Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Interaction-Based Computing in Physics

  • Franco BagnoliEmail author
Living reference work entry

Latest version View entry history

DOI: https://doi.org/10.1007/978-3-642-27737-5_291-6



The correlation between two variables is the difference between the joint probability that the two variables take some values and the product of the two probabilities (which is the joint probability of two uncorrelated variables), summed over all possible values. In an extended system, it is expected that the correlation among parts diminishes with their distance, typically in an exponential manner.

Critical phenomenon

A condition for which an extended system is correlated over extremely long distances.

Extended system

A system composed by many parts connected by a network of interactions that may be regular (lattice) or irregular (graph).

Graph, lattice, tree

A graph is set of nodes connected by links, oriented or not. If the graph is translationally invariant (it looks the same when changing nodes), it is called a (regular) lattice. A disordered lattice is a lattice with a fraction of removed links or nodes. An ordered set of nodes connected by links is called a...


Lyapunov Exponent Cellular Automaton Energy Landscape Extended System Couple Differential Equation 
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.
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Primary Literature

  1. Albert R, Barabasi AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. Bagnoli F, Rechtman R, Ruffo S (1991) Some facts of life. Physica A 171:249–264ADSMathSciNetCrossRefGoogle Scholar
  3. Bagnoli F (2000) Cellular automata. In: Bagnoli F, Ruffo S (eds) Dynamical modeling in biotechnologies. World Scientific, Singapore, p 1CrossRefGoogle Scholar
  4. Bagnoli F, Cecconi F (2001) Synchronization of non-chaotic dynamical systems. Phys Lett A 282(1–2):9–17ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. Bagnoli F, Rechtman R (1999) Synchronization and maximum Lyapunov exponents of cellular automata. Phys Rev E 59(2):R1307–R1310ADSCrossRefzbMATHGoogle Scholar
  6. Bagnoli F, Rechtman R (2009) Thermodynamic entropy and chaos in a discrete hydrodynamical system Phys Rev E 79:041115Google Scholar
  7. Bagnoli F, Rechtman R, Ruffo S (1992) Damage spreading and Lyapunov exponents in cellular automata. Phys Lett A 172:34ADSCrossRefzbMATHGoogle Scholar
  8. Bagnoli F, Boccara N, Rechtman R (2001) Nature of phase transitions in a probabilistic cellular automaton with two absorbing states. Phys Rev E 63(4):046116ADSCrossRefGoogle Scholar
  9. Bak P, Tang C, Weisenfeld K (1987) Self-organizing criticality: an explanation of 1/f noise. Phys Rev A 38:364–374ADSCrossRefGoogle Scholar
  10. Barkema GT, MacFarland T (1994) Parallel simulation of the ising model. Phys Rev E 50(2):1623–1628ADSCrossRefGoogle Scholar
  11. Berlekamp E, Conway J, Guy R (1982) What is life? Games in particular, vol 2. Academic, London. Chap. 25Google Scholar
  12. Binney J, Dowrick N, Fisher A, Newman MEJ (1993) The theory of critical phenomena. Oxford Science/Clarendon Press, OxfordzbMATHGoogle Scholar
  13. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424(4–5):175–308ADSMathSciNetCrossRefGoogle Scholar
  14. Broadbent S, Hammersley J (1957) Percolation processes I. Crystals and mazes. Proc Camb Philos Soc 53:629–641ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. Cailliau R (1995) A short history of the web. http://www.netvalley.com/archives/mirrors/robert_cailliau_speech.htm. Accessed 10 Apr 2017
  16. Car R, Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Phys Rev Lett 55(22):2471–2474ADSCrossRefGoogle Scholar
  17. Cecconi F, Livi R, Politi A (1998) Fuzzy transition region in a one-dimensional coupled-stable-map lattice. Phys Rev E 57(3):2703–2712ADSCrossRefGoogle Scholar
  18. Chopard B, Luthi P, Masselot A, Dupuis A (2002) Cellular automata and lattice Boltzmann techniques: an approach to model and simulate complex systems. Adv Complex Syst 5(2):103–246MathSciNetCrossRefzbMATHGoogle Scholar
  19. Crutchfield J, Kaneko K (1988) Are attractors relevant to turbulence? Phys Rev Lett 60(26):2715–2718ADSMathSciNetCrossRefGoogle Scholar
  20. Daxois T, Peyrard M, Ruffo S (2005) The Fermi-Pasta-Ulam ‘numerical experiment’: history and pedagogical perspectives. Eur J Phys 26:S3–S11MathSciNetCrossRefGoogle Scholar
  21. Domany E, Kinzel W (1984) Equivalence of cellular automata to Ising models and directed percolation. Phys Rev Lett 53(4):311–314ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. Dotsenko V (1994) An introduction to the theory of spin glasses and neural networks. World Scientific, SingaporezbMATHGoogle Scholar
  23. El Yacouby S, Chopard B, Bandini S (eds) (2006) Cellular automata, Lecture notes in computer science, vol 4173. Springer, BerlinGoogle Scholar
  24. European Grid Infrastructure. https://www.egi.eu/. Accessed 10 Apr 2017
  25. Fermi E, Pasta J, Ulam S (1955) Los alamos report la-1940. In: Segré E (ed) Collected papers of Enrico Fermi. University of Chicago Press, ChicagoGoogle Scholar
