Trajectory versus probability density entropy

Mauro Bologna, Paolo Grigolini, Markos Karagiorgis, and Angelo Rosa
Phys. Rev. E 64, 016223 – Published 27 June 2001
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Abstract

We show that the widely accepted conviction that a connection can be established between the probability density entropy and the Kolmogorov-Sinai (KS) entropy is questionable. We adopt the definition of density entropy as a functional of a distribution density whose time evolution is determined by a transport equation, conceived as the only prescription to use for the calculation. Although the transport equation is built up for the purpose of affording a picture equivalent to that stemming from trajectory dynamics, no direct use of trajectory time evolution is allowed, once the transport equation is defined. With this definition in mind we prove that the detection of a time regime of increase of the density entropy with a rate identical to the KS entropy is possible only in a limited number of cases. The proposals made by some authors to establish a connection between the two entropies in general, violate our definition of density entropy and imply the concept of trajectory, which is foreign to that of density entropy.

  • Received 3 March 2000

DOI:https://doi.org/10.1103/PhysRevE.64.016223

©2001 American Physical Society

Authors & Affiliations

Mauro Bologna1,*, Paolo Grigolini1,2,3,†, Markos Karagiorgis1, and Angelo Rosa3

  • 1Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427
  • 2Istituto di Biofisica del Consiglio Nazionale delle Ricerche, Area della Ricerca di Pisa, Via Alfieri 1, San Cataldo, 56010, Ghezzano-Pisa, Italy
  • 3Dipartimento di Fisica dell’Università di Pisa and INFM Piazza Torricelli 2, 56127 Pisa, Italy

  • *Email address: mb0015@unt.edu
  • Email address: grigo@jove.acs.unt.edu

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Vol. 64, Iss. 1 — July 2001

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