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.