  26. Frisch U, Hasslacher B, Pomeau Y (1986) Lattice-gas automata for the navier-stokes equation. Phys Rev Lett 56(14):1505–1508ADSCrossRefGoogle Scholar
  27. Gardiner CW (1994) Handbook of stochastic methods for physics, chemistry, and the natural sciences, Springer series in synergetics, vol 13. Springer, BerlinzbMATHGoogle Scholar
  28. Georges A, le Doussal P (1989) From equilibrium spin models to probabilistic cellular automata. J Stat Phys 54(3–4):1011–1064ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. Hardy J, Pomeau Y, de Pazzis O (1973) Time evolution of a two-dimensional classical lattice system. Phys Rev Lett 31(5):276–279ADSCrossRefGoogle Scholar
  30. Harlow H, Metropolis N (1983) Computing & computers – weapons simulation leads to the computer era. Los Alamos Sci 4(7):132Google Scholar
  31. Haw M (2005) Einstein’s random walk. Phys World 18:19–22CrossRefGoogle Scholar
  32. Hinrichsen H (1997) Stochastic lattice models with several absorbing states. Phys Rev E 55(1):219–226ADSCrossRefGoogle Scholar
  33. Jaynes E (1957) Information theory and statistical mechanics. Phys Rev 106(4):620–630ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. Kaneko K (1985) Spatiotemporal intermittency in coupled map lattices. Progr Theor Phys 74(5):1033–1044ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. Kawasaki K (1972) Kinetics of Ising model. In: Domb CM, Green MS (eds) Phase transitions and critical phenomena, vol 2. Academic, New York, p 443Google Scholar
  36. Kirkpatrick S, Gelatt CG Jr, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. Lawniczak A, Dab D, Kapral R, Boon JP (1991) Reactive lattice gas automata. Phys D 47(1–2):132–158MathSciNetCrossRefzbMATHGoogle Scholar
  38. Marinari E, Parisi G (1992) Simulated tempering: a new Monte Carlo scheme. Europhys Lett 19:451–458ADSCrossRefGoogle Scholar
  39. May R (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467ADSCrossRefGoogle Scholar
  40. Metropolis N, Hewlett J, Rota GC (eds) (1980) A history of computing in the twentieth century. Academic, New YorkzbMATHGoogle Scholar
  41. Mezard M, Parisi G, Virasoro MA (1987) Spin glass theory and beyond. World scientific lecture notes in physics, vol 9. World Scientific, SingaporezbMATHGoogle Scholar
  42. Newman ME (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46:323–351ADSCrossRefGoogle Scholar
  43. Niss M (2005) History of the Lenz-Ising model 1920–1950: from ferromagnetic to cooperative phenomena. Arch Hist Exact Sci 59(3):267–318MathSciNetCrossRefzbMATHGoogle Scholar
  44. Nordfalk J, Alstrøm P (1996) Phase transitions near the “game of life”. Phys Rev E 54(2):R1025–R1028ADSCrossRefGoogle Scholar
  45. Onsager L (1944) Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys Rev 65:117–149ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. Oestreicher C (2007) A history of chaos theory. Dialogues Clin Neurosci 9(3):279–289. Available online https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3202497/
  47. Politi A, Livi R, Oppo GL, Kapral R (1993) Unpredictable behaviour of stable systems. Europhys Lett 22(8):571–576ADSCrossRefGoogle Scholar
  48. Rabiner L (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proc IEEE 77(2):257–286CrossRefGoogle Scholar
  49. Rapaport DC (2004) The art of molecular dynamics simulation. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  50. Repast – recursive porus agent simulation toolkit (2008) http://repast.sourceforge.net/. Accessed 10 Apr 2017
  51. Rothman DH, Zaleski S (2004) Lattice-gas cellular automata. Monographs and texts in statistical physics. Collection Alea-Saclay, ParisGoogle Scholar
  52. Sornette D (2006) Critical phenomena in natural sciences, Springer series in synergetics. Springer, BerlinzbMATHGoogle Scholar
  53. Stauffer D, Aharony A (1994) Introduction to percolation theory. Taylor Francis, LondonzbMATHGoogle Scholar
  54. Succi S (2001) The lattice Boltzmann equation for fluid dynamics and beyond. Numerical mathematics and scientific computation. Oxford University Press, OxfordzbMATHGoogle Scholar
  55. Swendsen R, Wang JS (1987) Nonuniversal critical dynamics in Monte Carlo simulations. Phys Rev Lett 58(2):86–88ADSCrossRefGoogle Scholar
  56. van Kampen NG (1992) Stochastic processes in physics and chemistry. North-Holland, AmsterdamzbMATHGoogle Scholar
  57. von Neumann J, Burks AW (1966) Theory of self-reproducing automata. University of Illinois Press, Urbana/LondonGoogle Scholar
  58. Von Neumann universal constructor (2008) http://en.wikipedia.org/wiki/Von_Neumann_Universal_Constructor. Accessed 10 Apr 2017
  59. Watts D, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–441ADSCrossRefGoogle Scholar
  60. Wilensky U (1999) Netlogo. Center for connected learning and computer-based modeling, Northwestern University, Evanston. http://ccl.northwestern.edu/netlogo/. Accessed 10 Apr 2017
  61. Wolf-Gladrow D (2004) Lattice-gas cellular automata and lattice Boltzmann models: an introduction, Lecture notes in mathematics, vol 1725. Springer, BerlinzbMATHGoogle Scholar
  62. Wolfram S (1983) Statistical mechanics of cellular automata. Rev Mod Phys 55:601–644ADSMathSciNetCrossRefzbMATHGoogle Scholar

