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Trajectory versus probability density entropy.

M Bologna1, P Grigolini, M Karagiorgis

  • 1Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA. mb0015@unt.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 20, 2001
PubMed
Summary
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The link between probability density entropy and Kolmogorov-Sinai (KS) entropy is questionable. Our findings indicate this connection is only possible in limited cases, challenging widely held beliefs in statistical mechanics.

Area of Science:

  • Statistical Mechanics
  • Information Theory
  • Dynamical Systems

Background:

  • The relationship between probability density entropy and Kolmogorov-Sinai (KS) entropy is a widely accepted concept in statistical mechanics.
  • Previous studies have proposed a direct connection, assuming density entropy increases at a rate equivalent to KS entropy.

Purpose of the Study:

  • To critically evaluate the established connection between probability density entropy and KS entropy.
  • To rigorously define and utilize density entropy based solely on transport equations.

Main Methods:

  • Defined density entropy as a functional of distribution density, evolving via a transport equation.
  • Analyzed the time evolution of density entropy without direct use of trajectory dynamics.
  • Investigated conditions under which density entropy increases at a rate matching KS entropy.

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Main Results:

  • Demonstrated that a time regime of density entropy increase, matching KS entropy, occurs only in specific, limited cases.
  • Showed that many existing proposals for linking these entropies violate the strict definition of density entropy.
  • Highlighted that these proposals incorrectly incorporate trajectory concepts into density entropy calculations.

Conclusions:

  • The general connection between probability density entropy and KS entropy is not universally valid.
  • The strict definition of density entropy, based on transport equations, limits the applicability of the presumed link.
  • Future research should reconsider the theoretical underpinnings of entropy connections in dynamical systems.