CY Initiative project funded by Investissements d'Avenir ANR-16-IDEX-0008

July 2023 — June 2027

Team members Publications Conferences and seminar Positions Abstract

Team
members

The project involves five regular members and five associated researchers:

__Regular members__: Vojkan Jakšić, Flora
Koukiou, Claude-Alain Pillet, Armen
Shirikyan (coordinator of the project), Michał
Wrochna

__Associate researchers__: Noé Cuneo, Vahagn Nersesyan,
Thi
Hien Nguyen, Annalisa Panati, Renaud Raquépas

Publications

The following papers were written with the financial support of the project:

Conferences and seminar

A five-day closing conference will be held in June 2027. Further information will be available in due time. The project partially supports the working group on mathematical physics and the following workshops:

Positions

Gaia Pozzoli was hired as a one-year postdoc starting from December 1, 2023. Another one-year position will be available in 2025.

Abstract of the project

The project involves five regular members and five associated researchers:

Publications

The following papers were written with the financial support of the project:

__N. Barnfield, R. Grondin, G. Pozzoli, R. Raquépas__,*On the Ziv-Merhav theorem beyond Markovianity,*Canadian Mathematical Journal, to appear.__T. Benoist, L. Bruneau, V. Jaksic, A. Panati, C.-A. Pillet__,*A note on two-times measurement entropy production and modular theory*, Letters in Mathematical Physics**114**(2024), article 32.__T. Benoist, L. Bruneau, V. Jaksic, A. Panati, C.-A. Pillet__,*On the thermodynamic limit of two-times measurement entropy production*, submitted.

__A. Djurdjevac, A. Shirikyan__,*Exponential stability of the flow for a generalised Burgers equation on a circle*, Contemporary Mathematics. Fundamental Directions**69**(2023), No. 4, 588–598.__N. Barnfield, R. Grondin, G. Pozzoli, R. Raquépas__,*On the Ziv–Merhav theorem beyond Markovianity II:**A thermodynamic approach*, arxiv:2312.02098.__J. Földes, A. Shirikyan__,*Rayleigh–Bénard convection with stochastic forcing localised near the bottom*, Journal of Dynamics and Differential Equations, published online.

Conferences and seminar

A five-day closing conference will be held in June 2027. Further information will be available in due time. The project partially supports the working group on mathematical physics and the following workshops:

- Entropies for complex processes (videos of the talks are available here)

- 16th conference of
the GDR DynQua
*Quantum Dynamics* - Quantissima in the Serenissima V
- Session
*Recent Progress in Statistical Mechanics*at the 2023 CMS Winter meeting

Positions

Gaia Pozzoli was hired as a one-year postdoc starting from December 1, 2023. Another one-year position will be available in 2025.

Abstract of the project

The
objective of the project is to contribute (through the
development of mathematical tools for the study of
information theory in physics and statistics) to
strategic research related to the problems of
extraction, processing and protection of classical and
quantum information. We shall deal with three different,
but closely related topics.

The first group of problems concerns the Lempel–Ziv coding algorithm whose performance studies have led to deep insights into specific and relative entropies of stationary measures on shift spaces. Notable among those is the characterisation of the specific entropy of a stochastic source in terms of the asymptotics of recurrence times of a typical signal and the specific cross entropy in terms of waiting times. We shall investigate in depth the mathematical theory of entropic estimators with emphasis on their fluctuations and fractal dimension theory of the level sets.

The second group concerns bipartite systems with infinitely many degrees of freedom. For those systems, it is not clear what is a good measure of entanglement. Recently, much progress has been made in the setting of quantum fields on flat space, and computations in simple models are starting to reveal mechanisms responsible for geometric formulae for entanglement entropy. Furthermore, there are proposals on how to give meaning to information for subsystems such as a single wave. Our goal is to develop more realistic models in relativistic physics, with applications to QFT on AdS spaces and the black hole information paradox.

The third group deals with fluid flows in a pipe. An important question from the point of view of applications is the estimation of viscosity using long-time observations of the kinetic energy and vorticity. This requires investigation of probabilistic limit theorems for the observables and the expression of resulting quantities in terms of viscosity. Our goal is to find reliable estimators and to investigate their optimality.

The first group of problems concerns the Lempel–Ziv coding algorithm whose performance studies have led to deep insights into specific and relative entropies of stationary measures on shift spaces. Notable among those is the characterisation of the specific entropy of a stochastic source in terms of the asymptotics of recurrence times of a typical signal and the specific cross entropy in terms of waiting times. We shall investigate in depth the mathematical theory of entropic estimators with emphasis on their fluctuations and fractal dimension theory of the level sets.

The second group concerns bipartite systems with infinitely many degrees of freedom. For those systems, it is not clear what is a good measure of entanglement. Recently, much progress has been made in the setting of quantum fields on flat space, and computations in simple models are starting to reveal mechanisms responsible for geometric formulae for entanglement entropy. Furthermore, there are proposals on how to give meaning to information for subsystems such as a single wave. Our goal is to develop more realistic models in relativistic physics, with applications to QFT on AdS spaces and the black hole information paradox.

The third group deals with fluid flows in a pipe. An important question from the point of view of applications is the estimation of viscosity using long-time observations of the kinetic energy and vorticity. This requires investigation of probabilistic limit theorems for the observables and the expression of resulting quantities in terms of viscosity. Our goal is to find reliable estimators and to investigate their optimality.