Statistics of Thermal Avalanches in Driven Amorphous Systems

Abstract

Within the framework of the random first-order transition theory of glasses, we discuss the statistics of thermal avalanches, the large scale rearrangements in driven amorphous systems near their instability. Stringy excitations yield nonPoisson waiting time statistics. Embedding these statistics in a generalized Master equation captures the nonMarkovian, aging dynamics of avalanche clusters. We apply this framework to analyze nonequilibrium signatures of thermal avalanches, auto correlation functions and effective temperatures, under both quasi static shear and stochastic shaking protocols. We use full counting statistics to derive the complete distribution of both the avalanche magnitudes and avalanche counts, uncovering the intermediate time behavior.

Figures (6)

• Figure 1: The free energy $F(L)$ of a string of length $L$ consists of a deterministic linear contribution $\phi L$ and quenched fluctuations, leading to a rugged landscape with multiple metastable minima. A slow external ramp, modeled as a linear increase of the driving parameter $\phi \to \phi + \alpha t$, progressively tilts the landscape (dashed line), reducing local barriers. As the tilt increases, metastable states disappear via a spinodal-like instability, triggering irreversible string growth events (avalanches). The inset illustrates the local destabilization of a single metastable minimum under the applied ramp.
• Figure 2: Effective temperature $T_{\rm eff}$ under different protocols. (A) The external shear stress. $T_{\rm eff}/T$ is plotted versus $\Delta s_c$ and $\phi/|\alpha|$. (B) The random shaking protocol. $T_{\rm eff}/T_{\rm mix}$ is plotted versus the switching rates $\gamma_{l\to h}$ and $\gamma_{h\to l}$. Parameters are $\Delta C_p=1k_b$, $\eta=0$ and $R_0=0.01s^{-1}$. $T_g=250K$ is the glass transition temperature for cross-linked networks sharaf1980effects, $T_l=T_g$ and $T_h=2T_g$.
• Figure 3: The simulation results of the aging correlation function $C(t_w,t)$ as a function of the lag time $t$ for different aging time $t_w$. Here, $s_c^{\text{string}}=1.13k_B$. (a) $s_c=1.03k_B$; (b) $s_c=1.13k_B$; (a) $s_c=1.23k_B$. Other parameters are $\Delta C_p=1k_b$, $\eta=0$ and $R_0=0.01s^{-1}$.
• Figure 4: The conditional distribution of avalanches $P(L,t|l_a)$ when the number of avalanche initiations is $l_a=1$: Left panel: linear–linear plot; Right panel: semi-logarithmic plot with a linear horizontal axis and a logarithmic vertical axis. Dots: the full counting statistics [Eq.\ref{['Bessel']}]; lines: the Gaussian limit [Eq.\ref{['Gauss']}]. The system is prepared in the downhill flow situation with a frozen configurational state having $s_c^{\rm frozen}=1.1865k_B$ with the critical string entropy $s_c^{\rm string}=1.13k_B$. At the initial moment, the system was $-5\%$ away from instability. The free energy continued to be more and more downhill steadily under the same ramping rate. Over a longer observation period ($t\in\{100,150,200,250\}$), the system moved further away from the instability point (from $-30\%$ to $-67.5\%$). Other parameters are $\Delta C_p=1k_b$, $\eta=0$ and $R_0=0.01s^{-1}$.
• Figure 5: The conditional distribution of avalanches $P(l_a|L,t)$ of size being $L=50$. The system is prepared in the downhill flow situation with a frozen configurational state having $s_c^{\rm frozen}=1.1865k_B$ with the critical string entropy $s_c^{\rm string}=1.13k_B$. At the initial moment, the system was $-5\%$ away from instability. Under the stress ramping condition, the free energy continued to be more and more downhill steadily. Ultimately at time $t$, the system's stability deviation was $-5.5\%$ (green), $-6\%$ (orange), and $-7\%$ (blue). Other parameters are $\Delta C_p=1k_b$, $\eta=0$ and $R_0=0.01s^{-1}$. Left panel: fast ramping $t=100$; Right panel: slow ramping $t=200$. The arrows indicate the ranges, which quantify how microscopic initiation statistics are amplified to experimentally observable avalanche rates in both the cytoquakes and the molecular glass seismology.