December 20, 2022, 11:15
Dipartimento di Scienza e Alta Tecnologia
Università dell’Insubria
Via Valleggio 11
Como

## Program

- 11:15 –
**Annalisa Panati**
- 12:00 –
**Markus Lange**
- 12:45 – Lunch Break
- 14:45 –
**Andrea Mantile**
- 15:30 –
**Cristina Caraci**
- 16:15 – Closing

## Titles and Abstracts

**CARACI: Fluctuations of N-particle quantum dynamics around the Gross-Pitaevskii equation**

We consider the quantum dynamics of N interacting bosons in the Gross-Pitaevskii regime. We obtain a norm-approximation for the many-body evolution of initial states exhibiting Bose-Einstein condensation in terms of a unitary Fock space evolution with a quadratic generator for the fluctuations. In addition, using this result, we provide the proof of a central limit theorem for the fluctuations of bounded one-particle observables. This is a joint work with Jakob Oldenburg and Benjamin Schlein.

**MANTILE: Scattering theory with both regular and singular perturbations**

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of a scattering couple (A,B), where both A and B are self-adjoint operators and B formally corresponds to adding to A two terms, one regular and the other singular. This abstract construction applies to perturbations of the free Laplacian with a Kato-Rellich potential and a singular part modelling boundary or interface conditions at the boundary of a open, bounded Lipschitz domain. We will possibly discuss applications of these models to classical scattering problems.

**LANGE: Adiabatic Evolution of Low-Temperature Many-Body Systems**

We consider the evolution of thermal equilibrium state of finite-range, many-body fermionic lattice models after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. As an application, the expansion allows to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. The proof is based on a rigorous version of the Wick rotation.

**PANATI: Entropic Fluctuations in Quantum Two-time Measurement Framework**

Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, thank to the pioneering works of Evans, Cohen, Morris and Searles on the violation of the second law, soon followed by the ground-breaking formulation of the Fluctuation Theorem by Gallavotti and Cohen for entropy fluctuation in the early nineties. Their work was by vast literature, both theoretical and experimental. The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an undergoing effort. Kurchan’s seminal work (2000) showed the measurement role has to be taken in account, leading to the introduction of the so called two-time measurement statistics (also known as full counting statistics). However introducing this frameworks leads to surprising phenomena with no classical counterpart. In this talk, I will present some work in progress, where we attempt to introduce a quantum equivalent of Gallavotti-Cohen (steady) entropic functional and compare it with the Evans-Searls (transient) entropic functional. We show that, due to the invasive measurement role, the situation differs considerably to its classical counterpart.

Joint work with T. Benoist, L. Bruneau, V. Jakšic, C.A.Pillet.