The chaotic behaviour of classical non-linear systems is an ubiquitous effect which refers to the unpredictability of their complex dynamics, a phenomenon known as the butterfly effect. 

In mathematical terms this concept is characterized by the so-called Lyapunov exponent, a quantity which expresses the rate at which trajectories which start from close-by initial conditions (position and velocity), diverge with time. In quantum mechanics it is difficult to find good diagnostics of chaos and this question has been intensively debated since the beginnings of quantum mechanics. Recently, groundbreaking results from high energy physics [1] showed that quantum mechanics imposes a strict bound on the Lyapunov exponent.

This novel effect has motivated enormous interest over different fields, leaving, however, several outstanding open questions. In our work [2] we show that this bound can be regarded as a consequence of the quantum fluctuation-dissipation theorem, a fundamental result of statistical mechanics which connects the linear response to a perturbation with the equilibrium thermal fluctuations. These findings establish a direct connection between quantum chaos and other thermodynamic properties. They could give rise to new applications to different problems. 

In mathematical terms this concept is characterized by the so-called Lyapunov exponent, a quantity which expresses the rate at which trajectories which start from close-by initial conditions (position and velocity), diverge with time. In quantum mechanics it is difficult to find good diagnostics of chaos and this question has been intensively debated since the beginnings of quantum mechanics. Recently, groundbreaking results from high energy physics [1] showed that quantum mechanics imposes a strict bound on the Lyapunov exponent. This novel effect has motivated enormous interest over different fields, leaving, however, several outstanding open questions. In our work [2] we show that this bound can be regarded as a consequence of the quantum fluctuation-dissipation theorem, a fundamental result of statistical mechanics which connects the linear response to a perturbation with the equilibrium thermal fluctuations. These findings establish a direct connection between quantum chaos and other thermodynamic properties. They could give rise to new applications to different problems.

[1] J. Maldacena, S. H. Shenker and D. Stanford, Journal of High Energy Physics 2016(8) (2016).
[2] S. Pappalardi, L. Foini and J. Kurchan, SciPost Phys. 12, 130 (2022).

The next colloquium Rencontres de l'IPhT conference will be held in Autrans (Vercors) from 23 to 25 May. It will consist of conferences and lectures concerning both the major fields of research in theoretical physics and supporting activities.

The main themes of physics will be:
- Mathematical physics: group A
- Cosmology, particles and nuclei: group B
- Statistical physics - condensed matter: group C

Main goals:
- To share knowledge on scientific topics and strengthening cohesion within the Institute.
- To bring together all of the Institute's collaborators in order to discuss common subjects and activities and to strengthen interactions between members of the different teams, including support.
- To unite the teams around a common project.

See you soon in Autrans!

Given a set of deformable polygons with prescribed numbers of sides, in how many inequivalent ways can one glue them edge to edge so as to build a sphere? This mathematical question, which pertains to map combinatorics in random geometry, was partially answered by the mathematician William Tutte in a famous 1962 article "A census of slicings" where he solves the problem in the particular case where all the polygons, except at most two of them, have an even number of sides.

In a recent article, Jérémie Bouttier and Emmanuel Guitter (IPhT), in collaboration with the mathematician Grégory Miermont (ENS de Lyon), succeeded, 60 years later, in extending Tutte's formula to the most general case, namely for polygons whose numbers of sides have arbitrary parities. Even though more complex, their general explicit formula is still very elegant and conceals numerous symmetries. It is with no doubt the starting point for new combinatorial discoveries.

The image shows the 8 ways to assemble two 1-sided and two 3-sided polygons to build a sphere.

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