Many complex and disordered systems fail to reach equilibrium after they have been quenched or perturbed. Instead, they sluggishly relax toward equilibrium at an ever-slowing, history-dependent rate, a process termed physical aging. The microscopic processes underlying the dynamic slow-down during aging and the reason for its similar occurrence in different systems remain poorly understood. Here, we reveal the structural mechanism underlying logarithmic aging in disordered mechanical systems through experiments in crumpled sheets and simulations of a disordered network of bistable elastic elements. We show that under load, the system self-organizes to a metastable state poised on the verge of an instability, where it can remain for long, but finite, times. The system’s relaxation is intermittent, advancing via rapid sequences of instabilities, grouped into self-similar, aging avalanches. Crucially, the quiescent dwell times between avalanches grow in proportion to the system’s age, due to a slow increase of the lowest effective energy barrier, which leads to logarithmic aging.
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An example of the LAMMPS simulation script used in this study is available in the Supplementary Information. Other scripts and numerical data are available from the corresponding author upon reasonable request.
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We are grateful to D. Hexner, S. Reuveni, E. Bouchbinder, E. Lerner and K. González-Lopéz for insightful discussions and to Y. Shokef, H. Diamant and Y. Bar-Sinai for comments on the manuscript. We also thank B. Hirshberg for sharing insights on using LAMMPS and D. Hexner for providing the disordered network topologies. This work was supported by the Israel Science Foundation grants 2096/18 and 2117/22. D.S. acknowledges support from the Clore Israel Foundation.
The authors declare no competing interests.
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Supplementary Video 1, Figs. 1–11 and Sections 1–4.
In the supplementary video, we show the compaction of a network of n = 2,000 nodes under external forcing and weak thermal noise T = 10−3. While thermal fluctuations are barely apparent, they suffice to trigger large-scale avalanches of instabilities. The avalanches are separated by long quiescent waiting times in which the system is stationary, other than equilibrium-like fluctuations. These waiting timed grow longer as the systems ages. Bond colour in the video represents the potential energy of each bond U(δRij), as shown by the colourbar in Supplementary Fig. 1. Blue bonds are close to one of their minimal energy states, while red bonds are on the verge of instability. Note the distribution of unstable bonds in the network, the occurrence of avalanches separated by (growing) quiescent dwell times, and how avalanches generate new unstable bonds. This highlights the process by which the network self-organizes to a state on the verge of instability.
Processed experimental data of the acoustic emissions of crumpled sheets under external load. It also contained an example script for the LAMMPS simulations used in the study.
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Shohat, D., Friedman, Y. & Lahini, Y. Logarithmic aging via instability cascades in disordered systems. Nat. Phys. (2023). https://doi.org/10.1038/s41567-023-02220-2
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