Laurent Bruneau (Cergy): Repeated interaction quantum systems

A quantum system S interacts in a successive way with the elements E of a chain of identical and independant quantum subsystems. Each interaction lasts for a duration $\tau$ and is governed by a fixed coupling between S and E. We show that, in the limit of large times, the system approaches an asymptotic state which depends on the initial state of the chain but not of the system S. If the chain is initially at thermal equilibrium, then the asymptotic state satisfies an average 2nd law of thermodynamics.

Francois Castella (Rennes): Partial results on the derivation of the nonlinear quantum Boltzmann equations from large quantum systems of interacting particles

In this talk, we describe the problem of deriving a Boltzmann equation for a system of N interacting quantum particles, under the appropriate scaling limits. We mainly follow the approach developped by the authors in previous works. From a rigorous viewpoint, only partial results are available, even for short times, so that the complete problem is still open.
Joint work with D. Benedetto (Rome), R. Esposito (L'Aquila), M. Pulvirenti (Rome).

Wojciech De Roeck (Leuven): Quantum fluctuations: From Hamiltonian dynamics to unravelings of master equations

We present a rigorous scheme to calculate quantum fluctuations. One starts out with a hamiltonian dynamics for a small system coupled to reservoirs and one performs the weak coupling limit. As is well known since the work of E.B. Davies, the evolution of the small system in that limit is governed by a quantum master equation. One can however ask questions which are not answered by the solution of this master equation. Two such questions are : 1) Assume we are modelling a system in contact with reservoirs at different temperatures. What is the statistics of the heat currents? 2) Assume we are modelling an atom driven by a electromagnetic source. What is the statistics of emitted photons? On a heuristic level, one can (and in the case of question 2, physicists do) answer these questions by the technique of unravelings of master equations. Our aim is to derive this technique of unraveling from the original microscopic model by considering limits of reservoir fluctuations. Based on joint work with C. Maes and J. Derezinski.

Davide Gabrielli (L'Aquila): Current fluctuations in stochastic lattice gases

I will discuss current fluctuations in lattice gases in the hydrodynamic scaling limit. More precisely, I will illustrate a large deviation principle for the empirical current with a rate functional I(j) for a space-time fluctuation j. It is possible to estimate the probability of a fluctuation of the average current over a large time interval, by solving a variational problem for the functional I. We discuss several possible scenarios, interpreted as dynamical phase transitions, for this variational problem. They actually occur in specific models. This is a joint work with L. Bertini, A. De Sole, G. Jona-Lasinio, C. Landim.

Francois Germinet (Cergy): Kubo Formula and random Schrodinger operators

We review recent results obtained in collaboration with J.-M. Bouclet, A. Klein and J. Schenker. An expression of the conductance is obtained for unbounded Schrodinger operators, justifying the linear response theory in this case. Furthermore, we recover the well known expression of the Hall conductance when the Fermi energy lies in a region of (dynamical) localization.

Martin Hairer (Warwick): Ergodic theory for a class of non-Markovian processes

Most results on the long-time behaviour of systems perturbed by noise rely in a crucial way on the Markov property. We are going to present a general theory allowing to obtain ergodicity results for systems driven by non-Markovian noise. In particular, we will apply it to stochastic differential equations driven by fractional Brownian motion, where the driving process has long-range correlations.

Christian Maes (Leuven): Nonequilibrium fluctuations

All statistical mechanics starts from a fluctuation theory. The very notion of fluctuation is however depending on purpose and on observational context. The fluctuations can be static or dynamic, and they can involve time-symmetric or time-antisymmetric quantities. We discuss the relevant differences and how these relate to notions such as entropy production and escape rates.