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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1026-0226 1607-887X |