Random organization criticality with long-range hydrodynamic interactions

TL;DR

This work extends the Random Organization Model to include long-range hydrodynamic interactions, uncovering a continuous dependence of critical exponents on the interaction decay exponent $oldsymbol{\a}$ in $oldsymbol{d=2,3}$. It reveals a transition from CDP-like concave behavior at short range to convex, diffusion-dominated criticality at long range, accompanied by a loss of hyperuniformity in density fluctuations for small $oldsymbol{\a}$. A mean-field Lévy–HL framework is developed to describe diffusion-induced activity responsible for 2P→2A activation, linking the convex transition to the statistics of long-range mechanical noise and a pseudo-gap exponent $oldsymbol{ heta}$. The results imply distinct universality classes for bulk versus confined geometries and provide a potential bridge to yielding transitions through analogous diffusion-driven activation mechanisms, with implications for hyperuniformity control in soft matter systems.

Abstract

Driven soft athermal systems may display a reversible-irreversible transition between an absorbing, arrested state and an active phase where a steady-state dynamics sets in. A paradigmatic example consists in cyclically sheared suspensions under stroboscopic observation, for which in absence of contacts during a shear cycle particle trajectories are reversible and the stroboscopic dynamics is frozen, while contacts lead to diffusive stroboscopic motion. The Random Organization Model (ROM), which is a minimal model of the transition, shows a transition which falls into the Conserved Directed Percolation (CDP) universality class. However, the ROM ignores hydrodynamic interactions between suspended particles, which make contacts a source of long-range mechanical noise that in turn can create new contacts. Here, we generalize the ROM to include long-range interactions decaying like inverse power laws of the distance. Critical properties continuously depend on the decay exponent when it is smaller than the space dimension. Upon increasing the interaction range, the transition turns convex (that is, with an order parameter exponent $β> 1$), fluctuations turn from diverging to vanishing, and hyperuniformity at the transition disappears. We rationalize this critical behavior using a local mean-field model describing how particle contacts are created via mechanical noise, showing that diffusive motion induced by long-range interactions becomes dominant for slowly-decaying interactions.

Figures (18)

• Figure 1: (a) Description of the rules for the mediated ROM. Active particles undergo a random kick $\delta_a$ of typical size $\Delta_a$ while passive particles are given a random kick $\delta_p(\bm{r})$ of typical size $\Delta_p(\bm{r})$ that depends on their position. (b) Determination of the spatialized activity field for the computation of passive particles steps.
• Figure 2: Evolution of the global activity $A(t)$ in the system as a function of time $t$, for a range $\alpha=2$ in dimension $d=2$. Starting from a random initial state, activity decreases to a stationary value.
• Figure 3: Stationary average activity $\langle A \rangle$ as a function of $\delta\phi = \phi-\phi_\mathrm{c}$ for different ranges of interaction $\alpha$. Data are plotted on linear scales to emphasize the change from concave to convex curves when $\alpha$ is decreased.
• Figure 4: Log-log plots of the average activity $\langle A \rangle$ in stationary state as a function of $\delta\phi = \phi-\phi_\mathrm{c}$ for different ranges of interaction $\alpha$, for d=2 (a) and d=3 (b). Data in panel (a) are the same as in \ref{['fig:convexity']}.
• Figure 5: Variance of activity fluctuations $\langle \delta A^2 \rangle$ in stationary state as a function of $\delta\phi = \phi-\phi_\mathrm{c}$ for different ranges of interaction $\alpha$, for $d=2$ (a) and $d=3$ (b).