Brussels-Austin Group Research Articles

Quantum Time Arrows, Semigroups and Time-Reversal in Scattering

Two approaches toward the arrow of time for scattering processes have been proposed in rigged Hilbert space quantum mechanics. One, due to Arno Bohm, involves preparations and registrations in laboratory operations and results in two semigroups oriented in the forward direction of time. The other, employed by the Brussels-Austin group, is more general, involving excitations and de-excitations of systems, and apparently results in two semigroups oriented in opposite directions of time. It turns out that these two time arrows can be related to each other via Wigner's extensions of the spacetime symmetry group. Furthermore, their are subtle differences in causality as well as the possibilities for the existence and creation of time-reversed states depending on which time arrow is chosen.

Arrow of Time in Rigged Hilbert Space Quantum Mechanics

Arno Bohm and Ilya Prigogine's Brussels–Austin Group have been working on the quantum mechanical arrow of time and irreversibility in rigged Hilbert space quantum mechanics. A crucial notion in Bohm's approach is the so-called preparation/registration arrow. An analysis of this arrow and its role in Bohm's theory of scattering is given. Similarly, the Brussels–Austin Group uses an excitation/de-excitation arrow for ordering events, which is also analyzed. The relationship between the two approaches is initially discussed focusing on their semi-group operators and time arrows. Finally a possible realist interpretation of the rigged Hilbert space formulation of quantum mechanics is considered.

Nonequilibrium statistical mechanics Brussels–Austin style

The fundamental problem on which Ilya Prigogine and the Brussels–Austin Group have focused can be stated briefly as follows. Our observations indicate that there is an arrow of time in our experience of the world (e.g., decay of unstable radioactive atoms like uranium, or the mixing of cream in coffee). Most of the fundamental equations of physics are time reversible, however, presenting an apparent conflict between our theoretical descriptions and experimental observations. Many have thought that the observed arrow of time was either an artifact of our observations or due to very special initial conditions. An alternative approach, followed by the Brussels–Austin Group, is to consider the observed direction of time to be a basic physical phenomenon due to the dynamics of physical systems. This essay focuses mainly on recent developments in the Brussels–Austin Group after the mid-1980s. The fundamental concerns are the same as in their earlier approaches (subdynamics, similarity transformations), but the contemporary approach utilizes rigged Hilbert space (whereas the older approaches used Hilbert space). While the emphasis on nonequilibrium statistical mechanics remains the same, their more recent approach addresses the physical features of large Poincaré systems, nonlinear dynamics and the mathematical tools necessary to analyze them.

Ensemble dynamics of chaos

AbstractSystems of deterministic chaos demonstrate that the ensemble and trajectory descriptions of dynamics are not equivalent. This is shown through explicit constructions of the generalized spectral decompositions of the time evolution operator for probability densities. This realizes part of the long‐standing goal of the Brussels–Austin groups directed by Ilya Prigogine to elucidate the dynamical foundations of irreversibility. A brief review of the main aspects of the work are given here along with a new derivation of the fractality of eigenstates in some systems and a discussion of a Lyapunov functional in an unstable Hamiltonian system. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

A tribute to Ilya Prigogine (1917–2003)

Shortly before the publication of this tribute, Ilya Prigogine passed away on May 28, 2003. This is an enormous loss, not only to science but also for our culture and society at large. We have, nevertheless, decided to proceed with the tributary volume as originally planned. It is with great respect and devotion that we assert that Ilya Prigogine and his Brussels and Austin Schools have fundamentally transformed and revised the science of irreversible thermodynamic processes. We also acknowledge the way he has guided and led the scientific development, inspired the community at large. His original way of posing questions resulted in new fundamental knowledge from the most basic microscopic level to the most surrounding philosophical and humanistic organizations. We refer in particular to his fundamental presentation of the two conflicting views of nature—on one hand the deterministic time reversible exposition and on the other the evolutionary irreversible character of the universe. This insight calls for new perspectives concerning the description of natural phenomena and in this development the Brussels-Austin Groups have played a leading role as originator, designer, guide and inspirator. We celebrate Ilya Prigogine's pioneering and fundamental work on nonlinear dynamics, chaos and nonequilibrium phenomena, dissipative structures and irreversibility. Not everybody is convinced that the present inconsistencies derived from the two conflicting views of nature are serious, not to say they are illusive or deceptive. It is nevertheless highly unsatisfactory to validate fundamental natural laws as truly deterministic and time reversible saving irreversibility by incongruous initial conditions. As more and more realistic systems and their intrinsic properties are studied it is hoped that consensus will develop at the appropriate level of description. This would certainly be the legacy of the Brussels-Austin School even if we today are far from encompassing altogether the vision and foresight of Ilya Prigogine's predictions of the future of humanity.

