26.01.2022 - 28.01.2022

**http://pillet.univ-tln.fr/nonstops.html**

**Before attending the workshop starting on 26th January, you are kindly invited to participate in Prof Jaksic's guest lecture on Tuesday 25th January on:**

Shannon’s entropy, universality, and beyond

Abstract: I will review the concept of entropy as it appeared in Shannon’s foundational work on information theory, and discuss its universal aspects.

Date : **25 January 2022** from **12:30 **to** 14:00**

The video of **Prof Jaksic’s guest lecture** is available on the CY AS YouTube channel:

https://www.youtube.com/channel/UCv5Kk9SvqYdx4ZnEcKJPg9Q

**The workshop schedule :**

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**Wednesday the 26th**

12.00-13.30: Sandwiches

13.45-14.00 : Opening

14.00-15.00 : Noé Cunéo, *Repeated quantum measurement processes: time reversal and singularities*

15.00-15.30 : Coffee break

15.30-16-30 : Clément Tauber, *Approach to equilibrium in translation invariant quantum systems*

16.30-17.30 : Felix Rico, *Single molecule mechanics using high-speed atomic force microscopy*

Kindly find below the link to download the recording of the **first** day of workshop:

https://cyu-fr.zoom.us/rec/share/q4tokfWf9d2aLsuR3jcvpYAM6tPcmKv77wVoL-0w27Iygdgv3RMAaDaQ14R3NV4g.NtddfzUezBq4Rh3A

Passcode: $o4i9*HC

*The link will be available until 4 ^{th} February 2022*

**Thursday the 27th**

09.30-10.30 : Francesco Ticozzi, *Model reduction for Quantum and Classical Hidden Markov Dynamics*

10.30-11.00 : Coffee break

11.00-12.00 : Cécilia Lancien, *Spectrum of random quantum channels*

12.00-14.30 : Lunch

14.30-15.30 : Ana Djurdjevac, *Controllability for the (multi-dimensional) Burgers equation with localized one-dimensional control*

15.30-16.00 : Coffee break

16.00-17-00 : Mirko Degli Esposti, *Entropy and Language (natural and artificial)*

17.00-18.00 : Renaud Raquépas, *A program on limit theorems for entropy-reproducing random variables*

Kindly find below the link to download the recording of the **second** day of workshop:

Passcode di accesso: &96=jhbU

*The link will be available until 4 ^{th} February 2022*

**Friday the 28th**

09.30-10.30 : Giampaolo Cristadoro, *Symmetry and structure in genetic sequences*

10.30-11.00 : Coffee break

11.00-12.00 : Tristan Benoist, *A model for the appearance of tracks in particle detectors*

12.00-12 .15 : Closure

Kindly find below the link to download the recording of the **third** day of workshop:

Passcode di accesso: X9^Er^2#

*The link will be available until 4 ^{th} February 2022*

**Titles and abstracts :**

**Tristan Benoist : ***A model for the appearance of tracks in particle detectors*

While Heisenbergs's indefiniteness principle prevents quantum particles to be assigned definite position and momentum at a given time, their tracks in particle detectors such as bubble or wire chambers are close to classical deterministic path with random definite initial position and momentum. Solving this apparent contradiction is a long standing problem in quantum mechanics.

In a recent article [BBFF] and an ongoing work [BFF], with M. Ballesteros, M. Fraas and J. Fröhlich, we approach the question from a repeated measurements point of view. We model the particle detectors using repeated approximate measurements of the particle position. In between each measurement, the particle is considered isolated, thus its state evolves unitarily. In [BBFF] we study Gaussian approximate position measurements of particles driven by quadratic Hamiltonians. Under these assumptions we can carry out explicit enough computations showing that the approximate position measurement results form a random process close, in some sense, to the classical deterministic path of a particle with random initial definite position and momentum.

In [BFF] we generalize the model, considering arbitrary approximate position measurements and unitary dynamics that approximate classical dynamics in some appropriate semi-classical scaling. We typically consider heavy particle detected by light ones. Using Egorov's type theorems, we find again that the approximate position measurement results form a process close to a classical deterministic path of a particle with random initial definite position and momentum. For linear dynamics and translation invariant approximate measurements, we construct consistent estimators of this initial particle data.

In my talk I will present this work and our most recent results on the subject.

[BBFF] M. Ballesteros, T. Benoist, M. Fraas, and J. Fröhlich, "The appearance of particle tracks in detectors'', Commun. Math. Phys. 385 (2021) 429 – 463 (arXiv:2007.00785)

[BFF] T. Benoist, M. Fraas, and J. Fröhlich, in preparation.

**Giampaolo Cristadoro : ** *Symmetry and structure in genetic sequences*

Genetic sequences are known to possess non-random structures at different scales together with symmetries in the frequencies of their components. These features and their genesis are commonly investigated separately in the literature. At contrary, I will present evidence for a common origin for both symmetry and structure. Such unification unravels the existence of previously unknown additional symmetries, which are organized hierarchically through scales in which non-random structures are known to be present. Finally, I will propose a biologically-motivated dynamics for the evolution of symbolic sequences. In particular, the dynamical model shows that a metastable (long-lived) regime emerges in which the evolved sequences have symmetries and structure interlaced in a way that matches that of extant genomes.

