Symmetries and Dynamics of Discrete Systems

  • Vladimir V. Kornyak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4770)


We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program constructs and investigates phase portraits of discrete dynamical systems modulo groups of their symmetries, searches dynamical systems possessing specific properties, e.g.,reversibility, computes microcanonical partition functions and searches phase transitions in mesoscopic systems. Some computational results and observations are presented. In particular, we explain formation of moving soliton-like structures similar to “spaceships” in cellular automata.


Cellular Automaton Phase Portrait Ising Model Discrete System Discrete Dynamical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kornyak, V.V.: On Compatibility of Discrete Relations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 272–284. Springer, Heidelberg (2005), CrossRefGoogle Scholar
  2. 2.
    Kornyak, V.V.: Discrete Relations On Abstract Simplicial Complexes. Programming and Computer Software 32(2), 84–89 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kornyak, V.V.: Cellular Automata with Symmetric Local Rules. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 240–250. Springer, Heidelberg (2006), CrossRefGoogle Scholar
  4. 4.
    McKay, B.D.: Practical Graph Isomporphism. Congressus Numerantium 30, 45–87 (1981), MathSciNetGoogle Scholar
  5. 5.
    Gardner, M.: On Cellular Automata Self-reproduction, the Garden of Eden and the Game of Life. Sci. Am. 224, 112–117 (1971)Google Scholar
  6. 6.
    Hooft, G.: Quantum Gravity as a Dissipative Deterministic System. SPIN-1999/07, gr-qc/9903084; Class. Quant. Grav. 16, 3263 (1999); Also published in: Fundamental Interactions: from symmetries to black holes (Conference held on the occasion of the “Eméritat” of François Englert, 24-27 March 1999, Frère, J.-M., et al. (ed.) by Univ. Libre de Bruxelles, Belgium, pp. 221–240 (1999)Google Scholar
  7. 7.
    ’t Hooft, G.: The mathematical basis for deterministic quantum mechanics. ITP-UU-06/14, SPIN-06/12, quant-ph/0604008, pp. 1–17 (2006)Google Scholar
  8. 8.
    Imry, Y.: Introduction to Mesoscopic Physics (Mesoscopic Physics and Nanotechnology, 2), p. 256. Oxford University Press, USA (2002)Google Scholar
  9. 9.
    Gross, D.H.E.: Microcanonical thermodynamics: Phase transitions in “Small” Systems, p. 269. World Scientific, Singapore (2001)Google Scholar
  10. 10.
    Gross, D.H.E.: A New Thermodynamics from Nuclei to Stars. Entropy 6, 158–179 (2004)Google Scholar
  11. 11.
    Gross, D.H.E., Votyakov, E.V.: Phase Transitions in “Small” Systems. Eur. Phys. J. B 15, 115–126 (2000)Google Scholar
  12. 12.
    Ispolatov, I., Cohen, E.G.D.: On First-order Phase Transitions in Microcanonical and Canonical Non-extensive Systems. Physica A 295, 475–487 (2001)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Vladimir V. Kornyak
    • 1
  1. 1.Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 DubnaRussia

Personalised recommendations