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Abstract |
Disorder is ubiquitous in physical systems, and can radically alter their physical properties compared to their
‘pure’ counterparts. For instance, amorphous materials such as emulsions, foams, metallic
glasses or biological tissues are all structurally disordered, and this has key implications for their rheological,
mechanical or transport properties. Nevertheless, theoretical descriptions of such ‘driven’
amorphous materials remain challenging, despite decades of extensive analytical and computational studies.
The difficulties pertain to the interplay of competing sources of stochasticity, and to the
resulting out-of-equilibrium nature of these systems.
A standard model for amorphous materials, which allows one to focus on the key role of their structural
(positional) disorder, is provided by dense many-body systems of pairwise interacting particles.
Here I will introduce an exact Dynamical Mean-Field Theory (DMFT) for these many-body systems, derived in
the limit of infinite spatial dimension. In this framework, the many-body Langevin
dynamics of the whole problem can be exactly reduced to a single scalar effective stochastic process, and
dynamical observables such as pressure or shear stress can be computed for arbitrary
driving protocols. Using this DMFT, we were in particular able to establish a direct equivalence between a
global forcing (external shear) and a random local forcing (reminiscent of active matter), upon
a simple rescaling of the control parameter (the accumulated strain). In this framework, global shear is thus
simply a special case of a much broader family of local forcing that can be explored by
tuning its spatial correlations. Our predictions were moreover found to be in remarkably good agreement with
two-dimensional numerical simulations. 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 machine-learning algorithms.
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