**Noé Cuneo : ** *Repeated quantum measurement processes: time reversal and singularities*

After a short mathematical introduction to repeated quantum measurement processes, we will focus on the probability distribution of the sequence of measurement outcomes. We will limit ourselves to the case where all measurements take values in a finite set. As we shall see through a series of examples, the resulting distributions range from familiar (i.i.d., Markov, ...) measures to highly singular, non-Gibbsian ones. We will explore some of the singularities from the point of view of entropy production.

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**Mirko Degli Esposti : ***Entropy and Language (natural and artificial)*

Just a review of the more or less recent empirical entropy indicators used in several applications in natural language processing:

their mathematical definitions, their fluctuations (when known), their use in concrete scenarios and, mostly important, some future developments.

**Ana Durdevac : ***Controllability for the (multi-dimensional) Burgers equation with localized one-dimensional control*

We will consider the viscous Burgers equation driven by a localized one-dimensional control. The problem is considered in a bounded domain and issupplemented with the Dirichlet boundary condition. We will prove that any solution of the equation in question can be exponentially stabilized. Combining this result with an earlier result on local exact controllability we will show global exact controllability by a localized control. Furthermore, we will comment how these results can be utilized in showing a mixing property for the Markov process associated to the Burgers equation perturbed by a stochastic forcing. This is a joint work with A. Shirikyan and T. Cornelis.

**Cecilia Lancien : ** *Spectrum of random quantum channels*

The main question that we will investigate in this talk is the following: what does the spectrum of a quantum channel typically look like? We will see that various natural models of random quantum channels generically exhibit a large spectral gap, between their first and second largest eigenvalues. This is in tight analogy with what is observed for the spectral gap of transition matrices associated to random graphs. In both the classical and the quantum settings, establishing results of this kind is interesting as it has important consequences regarding the speed of convergence to equilibrium of the corresponding dynamics. We will also present implications of the quantum result in terms of typical decay of correlations in so-called matrix product states (which are used to describe the states of 1D many-body quantum systems with local interactions, appearing for instance in quantum condensed matter physics).

**Renaud Raquepas : ***A program on limit theorems for entropy-reproducing random variables*

* *In this informal talk, I will discuss a direction of research that we have been pursuing with Giampaolo, Mirko and Vojkan. To set the stage, think of the content of the Shannon–McMillan–Breiman theorem: if a measure

**Felix Rico : ***Single molecule mechanics using high-speed atomic force microscopy*

High-speed atomic force microscopy (HS-AFM) is a unique technology allowing sub-second, nanometric imaging of biological samples [1]. We have adapted HS-AFM to perform high-speed force spectroscopy (HS-FS) to probe single molecule processes with microsecond time resolution. This allows, for example, probing single protein unfolding at time scales reachable in all-atom molecular dynamics simulations. In this talk, I will describe the characteristics of HS-AFM and will show some applications [2–6]. HS-FS opens an avenue to better probe stochastic processes at the shortest timescales, and to address long standing and emerging questions.

1. T. Ando, N. Kodera, E. Takai, D. Maruyama, K. Saito, and A. Toda, "A high-speed atomic force microscope for studying biological macromolecules," Proceedings of the National Academy of Sciences **98**, 12468–12472 (2001).

2. F. Rico, L. Gonzalez, I. Casuso, M. Puig-Vidal, and S. Scheuring, "High-Speed Force Spectroscopy Unfolds Titin at the Velocity of Molecular Dynamics Simulations," Science **342**, 741–743 (2013).

3. H. Takahashi, F. Rico, C. Chipot, and S. Scheuring, "α-Helix Unwinding as Force Buffer in Spectrins," ACS Nano **12**, 2719–2727 (2018).

4. F. Rico, A. Russek, L. González, H. Grubmüller, and S. Scheuring, "Heterogeneous and rate-dependent streptavidin–biotin unbinding revealed by high-speed force spectroscopy and atomistic simulations," PNAS **116**, 6594–6601 (2019).

5. C. Valotteau, F. Sumbul, and F. Rico, "High-speed force spectroscopy: microsecond force measurements using ultrashort cantilevers," Biophys Rev (2019).

6. A. Rigato, A. Miyagi, S. Scheuring, and F. Rico, "High-frequency microrheology reveals cytoskeleton dynamics in living cells," Nat Phys **13**, 771–775 (2017).

**Clément Tauber :** *Approach to equilibrium in translation invariant quantum systems*

I will discuss some mathematical aspects of out-of-equilibrium quantum systems on a lattice, spin or fermions. The main question is : what happens to an equilibrium state when driven out of equilibrium by an infinite-volume but translation-invariant perturbation ? I will briefly review the C*-algebraic framework and then revisit some old results by Lanford and Robinson about quasi-free dynamics. Then I will present some structural results about the evolution of mean energy and entropy for a broader class of systems. The main result is a strict increase of entropy. I will finally mention several perspectives to tackle the general question that remains mainly open. This talk is based on a work in progress with Vojkan and Claude-Alain

**Francesco Ticozzi : ***Model reduction for Quantum and Classical Hidden Markov Dynamics*

A common framework can be used to describe a wide variety of dynamics of interest for quantum walks, algorithms, and open systems, as well as classical hidden-Markov models (HMM). All these can be written as a semigroup completely-positive trace-preserving dynamics (in discrete time for this talk), whose output of interest is a family of linear functionals of the state. We then leverage on linear system theory and the algebraic approach to realize a minimal such model whose output matches a target one. In doing this, we highlight the minimal resources needed to simulate a given process, probe its ``quantum-ness’’, and tackle an old open problem for HMM.

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