Le Doussal Pierre, Thiery Thimothée
Laboratoire de Physique de l'École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond, 75005 Paris, France.
Instituut voor Theoretische Fysica, KU Leuven, Leuven, Belgium.
Phys Rev E. 2020 Mar;101(3-1):032108. doi: 10.1103/PhysRevE.101.032108.
We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean-field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a ε=d_{c}-d expansion below the upper critical dimension d_{c} of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more generally, in two avalanches separated by a uniform displacement W of the interface. For (i) we obtain the correlations of the total sizes, of the local sizes, and of the total sizes with given seeds (starting points). For (ii) we obtain the correlations of the velocities, of the durations, and of the avalanche shapes. In general we find that the avalanches are anticorrelated, the occurrence of a larger avalanche making more likely the occurrence of a smaller one, and vice versa. Examining the universality of our results leads us to conjecture several exact scaling relations for the critical exponents that characterize the different distributions of correlations. The avalanche size predictions are confronted to numerical simulations for a d=1 interface with short range elasticity. They are also compared to our recent related work on static avalanches (shocks). Finally we show that the naive extrapolation of our result into the thermally activated creep regime at finite temperature predicts strong positive correlations between the forward motion events, as recently observed in numerical simulations.
我们研究了在随机基底上驱动的弹性界面脱钉动力学中雪崩之间的相关性。在平均场理论（布朗力模型）中，已知雪崩是不相关的。在此，我们得到了一个简单的场论，它描述了在模型的上临界维度(d_{c})以下的(\varepsilon = d_{c} - d)展开中，与这种不相关行为的首次偏差。我们将其应用于计算（i）雪崩大小之间、（ii）两个连续雪崩中，或者更一般地，在由界面均匀位移(W)分隔的两个雪崩中的雪崩动力学之间的相关性。对于（i），我们得到了总大小、局部大小以及给定种子（起始点）的总大小之间的相关性。对于（ii），我们得到了速度、持续时间以及雪崩形状之间的相关性。一般来说，我们发现雪崩是反相关的，较大雪崩的发生会使较小雪崩的发生更有可能，反之亦然。研究我们结果的普适性使我们推测了几个关于临界指数的精确标度关系，这些临界指数表征了不同的相关性分布。我们将雪崩大小的预测与具有短程弹性的(d = 1)界面的数值模拟进行了对比。它们还与我们最近关于静态雪崩（冲击）的相关工作进行了比较。最后，我们表明，将我们的结果简单外推到有限温度下的热激活蠕变区域，预测了正向运动事件之间有很强的正相关性，正如最近在数值模拟中所观察到的那样。