A Tale of Two Arrows

Two time arrows for scattering processes have been proposed in rigged Hilbert space quantum mechanics. One, due to Arno Bohm, involves preparations and registrations in laboratory operations and results in two semigroups oriented in the forward direction of time. The other, employed by the Brussels-Austin group, is more general, involving excitations and de-excitations of systems, and apparently results in two semigroups oriented in opposite directions of time. The relationship between these two arrows is discussed.

Caratheodory and the Foundations of Thermodynamics and Statistical Physics

Constantin Caratheodory offered the first systematic and contradiction free formulation of thermodynamics on the basis of his mathematical work on Pfaff forms. Moreover, his work on measure theory provided the basis for later improved formulations of thermodynamics and physics of continua where extensive variables are measures and intensive variables are densities. Caratheodory was the first to see that measure theory and not topology is the natural tool to understand the difficulties (ergodicity, approach to equilibrium, irreversibility) in the Foundations of Statistical Physics. He gave a measure-theoretic proof of Poincare's recurrence theorem in 1919. This work paved the way for Birkhoff to identify later ergodicity as metric transitivity and for Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert spaces. Mixing provided an explanation of the approach to equilibrium but not of irreversibility. The recent extension of spectral theory of dynamical systems to locally convex spaces, achieved by the Brussels–Austin groups, gives new nontrivial time asymmetric spectral decompositions for unstable and/or non-integrable systems. In this way irreversibility is resolved in a natural way.

Extension of the Dynamics of Unstable Systems

The work of the Brussels–Austin groups over the last six years has demonstrated that for unstable systems, classical or quantum, there exist spectral decompositions of the evolution In terms of resonances and resonance states which appear as eigenvalues and eigenprojections of the evolution operator. These new spectral decompositions are non-trivial only for unstable systems and define an extension of the evolution to suitable dual pairs. Duality here is the states/observables duality and the spectral decompositions extend the algebraic approach to dynamical systems to an intrinsically probabilistic and irreversible formulation. The extended formulation allows for probabilistic prediction and control beyond the traditional local techniques.

Nonstationary dressed states and effects of decay in nonadiabatic crossing of decaying levels

The rigorous equations for a nonadiabatic coupling of the amplitudes of decaying discrete states dressed by a continuum are derived by means of the complex spectral decomposition method developed by the Brussels-Austin group. The limitations of the previously used phenomenological equations are shown within the scope of the time-dependent N-level Friedrichs-Fano model. The novel nonadiabatic effects owing to the decay of discrete levels into a continuum are found: 1. (i) nonunitary nonadiabaticity caused from unitary one, 2. (ii) nonadiabaticity via time dependent coupling with a reservoir, and 3. (iii) additional cross-decay due to nonadiabaticity. Reasonable generalization of Berry's phase for decaying eigenstates and regeneration of driven decaying states via coherent mixing with reservoir are discussed. General results are applied to the quasienergy-state dynamics of a three-level molecule driven by DC and AC fields and to the non-steady Born-Oppenheimer approach of molecules electronically excited to decaying states. Also, a “resonance” approximation leading to the true non-Hermitian Hamiltonian, similar to the phenomenological one (used in quantum optics in the stationary case) and applicable to the nonstationary case, is formulated explicitly.

Quantum systems with diagonal singularity: Their logic and intrinsic irreversibility

The work of the Brussels-Austin group on irreversibility over the last years has shown that quantum large Poincaré 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, and the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity.

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 Poincaré 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.

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 Poincare 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.

Generalized spectral decompositions of mixing dynamical systems

AbstractWe introduce a method for the explicit computation of the eigenvalue problem of the evolution operator of mixing dynamical systems. The method is based on the subdynamics decomposition of the Brussels–Austin groups directed by Professor I. Prigogine. We apply the method to three different representatives of mixing systems, namely, the Renyi maps, baker's transformations, and the Friedrichs model. The obtained spectral decompositions acquire meaning in suitable rigged Hilbert spaces that we construct explicitly for the three models. The resulting spectral decompositions show explicitly the intrinsic irreversibility of baker's transformations and Friedrichs model and the intrinsically probabilistic characters of the Renyi maps and baker's transformations. The dynamical properties are reflected in the spectrum because the eigenvalues are the powers of the Lyapunov times for the Renyi and baker systems and include the lifetimes for the Friedrichs model. © 1993 John Wiley & Sons, Inc.

On entropy production in a stochastic model of open systems

Two expressions for the entropy production of a system arbitrarily far from equilibrium are considered for a stochastic model. The two expressions result from the consideration of two different definitions of the nonequilibrium entropy: the Gibbs form and the generalized form suggested by the Brussels–Austin group. The entropy principle is established briefly for isolated and then, in detail, for open systems, and the origin and form of the entropy flow is discussed.