Communications in Mathematical Physics

, Volume 272, Issue 1, pp 53–74 | Cite as

Continuity of Information Transport in Surjective Cellular Automata

  • Torbjørn Helvik
  • Kristian Lindgren
  • Mats G. Nordahl


We introduce a local version of the Shannon entropy in order to describe information transport in spatially extended dynamical systems, and to explore to what extent information can be viewed as a local quantity. Using an appropriately defined information current, this quantity is shown to obey a local conservation law in the case of one-dimensional reversible cellular automata with arbitrary initial measures. The result is also shown to apply to one-dimensional surjective cellular automata in the case of shift-invariant measures. Bounds on the information flow are also shown.


Local Information Cellular Automaton Cellular Automaton Shannon Entropy Information Current 
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.
    Bennett C.H. (1973). Logical reversibility of computation. IBM J. Res. Develop. 17(6): 525–532 zbMATHMathSciNetGoogle Scholar
  2. 2.
    Dab D., Lawniczak A., Boon J.P., Kapral R. (1990). Cellular-automaton model for reactive systems. Phys. Rev. Lett. 64: 2462–2465 CrossRefADSGoogle Scholar
  3. 3.
    Ferrari P., Maass A., Martínez S., Ney P. (2000). Cesàro mean distribution of group automata starting from measures with summable decay. Ergodic Theory Dynam. Systems. 20(6): 1657–1670 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Frisch U., Hasslacher B., Pomeau Y. (1986). Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett. 56: 1505–1508 CrossRefADSGoogle Scholar
  5. 5.
    Gänssler, P., Stute, W. (1977). Wahrscheinlichkeitstheorie. Springer Verlag, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  6. 6.
    Gray R.M. (1988). Probability, random processes and ergodic properties. Springer-Verlag, New York zbMATHGoogle Scholar
  7. 7.
    Hardy J., Pomeau Y., de Pazzis O. (1973). Time evolution of two-dimensional model system. I. invariant states and time correlation functions. J. Math. Phys. 14: 1746–1759 CrossRefADSGoogle Scholar
  8. 8.
    Hedlund G.A. (1969). Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory. 3: 320–375 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Host B., Maass A., Martínez S. (2003). Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6): 1423–1446 zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ito M., Osato N., Nasu M. (1983). Linear cellular automata over Zm. J. Comput. System Sci. 27(2): 125–140 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jaynes E.T. (1957). Information theory and statistical mechanics. Phys. Rev. 106: 620–630 CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Keller G. (1998). Equilibrium states in ergodic theory, Volume 42 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge Google Scholar
  13. 13.
    Kullback S., Leibler R.A. (1951). On information and sufficiency. Ann. Math. Stat. 22: 79–86 MathSciNetzbMATHGoogle Scholar
  14. 14.
    Landauer R. (1961). Irreversibility and heat generation in the computing process. IBM J. Res. Develop. 5(3): 183–191 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lind D.A. (1984). Applications of ergodic theory and sofic systems to cellular automata. Physica D. 10: 36–44 CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Lindgren K. (1987). Correlations and random information in cellular automata. Complex Systems. 1: 529–543 zbMATHMathSciNetGoogle Scholar
  17. 17.
    Lindgren K. (1988). Microscopic and macroscopic entropy. Phys. Rev. A. 38: 4794–4798 CrossRefADSGoogle Scholar
  18. 18.
    Pivato M., Yassawi R. (2002). Limit measures for affine cellular automata. Ergodic Theory Dynam. Systems. 22(4): 1269–1287 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pivato M., Yassawi R. (2004). Limit measures for affine cellular automata II. Ergodic Theory Dynam. Systems. 24(6): 1961–1980 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Richardson D. (1972). Tesselations with local transformations. J. Comput. System Sci. 5: 373–388 Google Scholar
  21. 21.
    Takesue S. (1987). Reversible cellular automata and statistical mechanics. Phys. Rev. Lett. 59: 2499–2502 CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Takesue S. (1990). Fourier’s law and the Green-Kubo formula in a cellular-automaton model. Phys. Rev. Lett. 64: 252–255 zbMATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Toffoli T. (1988). Information transport obeying the continuity equation. IBM J. Res. Develop. 32(1): 29–36 MathSciNetCrossRefGoogle Scholar
  24. 24.
    Toffoli T., Margolus N.H. (1990). Invertible cellular automata: a review. Physica D. 45(1–3): 229–253 zbMATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Vichniac G. (1984). Simulating physics with cellular automata. Physica D. 10: 96–115 CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Walters P. (1982). An Introduction to Ergodic Theory. Number 79 in Graduate Texts in Mathematics. Springer, Berlin Heidelberg New York Google Scholar
  27. 27.
    Wheeler J.A. (1989). Information, physics, quantum: The search for links. In: Zurek, WH (eds) Complexity, Entropy and the Physics of Information. Addison-Wesley, Redwood City, CAGoogle Scholar
  28. 28.
    Zurek W.H. (1989). Algorithmic randomness and physical entropy. Phys. Rev. A. 40: 4731–4751 CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Torbjørn Helvik
    • 1
  • Kristian Lindgren
    • 2
  • Mats G. Nordahl
    • 3
  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of Physical Resource TheoryChalmers University of Technology and Göteborg UniversityGöteborgSweden
  3. 3.Department of Applied Information TechnologyChalmers University of Technology and Göteborg UniversityGöteborgSweden

Personalised recommendations