Amorphous solids or yield stress fluids exhibit peculiar properties upon external drivings such as global shear. Dense assemblies of interacting particles are actually prototypes of such structurally disordered systems, which allow to focus on the key role generically played by structural disorder in their mechanical or rheological properties.

In infinite dimension the mean-field description of these simplified models becomes moreover exact, since spatial fluctuations as suppressed upon increasing dimensionality. Solving their equilibrium dynamics in this limit has been remarkably fruitful in capturing static properties of finite-dimensional systems as well. Here we address more generally their out-of-equilibrium dynamics, paving the way to obtaining a similar infinite-dimensional benchmark. More specifically, we derive the exact dynamical mean-field theory that describes a system of pairwise interacting particles, in infinite dimension and in the thermodynamic limit. Since we consider a very general setting, we can thus model a broad range of situations — equilibrium, quasi-statics, transients or steady-states — such as liquid and glass rheology or active self-propelled particles.

Here I will sketch the derivation of this effective dynamics, highlighting in particular the few key ingredients of the high-dimensional physics and their possible relevance for finite-dimensional systems. I will in addition discuss how we recover dynamically the ‘state-following’ equations describing the response of a glass under quasistatic perturbations (specifically thermal quench, random forces and a finite shear strain).

[1] E. Agoritsas, T. Maimbourg & F. Zamponi, ArXiv :1808.00236 / J. Phys. A (2019).
[2] E. Agoritsas, T. Maimbourg & F. Zamponi, follow-up with shear in preparation.