# LPTMS PhD Proposal: Self-assembly in space and time

Contact: Martin Lenz

Recent experimental developments have made assembling machines at the nanometer scales that mimic or even attempt to surpass the functions of biological objects an increasingly reasonable goal (see https://www.nobelprize.org/prizes/chemistry/2016/summary/). Despite remarkable progress in manufacturing individual nanometer-sized objects with controlled shapes however (see an example in the illustration), assembling many of them into larger structures remains an open challenge and an active field of research.

In this project we will undertake an additional challenge, namely to self-assemble such objects not only in space, but also in time. Specifically, we will explore the design principles for DNA origami

particles produced by our collaborator Seth Fraden (Brandeis University, USA) to assemble over a given sequence over time, which will allow for an actin-like treadmilling (coordinated polymerization from one end, depolymerization from the other) of a polymer-like structure under e.g., temperature cycling. Such mechanisms could be key in controlling the motor action of prospective molecular machines.

In a second stage you may develop simulations tools to optimize particle shapes for self-assembly of printed particles produced at PMMH in collaboration with Julien Heuvingh and Olivia du Roure. The successful applicant will have a taste for numerical simulations and working with experimentalists.

# LPTMS PhD proposal : Localization in open quantum systems

Contacts: Alberto Rosso (LPTMS) and Laura Foini (IphT)

Understanding how a many-body quantum system thermalises and when, at the opposite, it keeps memory of the initial preparation is an extraordinary challenge which has attracted enormous attention.

Nowadays, most of the efforts focus on closed systems where the competition between disorder and interactions leads either to thermalization or many body localisation (MBL). In this context the presence of an external bath is believed to induce always thermalisation and destroy any fingerprint of localisation. This is in general not true. The goal of this project (an internship that can lead to a thesis) is to study localisation effects in open systems (e.g. in interaction with a thermal bath and eventually a drive). Two directions will be investigated:

▪ The quench of a many-body system prepared in a state out-of equilibrium and let evolve in a bath of harmonic oscillators

▪ The stationary state of a system in contact with a thermal bath and driven out of equilibrium by irradiation.

In the first case we will focus on non-perturbative effects induced by the strong coupling with the bath.

In the second example we are interested in the nature of the stationary state using a weak coupling Lindblad approach. The work is both numerical and analytical and has strong connections with NMR experiments.

# LPTMS PhD Proposal: inhomogeneous systems out of equilibrium

### Responsable: Maurizio FAGOTTI + 33 (0)1 69 15 32 64

A fundamental concept in statistical physics is that the equilibrium properties of systems with a huge number of degrees of freedom can be described by few parameters, first and foremost the temperature. The latter can be tuned to modify the physical properties, and even the forms in which matter manifests itself, so-called phases of matter (e.g. solid, liquid, etc.). This generally requires a global control of the system, but there are also situations in which a local perturbation is sufficient to induce a phase transition. For example, pure water can be supercooled below its normal freezing point, remaining liquid; it is then sufficient to put the liquid in contact with a small piece of ice to induce global freezing.

When the system is not at equilibrium, its description becomes more complicated; nevertheless, a statistical description was shown to emerge when a quantum many-body system, isolated from the rest, is left to evolve for a long time. Being isolated, the system can not relax to an equilibrium state, but, when scrutinised locally, it appears as if it were prepared at an effective temperature or in some exotic state of matter. Arguably, the best understood situation is a quantum quench of a global parameter in a translationally invariant quantum many-body system.

In this thesis we will go beyond the assumption of translational invariance, studying the effects of inhomogeneities on the nonequilibrium dynamics after quantum quenches.

To apply, please refer to http://lptms.u-psud.fr/maurizio-fagotti/jobs/

# LPTMS PhD Proposal: Models and Time Series Analysis for Human Sports Performance

### Responsable: Thorsten Emig + 33 (0)1 69 15 31 80

This project is directed to students with a strong background in quantitative methods from statistical physics, and ideally some knowledge of machine learning, computational physiology and statistical analysis of large data. Interest in sports performance would be useful. Expected are both analytical and computer programming

skills.

Models for human sports performances of various complexities and underlying principles have been proposed, often combining data from world record performances and bio-energetic facts of human physiology. For running, we were the first to derive an observed logarithmic scaling between world record running speeds and times from basic principles of metabolic power supply. We showed that various female and male record performances (world, national) and also personal best performances of individual runners for distances from 800m to the marathon are excellently described by our approach, with mean errors of (often much) less than 1%.

