(Mean-Field) Disorder Dynamics in Driven Amorphous Materials

Amorphous materials are ubiquitous around us : emulsions as mayonnaise, foams, sandpiles or biological tissues are all structurally disordered. This has key implications for their mechanical, rheological and transport properties : statistically distributed from sample to sample, they stem from a microscopic disorder which depends on how the material has been prepared and on its subsequent deformation history. Understanding how to characterise this disorder and what are its experimental implications is thus paramount to harness –and even engineer– specific disorder-induced properties. Nevertheless, despite decades of extensive analytical and computational studies, theoretical descriptions of such ‘driven’ amorphous materials remain challenging.

A minimal model for amorphous materials, which allows to focus generically on the key role of their structural (positional) disorder, is provided by dense many-body 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 features of amorphous materials. Here I will introduce a ‘dynamical mean-field theory’ (DMFT) of these models, highlighting the physical ingredients that emerge from it. I will in particular rely on a direct connection that we were able to establish, between sheared passive systems and active matter mechanical response, to provide some intuition on the validity of these predictions for low-dimensional systems. These results hint at a unifying framework for establishing rigorous analogies, at the mean-field level, between different families of driven disordered systems, such as sheared granular materials and active matter, or even machine-learning algorithms.

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