Memory effects govern scale-free dynamics beyond universality classes

TL;DR

The paper addresses why avalanche exponents observed in experiments often exceed the mean-field value $\tau=\tfrac{3}{2}$ and are non-universal. Through extensive simulations of the OFC earthquake model, it identifies two regimes: memoryless S1 with $\tau_c\simeq 1.22$ in low-dissipation conditions, and memory-driven S2 where the exponent $\tau(\nu)$ grows with dissipation as avalanches pile on traces left by prior events. A simple quasi-critical framework shows that a power-law distribution of hovering distances to the critical point, $P(d_{cp})\sim d_{cp}^{-\gamma}$, yields $\tau(\nu)=\tau_c+1-\gamma(\nu)$, explaining larger observed exponents. This approach unifies critical and quasi-critical scale-free dynamics, with implications for natural seismicity and other driven dissipative systems where memory effects are intrinsic.

Abstract

Scale-invariant avalanches -- with events of all sizes following power-law distributions -- are considered critical. Above the upper critical dimension of four, the mean-field solution with a robust $3/2$ size exponent describes the dynamics. In two and three dimensions, spatial constraints yield smaller yet robust exponent values governed by universality classes. However, both earthquake data and experiments often show exponent values larger than $3/2$, challenging those theoretical arguments based on critical behavior. Through extensive simulations in the classical OFC earthquake model, here we show a clear transition from the theoretical expected behavior of a robust exponent value, to a regime of quasi-critical dynamics with larger than $3/2$ exponents that depend on dissipation. While the first critical regime exhibits an inherently memoryless behavior, both the transition and the second regime are driven by memory effects provoked by the growth of avalanches over the traces left by previous events, due to dissipative mechanisms. The system hovers at a distance $d_{cp}$ from the critical point, and accounting for a power-law distribution of $d_{cp}$, validated by susceptibility measurements, captures the transition. This framework provides a unified description of both critical and quasi-critical behavior, and thus of the full spectrum of scale-free dynamics observed in nature.

Figures (7)

• Figure 1: Two distinct behaviors. (a) (open symbols) Theoretical predictions of the exponent values $\tau$ of the avalanche size distribution in Abelian sandpile models (ASM1 Zhang1989 and ASM2 Lubeck1997), Manna model Chessa1999, Renormalization group Vespignani1995 and Functional Renormalization Group (FRG) LeDoussal2009 with: Random field (RF), Random bond (RB), Random periodic (RP) and Non-periodic (NP). In UCD=4 and above, the MF solution of 3/2 is obtained. For two- and three-dimensions exponent values cluster below the value 3/2 in different universality classes. (solid symbols) OFC model and selection of experimental results in granular faults in Lyon Lherminier2019 and Rennes Houdoux2021 Labs, fracture in Oslo FractureOslo2006 and Barcelona (BCN) Xu2019, and earthquake data Duplat2025Navas2019 displaying exponent values above 3/2. (b) Sketch of the two distinct behaviors of the avalanche size distributions when submitted to dissipation. S1: robust $\tau\leqslant 3/2$ and cut-off $s_{max}$ that retreats with dissipation. S2: robust cut-off $s_{max}$ proportional to the system size $N$ and $\tau$ values that increases with dissipation.
• Figure 2: Size distributions & trace dependence. (a) Avalanche size distribution in the OFC model for different dissipation values ranging from 0 to 0.6 (dark to bright colors) for a system size L = 256. (b) Power law exponents from the dashed lines in (a) as a function of dissipation. A vertical dashed line at $\nu = 2\%$ marks the threshold beyond which the exponent begins to increase. All insets provide a zoom into the low dissipation region. (c) Maximum avalanche size as a function of dissipation. The horizontal dashed line is a visual indicator of the size of the system. (d) Average linear size of the avalanches $\langle s\rangle^{1/2}$, scaling with the perimeter of the traces (color gradient) and mean value of the thickness $\delta z$ of the trace left by the avalanches (in blue). (e) Average dissipated energy, $\langle ntop \rangle \nu$, where $\langle ntop \rangle$ is the mean number of topplings as a function of dissipation (indicated by a color gradient). The graph matches that of the product $\langle \delta z \rangle \langle s \rangle^{1/2}$, (in blue) indicating a direct relationship between dissipation and the traces left by the avalanches.
• Figure 3: Memory effects & patch formation. (a) Inter-event time distribution normalized by the mean intertime $\langle T \rangle$, for different dissipation values ranging from 0 to 0.6 (dark to bright colors), in a system of size L = 256. The black dashed line represent power law fits for each dissipation value, while the red dashed line correspond to a pure exponential decay. (b) Exponent $\gamma$ of the power law fit from (a) as function of the dissipation. The inset provides a zoomed-in view of the low dissipation region. The vertical dashed line is a visual indicator for $2\%$ dissipation. Below $2\%$ distributions show a pure exponential decay, indicating a memoryless Poisson process, while above the transition, memory effects linked to temporal clustering of events, increased monotonically with dissipation. (c) Patches of neighboring z-values show an increase in contrast and a decrease in size with dissipation above the transition.
• Figure 4: Democratic algorithm. (a) Inter-event time distribution normalized by the mean inter-event time $\langle T \rangle$ for different dissipation values. The distribution for the conservative case in the non-modified OFC model (dashed line) is also plot for comparison. (b) Avalanche size distribution. A S1 behavior --with a robust $\tau = \tau_c \simeq 1.22$ and a cut-off that retreats with dissipation-- is obtained. (c) Snapshots of the structure under different dissipation values show that no patches appear.
• Figure 5: Transiting from critical to quasi-critical behavior. (a), Percentage of $s_c(\nu) \geq N/4$ for configurations immediately preceding large events ($s_{next}\geq N/4$), and those preceding very small events ($s_{next} = 1$) that are also far in time from any large event. For $\nu = 0$, large avalanches can be triggered from both types of configurations, indicating critical behavior. In contrast, for high dissipation values (S2 scenario), the percentage of $s_{next} = 1$ configurations is about $20\%$ just after the transition and then decreases monotonically to zero as dissipation increases. (b), For a given dissipation $\nu = 0.2$, the distribution of $z$-values --containing four peaks separated by $\alpha$-- is similar inside the patches (left panel) and in the case of one single domain (right panel), corresponding to a distribution of events with exponent $\tau_c \simeq 1.22$ in the democratic algorithm, suggesting a common intrinsic dynamics. (c), Distributions of $s_c$ for different dissipation values ranging from 0 to 0.6 (dark to bright colors). They follow power laws for large dissipation values. (d) Power-law exponents from the dashed lines in (c) as a function of dissipation (colors). Theoretical points correspond to $\varphi(\nu) = \tau(\nu) -\tau_c +1$, giving the expected behavior in the OFC model.