Open Quantum Systems

International Erwin-Schrödinger-Institute for Mathematical Physics
January 20 to March 31, 2005

organized by J. Yngvason, G. M. Graf and J. Derezinski.


2nd Workshop.

Mon, 14 March - Fri, 18 March, 2005.

Schedule

Abstracts

Claude Alain Pillet, from University of Toulon and CPT Marseille, will give a 5 hours mini-course under the title:

Transport properties of quasifree fermions

Independent electron models, as first approximation to the dynamics of electronic systems, are widely used in solid state physics. They turned out to be particularly useful in the study of mesoscopic devices. I shall present the mathematical framework adapted to the study of such models at positive density, in non-equilibrium situations. I will discuss the scattering approach to the construction of non-equilibrium steady states (NESS) suggested by Ruelle. I will show that this construction is equivalent to the well known Landauer-Buettiker approach to the study of transport properties of mesoscopic devices. I shall also discuss the connections between Ruelle's approach and other subjects like the Kubo formula and the Onsager reciprocity relations.

Walid Abou Salem, ETH-Zurich

An adiabatic theorem for non-equilibrium steady states

I will discuss an adiabatic type theorem for states close to the non-equilibrium steady states of a small subsystem coupled to two fermionic reservoirs at different temperatures.

Juerg Froehlich, Zuerich

Nonequilibrium statistical mechanics and thermodynamics

I review results concerning the relationship between statistical mechanics and thermodynamics. Among other topics, I discuss partial derivations of the fundamental laws of thermodynamics from nonequilibrium statistical mechanics and matters related to entanglement and decoherence.

Christian Hainzl, Copenhagen

A non-linear model for relativistic electrons interacting with Dirac's vacuum

We study the Bogolubov-Dirac-Fock (BDF) model, which is a mean-field theory deduced from QED. Contrary to usual Dirac-type theories the associated BDF-functional is bounded from below. Its variables are infinite rank projectors describing the electrons, the observable ones as well as those filling up the Dirac sea. We prove that for any ultra-violet momentum cut-off $\Lambda$ the BDF-functional attains its minimum. The minimizer fulfills a self-consistent equation representing the polarized vacuum. Moreover we show that for $\Lambda$ to infinity the theory gets "nullified" as predicted by Landau.Additionally we also state the existence of global-in-time solutions to the associated time-dependent equation. Finally, our framework allows to minimize a general first principle Hamiltonian, although it is a-priori ill defined. We derive a link between its minimizer and the BDF-minimizer. The corresponding self-consistent solution is of Schwinger-Dyson-type.

Masao Hirokawa, Okayama

Some bipolaron problems

We consider some problems on a Froehlich bipolaron, such as the tug between the Coulomb repulsion and the attraction caused by LO phonons.

Fumio Hiroshima, Setsunan University

Asymptotic fields and ground states of a quantum field model.

Applying asymptotic fields, we show the regularity of ground states and give an upper bound of its multiplicity. The ground state of a fibered Hamiltonian is also discussed.

Christian Jaekel, Sao Paulo

On the relativistic KMS condition for the $P(\phi)_2$ model

The relativistic KMS condition introduced by Bros and Buchholz provides a missing link between quantum statistical mechanics and quantum field theory. We show that the $P(\phi)_2$ model at positive temperature satisfies the relativistic KMS condition.

Gianni Jona-Lasinio Rome

Signatures of time reversal violation in the statistical mechanics of stochastic lattice gases

Violation of space reflection symmetry and/or long range space correlations seem to be generically associated to microscopic dynamics which are not time reversal invariant.

Andrzej Kossakowski, Torun

Remarks on the weak coupling limit

A possibility to go beyond the weak coupling limit is discussed. Time evolutions in weak coupling limit and the exact one are compared.

Taku Matsui, Kyushu University

BEC and NESS on Lattices

We consider (quasi)free Bose on lattices and its condensation. Physical situation we have in our mind is the super-conducting network. The conditions for Bose condensation on general infinite graphs is given in terms of random walks. Ness and Josephsson current are studied in case of integer lattices.

Marco Merkli, McGill University, Montreal

Stability of multiple equilibria

We consider a quantum system composed of a spatially infinitely extended free Bose gas with a condensate, interacting with a small system (quantum dot) which can trap finitely many Bosons. Due to spontaneous symmetry breaking in the presence of the condensate, the system has many equilibrium states for each fixed temperature. We extend the notion of Return to Equilibrium to systems possessing a multitude of equilibrium states and show in particular that a condensate coupled to a quantum dot has the property of Return to Equilibrium in a weak coupling sense: any local perturbation of an equilibrium state of the coupled system, evolving under the interacting dynamics, converges in the long time limit to an asymptotic state. The latter is, modulo an error term, an equilibrium state which depends (in an explicit way) on the local perturbation (an effect due to long-range correlations). The error term vanishes in the small coupling limit.

Heidi Narnhofer Vienna University

Effective quantum dynamics on the mesoscopic level

The fluctuation algebra serves as a nonlocal and at the same time noncommutative algebra. In mean field theories the time evolution differs from the state dependent local time evolution. For phase transitions the scaling has to be adjusted. We discuss the connection of the mesoscopic algebra in the BCS-model with the Josephson effect and the corresponding phase transition.

Jeoffrey Schenker

Equality of the edge and bulk Hall conductances in a mobility gap

I will discuss the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. An appropriate definition of the edge conductance may be obtained by including a contribution from states in the localized band. When this contribution is included the edge and bulk conductances are equal. Furthermore the contribution is essential, as it may be shown to be non-zero for the Harper Hamiltonian with Cauchy randomness. [Joint work with A. Elgart and G. M. Graf]

Geoffrey Sewell, London

Quantum macrostatistical theory of nonequilibrium steady states.

I shall provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centred on the large scale properties of locally conserved hydrodynamical observables, and is designed to form a bridge between the quantum microscopic and classical macroscopic continuum mechanical descriptions of many-particle systems, rather than a derivation of the latter from the former. The key physical assumptions of the theory are (a) a chaoticity hypothesis for the nonconserved currents carried by the locally conserved hydrodynamical observables, (b) an extension of Onsager's regression hypothesis to fluctuations about nonequilibrium steady states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis, I obtain a picture wherein the spatial correlations of the hydrodynamical variables are generically of long range in nonequilibrium steady states. This result constitutes a quantum mechanical generalisation of those obtained by microscopic treatments of some special classical stochastic models, and marks a striking difference beween the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations

Jan Philip. Solovej University of Copenhagen

On a mean field approximation to Quantum Electrodynamics

Tentative list of participants of the 2nd workshop