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Coarse-grained probabilistic automata mimicking chaotic systems.

Massimo Falcioni1, Angelo Vulpiani, Giorgio Mantica

  • 1Dipartimento di Fisica, Università di Roma La Sapienza, INFM, Unità di Roma1 and SMC Center, Piazzale Aldo Moro 2, 00185 Roma, Italy.

Physical Review Letters
|August 9, 2003
PubMed
Summary

Discretization typically eliminates chaos in dynamical systems. However, this study demonstrates that incorporating randomness into discretization allows chaotic dynamics to persist, remaining indistinguishable from the original system via entropic analysis.

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Area of Science:

  • * Physics, specifically focusing on dynamical systems and chaos theory.
  • * Explores the intersection of classical and quantum mechanics through the lens of chaos.
  • * Contributes to the field of computational physics and numerical analysis of complex systems.

Background:

  • * The process of discretizing phase space in dynamical systems generally leads to the suppression of chaotic behavior.
  • * Understanding the conditions under which chaos can be preserved or mimicked in discrete systems is crucial for accurate modeling.
  • * The relationship between classical chaos and quantum chaos remains an active area of research.

Purpose of the Study:

  • * To investigate whether dynamical chaos can survive the discretization of phase space when randomness is introduced.
  • * To determine if such discretized chaotic systems can be operationally indistinguishable from their continuous counterparts using specific analytical methods.

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  • * To explore the implications of these findings for the study of quantum chaos.

    • Main Methods:

    • * Numerical simulation of dynamical systems undergoing phase space discretization.
    • * Introduction of randomness as a component of the discretization process.
    • * Application of entropic, coarse-grained analysis to compare discrete and continuous system dynamics.

    Main Results:

    • * Dynamical chaos can indeed survive discretization when randomness is incorporated.
    • * The entropic, coarse-grained analysis shows that the discretized chaotic system is indistinguishable from the original continuous chaotic system.
    • * The survival of chaos in discrete systems is demonstrated to be robust under the applied analytical framework.

    Conclusions:

    • * Discretization does not necessarily nullify chaos in dynamical systems, provided randomness is included.
    • * The phenomenon of surviving chaos has significant implications for understanding quantum chaos.
    • * This research offers a new perspective on the challenges of simulating chaotic systems and their quantum counterparts.