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Discrete Dynamics in Nature and Society
Volume 2004 (2004), Issue 1, Pages 251-272
http://dx.doi.org/10.1155/S1026022604312069

On Prigogine's approaches to irreversibility: a case study by the baker map

Advanced Institute for Complex Systems and Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

Received 30 December 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The baker map is investigated by two different theories of irreversibility by Prigogine and his colleagues, namely, the Λ-transformation and complex spectral theories, and their structures are compared. In both theories, the evolution operator U of observables (the Koopman operator) is found to acquire dissipativity by restricting observables to an appropriate subspace Φ of the Hilbert space L2 of square integrable functions. Consequently, its spectral set contains an annulus in the unit disc. However, the two theories are not equivalent. In the Λ-transformation theory, a bijective map Λ1:ΦL2 is looked for and the evolution operator U of densities (the Frobenius-Perron operator) is transformed to a dissipative operator W=ΛUΛ1. In the complex spectral theory, the class of densities is restricted further so that most values in the interior of the annulus are removed from the spectrum, and the relaxation of expectation values is described in terms of a few point spectra in the annulus (Pollicott-Ruelle resonances) and faster decaying terms.