Books and Reviews

  1. Boccara N (2004) Modeling complex systems. In: Graduate texts in contemporary physics. Springer, BerlinGoogle Scholar
  2. Bungartz H-J, Mundani R-P, Frank AC (2005) Bubbles, jaws, moose tests, and more: the wonderful world of numerical simulation, Springer VideoMATH. Springer, Berlin. (DVD)zbMATHGoogle Scholar
  3. Chopard B, Droz M (2005) Cellular automata modeling of physical systems. In: Collection Alea-Saclay: monographs and texts in statistical physics. Cambridge University Press, CambridgeGoogle Scholar
  4. Deisboeck S, Kresh JY (2006) Complex systems science in biomedicine. In: Deisboeck S, Kresh JY (eds) Topics in biomedical engineering. Springer, New YorkGoogle Scholar
  5. Gould H, Tobochnik J, Christian W (2007) An introduction to computer simulation methods: applications to physical systems. Addison-Wesley, New YorkGoogle Scholar
  6. Landau RH (2005) A first course in scientific computing: symbolic, graphic, and numeric modeling using maple, java, Mathematica, and Fortran90. Princeton University Press, PrincetonzbMATHGoogle Scholar
  7. Open Source Physics. http://www.opensourcephysics.org/. Accessed 10 Apr 2017
  8. Resnick M (1994) Turtles, termites, and traffic jams. Explorations in massively parallel microworlds. In: Complex adaptive systems. MIT Press, CambridgeGoogle Scholar
  9. Shalizi C Cosma’s home page. http://www.cscs.umich.edu/~crshalizi/. Accessed 10 Apr 2017

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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department Physics and Astronomy and CSDCUniversity of FlorenceFlorenceItaly