Mean-field dynamics of infinite-dimensional particle systems: global shear versus random local forcing

TL;DR

This paper demonstrates that in infinite dimensions, the mean-field dynamics of particle systems under random local forcing are equivalent to those under global shear, revealing a unifying framework for driven disordered systems.

Contribution

It shows that local random forcing with finite spatial correlation can be mapped to global shear dynamics through simple rescaling in the mean-field limit.

Findings

Mean-field dynamics under local forcing are equivalent to global shear after rescaling.

The rescaling factor depends on the variance of local displacements, linking spatial correlations to global response.

Results suggest a broader unifying framework for driven disordered systems at the mean-field level.

Abstract

In infinite dimension, many-body systems of pairwise interacting particles provide exact analytical benchmarks for features of amorphous materials, such as the stress-strain curve of glasses under quasistatic shear. Here, instead of a global shear, we consider an alternative driving protocol as recently introduced in Ref. [1], which consists of randomly assigning a constant local displacement on each particle, with a finite spatial correlation length. We show that, in the infinite-dimension limit, the mean-field dynamics under such a random forcing is strictly equivalent to that under global shear, upon a simple rescaling of the accumulated strain. Moreover, the scaling factor is essentially given by the variance of the relative local displacements on interacting pairs of particles, which encodes the presence of a finite spatial correlation. In this framework, global shear is simply a special case of a much broader family of local forcing, that can be explored by tuning its spatial correlations. We discuss specifically the implications on the quasistatic driving of glasses -- initially prepared at a replica-symmetric equilibrium -- and how the corresponding 'stress-strain'-like curves and the elastic moduli can be rescaled onto their quasistatic-shear counterparts. These results hint at a unifying framework for establishing rigourous analogies, at the mean-field level, between different driven disordered systems.