Yielding versus random organization: convex absorbing transitions in soft matter

Abstract

We compare two different soft matter models, a generalized Random Organization Model (ROM) describing the stroboscopic dynamics of cyclically sheared suspensions, and an elastoplastic model describing the mesoscale dynamics of a yield-stress fluid under imposed stress. Both show absorbing phase transitions, sharing a peculiar mechanism: activity induces an internal noise which is transmitted over large distances by long-ranged mediated interactions, either hydrodynamic or elastic, which results in non-local creation of activity. They also both show convex transitions (i.e., the exponent $\beta >1$), in stark contrast with usual absorbing phase transitions, like (Conserved) Directed Percolation, which are concave ($\beta <1$). We further compare the dependence of the critical properties (activity mean value and fluctuations, avalanche statistics, low-wavenumber structure factor) on the decay exponent $\alpha$ of long-range interactions in both models, finding a qualitatively similar scenario. A smooth crossover is observed as a function of $\alpha$ between a concave transition regime for short-range interactions, with diverging fluctuations and compact avalanches, and a convex transition regime, with vanishing fluctuations and non-compact avalanches, for longer-range interactions. Although for a given range exponent $\alpha$, the values of critical exponents for both models differ, a good agreement between the models is found by parametrically plotting the different critical exponents as a function of the exponent $\beta$ of the mean activity. In this parametric representation, the concave regime is consistent with the behavior of the Long-Range Conserved Directed Percolation class, while the convex regime can be accounted for by a mean-field-type scenario with anomalous diffusion close to an absorbing boundary, inspired by the H\'ebraud-Lequeux model for the yielding transition.