Instability cascades in crumpling mylar sheets follow a log-Poisson statistic

TL;DR

The paper investigates aging in disordered materials, focusing on crumpling mylar under load, and tests whether record dynamics (RD) with a log-Poisson timing statistics can universally describe logarithmic aging. RD posits that relaxation proceeds via intermittent, activated, record-sized barrier crossings, yielding an event rate $\lambda(t) \sim r/t$ and a clock that advances through $\langle n_I\rangle \sim r\ln(t/t_w)$. Through experiments and large-scale simulations of a bistable network, the authors demonstrate a robust log-Poisson distribution for inter-event intervals in $\Delta\ln t_i$ and collapses with system size $N$, supporting RD over trap-like or Poisson-in-time theories. A mean-field cluster model shows that the microscopic avalanche details are unnecessary for RD, reinforcing RD as a universal descriptor of logarithmic aging in glassy systems and providing a discriminant against competing aging theories.

Abstract

The process of aging following a hard quench into a glassy state is characterized universally, for a wide class of materials, by logarithmic evolution of state variables and a power-law decay of two-time correlation functions that collapse only for the ratio of those times. This stands in stark contrast with relaxation in equilibrium materials, where time-translational invariance holds. It is by now widely recognized that these aging processes, which ever so slowly relax a complex disordered material after a quench, are facilitated by activated events. Yet, theories often cited to describe such a non-equilibrium process can be shown to miss pertinent aspects that are inherent to many experiments. A case in point are recent experiments on crumpling sheets of mylar loaded by a weight whose acoustic emissions are measured while the material buckles. Using extensive simulations to generate long time-series of such buckling events, we show that crumpling is a log-Poisson process activated by increasingly rare record-sized fluctuations in a slowly stiffening material characterized by a logarithmically growing length-scale. Crumpling thus adds to a range of glassy materials exhibiting the log-Poisson property, which can be used to discriminate between theories.

Figures (7)

• Figure 1: Sketch of the hierarchical free-energy landscape of a disordered system, with a typical trajectory of an aging dynamics (blue and red) Robe16. With increasing free energy $F$, local minima proliferate rapidly but also become shallower. Relaxing downwards (red-dashed arrow), the dynamics evolves through a sequence of quasi-equilibrium explorations (blue) and intermittent, irreversible avalanches over record barriers (red) that access an exponentially expanding portion of configuration space (black-dashed arrow) BoSi.
• Figure 2: Decay in the rate $\lambda(t)$ of intermittent events ("clicks") after applying a load to crumpled sheets in the experiments (circles, in arbitrary units) and for the MD simulations (squares for $N=2000$ and diamonds for $N=8000$ nodes, averaged over 2000 instances) of the bistable model, both provided in Ref. Shohat23. Counting all events in the experiments ($\Circle$) appears to result in a logarithmic overcount on record statistic, i.e. $\lambda(t)\sim\log(t)/t$ (green line), which is removed by combining all subsequent events within a time window of $\delta t/t=0.001$ after the onset into a single avalanche ($\CIRCLE$) that occur at rate of $\lambda(t)\sim1/t$ (red line). In the simulations, the rate for all events are consistent with the hyperbolic behavior predicted by RD, even without corrections.
• Figure 3: Log-Poisson statistic for the inter-event intervals $\Delta\ln t_{i}=\ln\left(1+\Delta t_{i}/t_{i}\right)$ for the $i^{{\rm th}}$ event, measured within two different system sizes, $N=2000$ and $N=8000$, for the MD simulations as well as the experimental data provided by Ref. Shohat23. The main plot shows the raw data for each PDF, while the collapses of that data is shown in the inset, rescaled by the system size $N$. Accordingly, the experimental data would correspond to a simulation with $N_{{\rm exp}}\approx4.2\ 10^{5}$ nodes. Note that the data for the small systems extends to well above $\Delta t_{i}/t_{i}=1$, while for the experimental data it is limited to $\Delta t_{i}/t_{i}\ll1$Lahini23. The simulation data in Ref. Shohat23 is plotted for $\Delta t_{i}/t_{i+1}$, which is strictly $\leq1$.
• Figure 4: Average spatial extend $\left\langle \Delta r\right\rangle$ of the clusters of events within a time window $\delta t$ following an initial event at time $t$ in MD simulations of the bistable model with $N=8000$ nodes within a 900-by-900 box (see inset), here for $\delta t/t=0.001$. For short times immediately after applying the load ($t<200$), too many otherwise independent events that are spread randomly over the box overlap to artificially enlarge $\left\langle \Delta r\right\rangle$. With decelerating activity, these overlaps fade to reveal the logarithmic growth of authentically correlated behavior within those windows, as predicted for RD Becker14Robe16GB20. The inset illustrates such a spatially correlated cluster for a typical (albeit unusually active) sequence of events, initiated at a late time $t=6484$ at coordinate $\left(104,473\right)$. Depending on window size $\delta t$ (see legend), an increasing set of additional events -- increasingly farther apart -- are taken as correlated with the first event, suggesting reasonable intermediate choices for $\delta t$ that provide robust log-scaling (not shown).
• Figure 5: Generic results for the mean-field cluster model BoSi09Becker14 of size $N$, here obtained by randomly partitioning clusters of size $n_{k}>1$ into two. (a) Hyperbolically decelerating rate of break-up events, whether all events are counted (solid symbols) or discounting those within a window of $\delta t=0.01t$ following an event at $t_{i}=t$ (open symbols). (b) Distribution $p_{n}$ of clusters of size $n$, measured at an exponential progression of times $t=2^{j}$ for $j=6,9,\ldots,42$ at $N=8000$, that shows the log-time expansion of a "gap" below the smallest of the large clusters at any $t\gg1$, such that the average cluster size grows as $\left\langle n\right\rangle \sim\ln t$ independent of $N$ (see inset), similar to Fig. \ref{['fig:clustersize']}. (c) Log-Poisson statistic, and its collapse when rescaled with $N$ (inset), comparable with the crumpling simulations in Fig. \ref{['fig:logPoissonCrumble']}. Note that the overcount of many events with small $\Delta t$ inside avalanches at large $t$ merely result in a spurious peak at the origin.