On Prigogine's approaches to irreversibility: a case study by the baker map
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 acq...
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| Format: | Article |
| Language: | English |
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Wiley
2004-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/S1026022604312069 |
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| author | S. Tasaki |
| author_facet | S. Tasaki |
| author_sort | S. Tasaki |
| collection | DOAJ |
| description | 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 Λ†−1:Φ→L2 is looked for and the evolution operator
U of densities (the Frobenius-Perron operator) is transformed
to a dissipative operator W=ΛUΛ−1. 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. |
| format | Article |
| id | doaj-art-02eba8880e2c4e6989fc15ec355ebd83 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2004-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-02eba8880e2c4e6989fc15ec355ebd832025-08-20T03:55:23ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2004-01-012004125127210.1155/S1026022604312069On Prigogine's approaches to irreversibility: a case study by the baker mapS. Tasaki0Advanced 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, JapanThe 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 Λ†−1:Φ→L2 is looked for and the evolution operator U of densities (the Frobenius-Perron operator) is transformed to a dissipative operator W=ΛUΛ−1. 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.http://dx.doi.org/10.1155/S1026022604312069 |
| spellingShingle | S. Tasaki On Prigogine's approaches to irreversibility: a case study by the baker map Discrete Dynamics in Nature and Society |
| title | On Prigogine's approaches to irreversibility: a case study by the
baker map |
| title_full | On Prigogine's approaches to irreversibility: a case study by the
baker map |
| title_fullStr | On Prigogine's approaches to irreversibility: a case study by the
baker map |
| title_full_unstemmed | On Prigogine's approaches to irreversibility: a case study by the
baker map |
| title_short | On Prigogine's approaches to irreversibility: a case study by the
baker map |
| title_sort | on prigogine s approaches to irreversibility a case study by the baker map |
| url | http://dx.doi.org/10.1155/S1026022604312069 |
| work_keys_str_mv | AT stasaki onprigoginesapproachestoirreversibilityacasestudybythebakermap |