Zdzislaw Suchanecki

Intrinsic irreversibility of quantum systems with diagonal singularity

The work of the Brussels-Austin group on irreversibility over the last years has shown that quantum large Poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional Hilbert space framework and logic. We characterize the algebra of observables, the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity. We illustrate the general ideas for the Friedrichs model.

Spectral decomposition and fractal eigenvectors for a class of piecewise linear maps

Abstract For a class of non-uniformly expanding piecewise linear maps, the spectral properties of the Frobenius-Perron operator are studied. We construct integral operators which intertwine two Frobenius-Perron operators corresponding to different invariant measures. With the aid of the intertwining integral operators, the right and left eigenvectors corresponding to the Pollicott-Ruelle resonances are obtained, and the decrease of the essential spectral radius due to the increase of the smoothness of the domain is shown. The Frobenius-Perron operator is also shown to admit a generalized spectral decomposition consisting of only isolated point spectra on suitable test function spaces. The Riesz representation of the left eigenvectors is given by singular continuous functions, which are solutions of a generalized De Rham equation. The singular functions have fractal properties reflecting the complexity of the dynamics due to stretching and folding.

On lambda and internal time operators

An extended construction of Λ transformations leading from conservative to dissipative evolutions is presented and new sets of admissible states are determined. Then is presented a probabilistic approach to the concept of internal time operator. In this probabilistic context it is shown that Λ is the expected value of the new time operator, the scaling function of Λ its tail distribution and that the choice of a profile function reduces to at most three possibilities.

Nonlocality of the Misra-Prigogine-Courbage semigroup

We show that the Markov semigroups constructed by Misra, Prigogine, and Courbage through nonunitary similarity transformations of Kolmogorov systems are not implementable by local point transformations, i.e., they are not the Frobenius-Perron semigroups associated with noninvertible point transformations, in contrast with the semigroups obtained by coarse-graining projections. Our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by Goldstein, Misra, and Courbage and also of the previous illustration by Misra and Prigogine for the baker transformation and completes the characterization of the Misra-Prigogine-Courbage semigroups.

Rigged Hilbert spaces for chaotic dynamical systems

We consider the problem of rigging for the Koopman operators of the Renyi and the baker maps. We show that the rigged Hilbert space for the Renyi maps has some of the properties of a strict inductive limit and give a detailed description of the rigged Hilbert space for the baker maps.

The spectrum of the Liouville-von Neumann operator in the Hilbert-Schmidt space

The singular continuous spectrum of the Liouville operator of quantum statistical physics is, in general, properly included in the difference of the spectral values of the singular continuous spectrum of the associated Hamiltonian. The absolutely continuous spectrum of the Liouvillian may arise from a purely singular continuous Hamiltonian. We provide the correct formulas for the spectrum of the Liouville operator and show that the decaying states of the singular continuous subspace of the Hamiltonian do not necessarily contribute to the absolutely continuous subspace of the Liouvillian.

General Properties of the Liouville Operator

We study the self-adjointness of the Liouvillianof a symmetric operator. We also discuss some cases ofthe spectrum of the Liouville operator of a self-adjointHamiltonian with purely continuous singular spectrum. The presence of an absolutelycontinuous part for the spectrum of Liouvillianscorresponding to Hamiltonians with purely continuoussingular spectrum shows that quantum theory in Hilbertand Liouville spaces is not equivalent.

The logic of quantum systems with diagonal singularities

The work of the Brussels-Austin groups on irreversibility over the last years has shown that Quantum Large Poincaré systems with diagonal singularity lead to an extension of the conventional formulation of dynamics at the level of mixtures which is manifestly time asymmetric. States with diagonal singularity acquire meaning as linear fractionals over the involutive Banach algebra of operators with diagonal singularity. We show in this paper that the logic of quantum systems with diagonal singularity is not the conventional logic of Hilbert space, because only finite combinations of prepositions are allowed.

Analysis of resources distribution in economics based on entropy

We propose a new approach to the problem of efficient resources distribution in different types of economic systems. We also propose to use entropy as an indicator of the efficiency of resources distribution. Our approach is based on methods of statistical physics in which the states of economic systems are described in terms of the density functions ρ(g,α) of the variable g parametrized by α. The parameter α plays a role of the integral characteristic of the state of the economic system. Having the density function ρ(g,α) we can use the corresponding entropy to evaluate the efficiency of the resources distribution. Our theoretical study have been tested on real data related to the portfolio investment.

Generalized spectral decomposition and intrinsic irreversibility of the Arnold Cat Map

Abstract We construct a generalized spectral decomposition of the Frobenius-Perron and Koopman operators of the Arnold Cat map. We define a suitable dual pair or rigged Hubert space which provides mathematical meaning to the spectral decomposition. The eigenvalues in the decomposition are the resonances of the power spectrum which determine the decay rates of the correlation functions and the rate of approach to equilibrium. The extended unitary evolution splits into two distinct semigroups which express the intrinsic irreversibility of the Cat map resulting from the strong chaotic properties.