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

Abstract

The baker map is investigated by two different theories of irreversibility by Prigogine and his colleagues, namely, theΛ-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 dissipativityby restrictingobservables to an appropriate subspaceΦof the Hilbert spaceL2of square integrable functions. Consequently, its spectral set contains an annulus in the unit disc. However, the two theories are not equivalent. In theΛ-transformation theory, a bijective mapΛ†−1:Φ→L2is looked for and the evolution operatorUof densities (the Frobenius-Perron operator) is transformed to a dissipative operatorW=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.