Main goal of this thesis project is the data-driven modeling of physiological and biomechanical processes in endurance sports, in particular running. The physiological and mechanical response of humans to exercise constitutes a complex system that involves many dynamical variables. Examples are the beat-to-beat intervals between heart beats, oxygen uptake, and stride frequency to name a few. These variables show inherent fluctuations that can be correlated.

Time series analysis can be used to detect these correlations which can show fractal scaling. This has been demonstrated for patients with cardiac diseases by Goldberger (see references below). Methods include detrended fluctuation analysis (DFA), multifractal DFA, EMD, multiscale entropy, and transfer entropy.

Models for complex physiological systems shall be constructed by learning from data. For example, running performance has been studied using recent advances in machine learning (see reference by Blythe and Kiraly). One aspect of this project is to apply machine learning to complex physiological data for endurance exercise and compare the so obtained results to findings from other methods.

This project potentially involves collaborations with Prof. A. Goldberger (Harvard Medical School) and Prof. E. Räsänen (TUT, Finland).

The official application can be found on the web site of Ecole Doctorale at https://www.edpif.org/fr/recrutement/prop.php

You can also contact me directly at thorsten.emig@u-psud.fr or at 01.69.15.31.80.

# LPTMS PhD Proposal: Mean field games

### Responsable: Denis ULLMO + 33 (0)1 69 15 74 76

Mean field games present a new area of research at the boundary between applied mathematics, social sciences, engineering sciences and physics. It has been initiated a decade ago by Pierre-Louis Lions (recipient of the 94 Fields medal) and Jean-Michel Lasry as a new and promising tool to study many problem of social sciences, and with an explicit mention of the influence of concepts coming from physics (the notion of “mean field approximation”). This field has since then grown significantly, and after a period where mainly stylized models where introduced, we witness now the appearance of (necessarily more involved) mean field game models closer to practical applications in finance, vaccination policies, or energy management through smart electronics.

Up to now, the development of Mean Field Games has mainly originated from the mathematics and economic communities. Mean Field Games theory is, however, by essence a multi-disciplinary field for which the input of physicists is much needed. Indeed, as important as they are, the studies of internal consistency and the numerical schemes developed by mathematicians cannot replace the deeper

understanding of the behavior of these models, obtained in particular through powerful approximation schemes, that physicists (and essentially only them) know how to provide.

For physicists a good “entry point” to the problematic of Mean Field Games is through the formal, but deep, connection between Mean Field Games and the nonlinear Schroedinger (or Gross-Pitaevskii) equation. This connection makes it possible to import to the field of Mean Field Games a variety of tools (ranging from exact methods and approximation schemes to intuitive qualitative descriptions) which have been developed along the year by physicists when studying interacting bosons or gravity waves in inviscid fluids.

The general subject of the proposed thesis is the study of Mean Field Games from a physicist point of view, that is with an objective to provide a true understanding (through the identification of the relevant parameters and scale and the development of approximation schemes in the regimes of interest) of the solutions of Mean Field Games equations. More specifically, two possible directions the proposed PhD could take would be:

1. The study of phase transition in Mean Filed games.

2. To use the knowledge obtained on simple models to study more complicated Mean Field Games, and in particular address more realistic (less stylized) Mean Field Games.

These studies should imply a mix between analytical and numerical works, somewhat more shifted on the analytical side.

# LPTMS PhD Proposal: Exclusion statistics and lattice random walks

### Responsable: OUVRY Stéphane + 33 (0)1 69 15 36 30

Thesis proposal :Recently [1] a formula for the algebraic area enumeration of closed random walks on a square lattice has been obtained from the Kreft coefficients which encode the Schrodinger equation of the quantum Hofstadter model.

The Hofstadter model (a charged particle hopping on a square lattice coupled to a perpendicular magnetic field) has a spectrum which is a rare example of a quantum fractal. It happens to be related to closed random walks on a square lattice via a mapping between the n-th moment of the Hofstadter Hamiltonian and the generating function for the enumeration of close lattice walks making n steps and enclosing a given algebraic area. More recently [2] the algebraic area enumeration was generalized to a wider class of random walks and lattices by recognizing the underlying role of exclusion statistics in the enumeration. Several key observations both in [1] and [2] happen to be still incompletely understood and not yet seated on solid mathematical grounds. The enumeration itself has a complexity which increases exponentially with n making it difficult to be used for walks with a large number of steps. The thesis will focus on a better understanding and improving of [1] and [2], in particular simplifying the formula to make it more tractable for large n. Also the investigation of various lattices and random walks will be pushed forward.

[1] S. Ouvry and S. Wu, «The algebraic area of closed lattice random walks » arXiv:1810.04098

[2] S. Ouvry and A. Polychronakos, «Exclusion statistics and lattice random walks » arXiv:1908.00990