Volume 55, Issue 7, July 2014

We consider an ideal Fermi gas confined to a geometric structure consisting of a central region – the sample – connected to several infinitely extended ends—the reservoirs. Under physically reasonable assumptions on the propagation properties of the oneparticle dynamics within these reservoirs, we show that the state of the Fermi gas relaxes to a steady state. We compute the expected value of various current observables in this steady state and express the result in terms of scattering data, thus obtaining a geometric version of the celebrated LandauerBüttiker formula.
 SPECIAL ISSUE: NONEQUILIBRIUM STATISTICAL MECHANICS



Quantum fields with classical perturbations
View Description Hide DescriptionThe main purpose of these notes is a review of various models of Quantum Field Theory (QFT) involving quadratic Lagrangians. We discuss scalar and vector bosons, spin fermions, both neutral and charged. Beside free theories, we study their interactions with classical perturbations, called, depending on the context, an external linear source, masslike term, current or electromagnetic potential. The notes may serve as a first introduction to QFT.

A geometric approach to the LandauerBüttiker formula
View Description Hide DescriptionWe consider an ideal Fermi gas confined to a geometric structure consisting of a central region – the sample – connected to several infinitely extended ends—the reservoirs. Under physically reasonable assumptions on the propagation properties of the oneparticle dynamics within these reservoirs, we show that the state of the Fermi gas relaxes to a steady state. We compute the expected value of various current observables in this steady state and express the result in terms of scattering data, thus obtaining a geometric version of the celebrated LandauerBüttiker formula.

Fully coupled PauliFierz systems at zero and positive temperature
View Description Hide DescriptionThese notes provide an introduction to the spectral analysis of PauliFierz systems at zero and positive temperature. More precisely, we study finite dimensional quantum systems linearly coupled to a single reservoir, a massless scalar quantum field. We emphasize structure results valid at arbitrary systemreservoir coupling strength. The notes contain a mixture of known, refined, and new results and each section ends with a discussion of open problems.

Repeated interactions in open quantum systems
View Description Hide DescriptionAnalyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in nonequilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of nonequilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.

Dynamics, stability, and statistics on lattices and networks
View Description Hide DescriptionThese lectures aim at surveying some dynamical models that have been widely explored in the recent scientific literature as case studies of complex dynamical evolution, emerging from the spatiotemporal organization of several coupled dynamical variables. The first message is that a suitable mathematical description of such models needs tools and concepts borrowed from the general theory of dynamical systems and from outofequilibrium statistical mechanics. The second message is that the overall scenario is definitely reacher than the standard problems in these fields. For instance, systems exhibiting complex unpredictable evolution do not necessarily exhibit deterministic chaotic behavior (i.e., Lyapunov chaos) as it happens for dynamical models made of a few degrees of freedom. In fact, a very large number of spatially organized dynamical variables may yield unpredictable evolution even in the absence of Lyapunov instability. Such a mechanism may emerge from the combination of spatial extension and nonlinearity. Moreover, spatial extension allows one to introduce naturally disorder, or heterogeneity of the interactions as important ingredients for complex evolution. It is worth to point out that the models discussed in these lectures share such features, despite they have been inspired by quite different physical and biological problems. Along these lectures we describe also some of the technical tools employed for the study of such models, e.g., Lyapunov stability analysis, unpredictability indicators for “stable chaos,” hydrodynamic description of transport in low spatial dimension, spectral decomposition of stochastic dynamics on directed networks, etc.

Quantum diffusion with drift and the Einstein relation. I
View Description Hide DescriptionWe study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a nonequilibrium steady state describing a nonzero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.

Quantum diffusion with drift and the Einstein relation. II
View Description Hide DescriptionThis paper is a companion to Paper I [W. De Roeck, J. Fröhlich, and K. Schnelli, “Quantum diffusion with drift and the Einstein relation. I,” J. Math. Phys.55, 075206 (2014)]. The purpose of this paper is to describe and prove a certain number of technical results used in Paper I, but not proven there. Both papers concern longtime properties (diffusion, drift) of the motion of a driven quantum particle coupled to an array of thermal reservoirs. The main technical results derived in the present paper are: (1) an asymptotic perturbation theory applicable for small driving force, and (2) the construction of timedependent correlation functions of particle observables.

Random paths and current fluctuations in nonequilibrium statistical mechanics
View Description Hide DescriptionAn overview is given of recent advances in nonequilibrium statistical mechanics about the statistics of random paths and current fluctuations. Although statistics is carried out in space for equilibrium statistical mechanics, statistics is considered in time or spacetime for nonequilibrium systems. In this approach, relationships have been established between nonequilibrium properties such as the transport coefficients, the thermodynamic entropy production, or the affinities, and quantities characterizing the microscopic Hamiltonian dynamics and the chaos or fluctuations it may generate. This overview presents results for classical systems in the escaperate formalism, stochastic processes, and open quantum systems.

Bose gases, Bose–Einstein condensation, and the Bogoliubov approximation
View Description Hide DescriptionWe review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum manybody systems. We focus, in particular, on the excitation spectrum of a Bose gas in the meanfield (Hartree) limit. A list of open problems will be discussed at the end.

A note on the Landauer principle in quantum statistical mechanics
View Description Hide DescriptionThe Landauer principle asserts that the energy cost of erasure of one bit of information by the action of a thermal reservoir in equilibrium at temperature T is never less than k B T log 2. We discuss Landauer's principle for quantum statistical models describing a finite level quantum system coupled to an infinitely extended thermal reservoir . Using Araki's perturbation theory of KMS states and the AvronElgart adiabatic theorem we prove, under a natural ergodicity assumption on the joint system , that Landauer's bound saturates for adiabatically switched interactions. The recent work [Reeb, D. and Wolf M. M., “(Im)proving Landauer's principle,” preprint arXiv:1306.4352v2 (2013)] on the subject is discussed and compared.

Quantum correlations and distinguishability of quantum states
View Description Hide DescriptionA survey of various concepts in quantum information is given, with a main emphasis on the distinguishability of quantum states and quantum correlations. Covered topics include generalized and least square measurements, state discrimination, quantum relative entropies, the Bures distance on the set of quantum states, the quantum Fisher information, the quantum Chernoff bound, bipartite entanglement, the quantum discord, and geometrical measures of quantum correlations. The article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them, and for mathematicians looking for an uptodate introductory course on these subjects, which are mainly developed in the physics literature.
