Garden of Eden and the Game of “Life,” Sci. Am., 1971, vol. 224, pp. 112–117.
Kornyak, V.V., Cellular Automata with Symmetric Local Rules, Computer Algebra in Scientific Computing, Ganzha, V.G., Mayr, E.W., and Vorozhtsov, E.V., Eds., Berlin-Heidelberg: Springer, 2006, pp. 240–250, http://arXiv.org/abs/math-ph/0605040.
Kornyak, V.V., Symmetric Cellular Automata, Programmirovanie, 2007, no. 2, pp. 41–49 [Programming Comput. Software (Engl. Transl.), 2007, vol. 33, no. 2, pp. 87–93].
’t Hooft, G., Quantum Gravity as a Dissipative Deterministic System, Class. Quant. Grav., 1999, vol. 16, p. 3263.
’t Hooft, G., The Mathematical Basis for Deterministic Quantum Mechanics, ITP-UU-06/14, SPIN-06/12, quantph/0604008, 2006, pp. 1–17.
Imry, Y., Introduction to Mesoscopic Physics (Mesoscopic Physics and Nanotechnology, 2), Oxford University Press, USA, 2002.
Gross, D.H.E., Microcanonical Thermodynamics: Phase Transitions in “Small” Systems, Singapore: World Scientific, 2001.
Gross, D.H.E., A New Thermodynamics from Nuclei to Stars, Entropy, 2004, vol. 6, pp. 158–179.
Gross, D.H.E. and Votyakov, E.V., Phase Transitions on “Small” Systems, European Physics J. B., 2000, vol. 15, pp. 115–126.
Ispolatov, I. and Cohen, E.G.D., On First-Order Phase Transition in Microcanonical and Canonical Non-extensive Systems, Physica A., 2001, vol. 295, pp. 475–487.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.V. Kornyak, 2008, published in Programmirovanie, 2008, Vol. 34, No. 2.
Rights and permissions
About this article
Cite this article
Kornyak, V.V. Discrete dynamical systems with symmetries: Computer analysis. Program Comput Soft 34, 84–94 (2008). https://doi.org/10.1134/S0361768808020059
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0361768808020059