Symmetries and Dynamics of Discrete Systems
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.
KeywordsCellular Automaton Phase Portrait Ising Model Discrete System Discrete Dynamical System
Unable to display preview. Download preview PDF.
- 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.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.’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.Imry, Y.: Introduction to Mesoscopic Physics (Mesoscopic Physics and Nanotechnology, 2), p. 256. Oxford University Press, USA (2002)Google Scholar
- 9.Gross, D.H.E.: Microcanonical thermodynamics: Phase transitions in “Small” Systems, p. 269. World Scientific, Singapore (2001)Google Scholar
- 10.Gross, D.H.E.: A New Thermodynamics from Nuclei to Stars. Entropy 6, 158–179 (2004)Google Scholar
- 11.Gross, D.H.E., Votyakov, E.V.: Phase Transitions in “Small” Systems. Eur. Phys. J. B 15, 115–126 (2000)Google Scholar