# LPTMS PhD Proposal: Non stationary avalanches in disordered systems

### Responsable: Alberto ROSSO 01 69 15 31 79

In many macroscopic systems the response to a constant energy input can be strongly non-linear: stress slowly accumulates over time up to a sudden and unpredictable energy release is induced by an extended reorganisations called avalanches. Economical crisis, earthquakes or epidemic outbreaks are examples of avalanche dynamics. Also the intermittent flow of soft materials, like mayonnaise or foams, can be understood in terms of avalanches. For almost 20 years, there have been many attempts to understand avalanches in the framework of critical and collective phenomena. An important result has been to show that avalanches display universal statistics. However we now know that the physics behind these latter phenomena is much richer than their equilibrium counterpart. In particular avalanches have a strong memory of the story of the material. As remarquable examples one can cite the aftershocks after an earthquake or the occurrence of a macroscopic failure in well aged materials. In this thesis we will focus on avalanches produced using non-stationary protocols in connection with experimental observations and using the tools developed for disordered systems.

# LPTMS PhD Proposal: Phase transition in Mean Field Games

### Responsable: Denis ULLMO 01 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.

In this general context, the goal of this internship will be to study "phase transitions" in MFG, that is a discontinuous changes of behavior as a parameter is varied. During the internship, this study will be limited to a class of Mean Field Games for which there exist a formal, but deep, connection with the non-linear Schrödinger equation, which is making their analysis, and in particular the origin of these phase transitions, more transparent.

# LPTMS PhD Proposal: Emergence of fibers in frustrated self-assembly

### Responsable: Martin Lenz 01 69 15 32 62

Self-organization is key to the function of living cells – but sometimes goes wrong! In Alzheimer’s

and many other diseases, normally soluble proteins thus clump up into pathological fiber-like

aggregates. While biologists typically explain this on the grounds of detailed molecular interactions, we

have started proving that such fibers are actually expected from very general physical principles. We

thus show that geometrical frustration builds up when mismatched objects self-assemble, and

leads to non-trivial aggregate morphologies, including fibers.

Despite several examples of this in our numerical simulations (see illustration), we have yet to

better understand the underlying physics. Is fiber formation based on a well-defined phase transition? Is

this transition fundamentally out of equilibrium as some of our results suggest? To what extent can it be

mapped onto the standard geometrical understanding of frustration as the embedding of a manifold into

a space with an incompatible metric? We will tackle these questions using two minimal models of

frustration where we hope to combine analytical and numerical insights. One of these is akin to

historical descriptions of Josephson junctions, where lattice-based Heisenberg spins want to realize a

certain fixed mismatch in their alignments between neighboring sites.

This project offers opportunities for collaborations with the theoretical group of Gregory Grason at

U. Mass. Amherst (USA), as well as with several groups that are currently initiating experiments on

frustration-driven fiber formation.

Informal inquiries welcome.