Session Y44: DMFT for Classical Systems of Interacting Objects

On the memory of the initial state during gradient descent dynamics in random energy landscapes

The weak ergodicity breaking hypothesis postulates that, after a quench from high to low temperature, out-of-equilibrium glassy systems lose memory of their initial state despite being unable to reach an equilibrium stationary state. It is a milestone of glass physics, and has provided a lot of insight on the physical properties of glass aging. Despite its undoubted usefulness as a guiding principle, its general validity remains a subject of debate. I will discuss evidence that this hypothesis does not hold in the special case of gradient descent dynamics in a specific class of random energy landscapes, corresponding to mean-field spin glass models. I will discuss how this impacts our understanding of gradient descent dynamics in more general disordered landscapes, such as glassy energy landscape or cost functions in random optimization problems.   

Regaining physical intuition on the infinite-dimensional limit of driven amorphous materials

Amorphous materials are ubiquitous around us: emulsions as mayonnaise, foams, sandpiles or biological tissues are all structurally disordered, and this has key implications for their mechanical, rheological and transport properties. A minimal model for amorphous materials, which allows to focus generically on the key role of this disorder, is provided by dense systems of pairwise-interacting particles. The limit of infinite spatial dimension then plays a very special role: it uniquely provides exact analytical benchmarks, otherwise scarce, for static and dynamic features of such many-body systems. Those include for instance the critical scalings in the vicinity of the jamming transition, the stress-strain curve of glasses under quasistatic shear, or their equilibrium phase diagram depending on their temperature and packing fraction. In the last couple of years, we also derived an exact set of equations for the out-of-equilibrium ‘dynamical mean-field theory’ (DMFT) of these models. These pave the way to a dynamical understanding of previous static results, and more importantly, towards a characterization of the out-of-equilibrium properties of driven amorphous materials.

Yet, the limit of infinite spatial dimension remains quite abstract. What is the relevance of these benchmarks for our two- or three- dimensional physical world? We have in fact some freedom, when going to arbitrary large dimension, on how to choose to generalize the pairwise interaction potential. What are the implications on their corresponding DMFT, and what is special about the specific choice which is usually made? Furthermore, the corresponding analytical characterization remains a tour de force, already at a static level, rending these approaches quite hermetic especially for non-experts. Here my aim will be to provide a more intuitive and compact way to understand this special limit and the above related issues. I will in particular discuss the case of dense active matter, where particles have their own local drive, and how they compare to dense amorphous materials under global shear.   

Searching for solutions to the dynamical mean-field theory of glasses by simulating minimally structured glass formers

The glass problem is fundamentally one of dynamics. The recently formulated dynamical mean-field theory (DMFT) of glasses, which is exact in the infinite-dimensional limit d→∞, therefore has the potential to answer several questions about jamming as well as about the non-equilibrium dynamics and rheology of glasses. As yet, however, solutions of the DMFT equations can only be obtained within a narrow range of equilibrium and low-density conditions. Fundamental questions about the nature of jamming and relaxation thus hinge on our understanding of low d systems, which when extrapolated to d→∞, should help distinguish mean-field physics from activated and other non-perturbative processes. Because the rich structure of low-d liquids interferes with such extrapolations, we here turn to two minimally structured glass formers whose descriptions converge to the DMFT in the infinite dimensional limit: the random Lorentz gas (RLG) and hard spheres with the Mari-Kurchan interaction (MK). Surprisingly, extrapolation differences persist between the two models, most notably in the values of the jamming density. Resolving the origin of such discrepancies is essential to understanding the structure of the glass landscape, in particular, and the solutions of the DMFT more generally.   

Dynamical mean field theory of learning

Training algorithms are key to the success of artificial and recurrent neural networks. However we know very little about them. The main example of these algorithms is stochastic gradient descent which is the workhorse of the deep learning technology. I will review how dynamical mean field theory can be used to gain some understanding of training algorithms in prototypical settings.   

Dynamical Mean-Field Theory: from glassy systems to large ecological communities with asymmetric interactions

The emergence of numerous metastable states, slow relaxation dynamics, and aging are hallmarks of glassiness. By focusing on systems with soft disordered interactions, I will show how to incorporate aging within Dynamical Mean-Field Theory and deduce the main quantities of interest – such as effective temperatures, correlations, and responses –  in the slow regime. I will present both the case with only one slow timescale and that with an infinite hierarchy of slower and slower timescales. Remarkably, for the latter, aging dynamics is governed by a self-consistent stochastic equation, which can be shown to precisely correspond to the ultrametric protocol proposed by Parisi for the static counterpart.

These findings extend the realm of out-of-equilibrium mean-field scenarios to a broader spectrum of contexts where DMFT is applicable.

As a final highlight, I will discuss a promising approach to theoretical ecology aiming to capture the complexity of large ecosystems. I will then stick to a Generalized Lotka-Volterra model with random interactions between species and finite demographic fluctuations.