We study the energy minimization problem for an elastic interface in a random potential plus a quadratic well. As the position of the well is varied, the ground state undergoes jumps, called shocks or static avalanches. We introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements. The tree-level calculation, i.e. mean-field limit, is obtained by solving a saddle-point equation. Graphically, it can be interpreted as a the sum of all tree graphs. The 1-loop corrections are computed using results from the functional renormalization group. At the upper critical dimension the shock statistics is described by the Brownian Force model (BFM), the static version of the so-called ABBM model in the non-equilibrium context of depinning. This model can itself be treated exactly in any dimension and its shock statistics is that of a Levy process. Contact is made with classical results in probability theory on the Burgers equation with Brownian initial conditions. In particular we obtain a functional extension of an evolution equation introduced by Carraro and Duchon, which recursively constructs the tree diagrams in the field theory.
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 disp…
Avalanches in mean-field models and the Barkhausen noise in spin-glasses
我们得到一个一般公式的大小分布的“静态雪崩”，或冲击，在一般的平均场玻璃复制对称性破坏鞍点。对于Sherrington-Kirkpatrick（SK）自旋玻璃，它给出了零温下沿平衡磁化曲线的磁化跃变大小S的密度ρ（S）。连续的复本对称性破缺允许幂律行为rho（S）~ 1/（S）^tau，其中SK的指数tau=1，与自旋玻璃相的临界性（边缘稳定性）相关。超度量相空间的所有尺度都与跳跃事件有关。对于弹性系统的静态跳跃的大小S，或Burgers湍流中的大尺度激波的大小S，也得到了类似的结果。在所有一步解法的情况下，rho（S）~ S exp（-A S^2）。一个简单的解释有关的液滴冲击，并在有限维自旋玻璃巴克豪森噪声的平衡模拟标度理论进行了讨论。
We investigate the effect of the amount of disorder on the statistics of breaking bursts during the quasi-static fracture of heterogeneous materials. We consider a fiber bundle model where the strength of single fibers is sampled from a power law distribution over a finite range, so that the amount of materials' disorder can be controlled by varying the power law exponent and the upper cutoff of fibers' strength. Analytical calculations and computer simulations, performed in the limit of equal load sharing, revealed that depending on the disorder parameters the mechanical response of the bundl…
Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modelled as an elastic system subject to quenched disorder. Its field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group flow involves a function, the disorder correlator $Δ(w)$, therefore termed the functional renormalization group (FRG). $Δ(w)$ is a physical observable, the auto-correlation function of the centre of mass of the elastic manifold. In this review, we give a …
