Avalanches in the Random Organization Model with long-range interactions

Abstract

Oscillatory sheared suspensions, when observed stroboscopically, exhibit a reversible-irreversible transition as a function of the strain amplitude, which is a kind of absorbing phase transition. So far studies of this transition focused on global quantities, e.g. quantifying the irreversibility on one side of the transition or the time to reach a reversible state on the other side. Here, motivated by the kin depinning transition, we focus on the intermittent dynamics near the transition. We perform simulations of a modified Random Organization Model (ROM), a minimal particle model which we recently adapted to take into account the generic presence of long-range interactions mediated by the fluid, taking the power-law-decay exponent $α$ as an additional control parameter of the model. We show that at the absorbing phase transition, this model displays power-law-distributed avalanches. We characterize the avalanche statistics in terms of avalanche size, duration and number of particles involved, and we determine the associated exponents. By varying the exponent $α$, the fractal dimension of avalanches crosses space dimension $d$, inducing a qualitative change of the spatial structure of avalanches, from compact avalanches when interactions have a short range, to sparse avalanches when interactions are long-ranged. Finally, we characterize the clusters within the avalanches, which we also find power-law distributed.

Figures (17)

• Figure 1: Avalanche size distribution $P(S)$ for $\alpha=3$ (a,b) and $\alpha=0.5$ (c,d). Left panels (a,c): $P(S)$ rescaled by system size $L$, showing a good data collapse on a power-law distribution $\propto S^{-\tau}$ followed by a size-dependent cutoff around $S_c \sim L^{d_f}$. Right panels (b,d): compensated plots obtained by dividing the distribution $P(S)$ by the estimated power law $S^{-\tau}$.
• Figure 2: Avalanche duration distribution $P(T)$ for $\alpha=3$ (a,b) and $\alpha=0.5$ (c,d). Left panels (a,c): $P(T)$ rescaled by system size $L$, showing a good data collapse on a power-law distribution $\propto T^{-\tau'}$ followed by a size-dependent cutoff $T_c \sim L^z$. Right panels (b,d): compensated plots obtained by dividing the distribution $P(T)$ by the estimated power law $T^{-\tau'}$.
• Figure 3: Distribution $P(N)$ of the number $N$ of particles involved in an avalanche, for $\alpha=3$ (a,b) and $\alpha=0.5$ (c,d). Left panels (a,c): $P(N)$ rescaled by system size $L$, showing a good data collapse on a power-law distribution $\propto N^{-\tau"}$ followed by a size-dependent cutoff $N_c \sim L^{\chi}$. Right panels (b,d): compensated plots obtained by dividing the distribution $P(N)$ by the estimated power law $N^{-\tau"}$.
• Figure 4: Evolution of the exponents characterizing the avalanche distributions $P(S)$, $P(T)$ and $P(N)$ [see Eq. (\ref{['eq:AvDistribSusp']})] as a function of the decay exponent $\alpha$ of long-range interactions. Top: Power-law exponents $\tau$, $\tau'$ and $\tau"$. Bottom: Cutoff scaling exponents $d_f$, $z$ and $\chi$.
• Figure 5: Space-time plots of a typical avalanche, for $\alpha=0.5$ (a), showing a non-compact avalanche, and $\alpha=3$ (b), showing a compact avalanche.