On Prigogine's approaches to irreversibility: a case study by the baker map
One of the long standing problems in statistical mechanics is the consistent description of macroscopic irreversibility in terms of reversible microscopic dynamics. Two different approaches to irreversibility by I. Prigogine school are known: the \(\Lambda\)-transformation and complex spectral theories. The author studies the baker map on these two approaches and compares the results. It is shown that in both theories the evolution operator \(U^{+}\) becomes dissipative by restriction obvervables to an appropriate subspace \(\Phi\) of the Hilbert space \(L_2\), and its spectral set contains an annulus in the unit disc. But these theories are not equivalent. According to the first one a bijective map \(\Lambda^{\dag -1}:\Phi\rightarrow L^2\) is looked for and the evolution operator \(U\) of densities is transformed to a dissipative operator \(W=\Lambda U\Lambda^{-1}\). On the second one the class of densities is restricted further so that the most values in the interior of the annulus are removed from the spectrum. Here the relaxation of expectation values is described by the Pollicot-Ruelle decomposition.
scientific article; zbMATH DE number 2148593