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 operatorU† of observables (the Koopman operator) is found to acquire dissipativity by restricting observables to an appropriate subspaceΦ 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.