论文标题
有关布朗部队模型的更多信息：雪崩形状，尖端驱动，更高的$ d $
More on the Brownian force model: avalanche shapes, tip driven, higher $d$
论文作者
论文摘要
布朗力模型（BFM）是弹性界面雪崩的平均场模型，以随机介质缓慢驱动。它描述了速度场的时空统计数据，并且在某种程度上是可以分析的。我们扩展了以前的研究，以获取具有短距离弹性的BFM的几个可观察力，与本地跳跃尺寸$ s（x）$以及与$ d = 1 $的雪崩空间扩展相关，或者在$ d> 1 $中的雪崩跨度。在$ d = 1 $中，我们考虑通过施加的力（i）驾驶（i）通过“尖端”施加的位移（ii）驾驶，并在每种情况下获得平均空间形状$ \ langle s（x）\ rangle $在固定延长处或在固定种子到边缘距离。我们发现，在边缘附近$ x_e $，$ s（x）\simeqσ| x-x_e |^3 $，其中$σ$具有我们获得的通用分布。我们还获得了边缘附近的时空形状。在$ d> 1 $中，我们获得（i）平均形状$ \ langle s（x_1，x_ \ perp）\ rangle $在固定跨度中，在横向距离$ x_ \ perp $（ii）的横向距离中表现出非平凡的依赖性该中心为$ | x |^{b_d} $，带有非琐碎指数，$ b_1 = 4 $，$ b_2 = 2 \ sqrt {2} $和$ b_3 = \ frac {1} {2} {2} {2}（\ sqrt {17} -1）$。我们在任何$ d $中获得雪崩最大半径的任何概率分布，而接近给定点的最小距离，以及不以$ d = 2 $击中圆锥体的概率。这些结果同样适用于某些分支扩散的连续限制，如伴侣纸中所述。
The Brownian force model (BFM) is the mean-field model for the avalanches of an elastic interface slowly driven in a random medium. It describes the spatio-temporal statistics of the velocity field, and, to some extent is analytically tractable. We extend our previous studies to obtain several observables for the BFM with short range elasticity, related to the local jump sizes $S(x)$ and to the avalanche spatial extension in $d=1$, or the avalanche span in $d>1$. In $d=1$ we consider both driving (i) by an imposed force (ii) by an imposed displacement "at the tip" and obtain in each case the mean spatial shape $\langle S(x) \rangle$ at fixed extension, or at fixed seed to edge distance. We find that near an edge $x_e$, $S(x) \simeq σ|x-x_e|^3$ where $σ$ has a universal distribution that we obtain. We also obtain the spatiotemporal shape near the edge. In $d>1$ we obtain (i) the mean shape $\langle S(x_1,x_\perp) \rangle$ for a fixed span, which exhibits a non-trivial dependence in the transverse distance to the seed $x_\perp$ (ii) the mean shape around a point which has not moved, $\langle S(x) \rangle_{S(0)=0}$, which vanishes at the center as $|x|^{b_d}$ with non trivial exponents, $b_1=4$, $b_2=2 \sqrt{2}$ and $b_3=\frac{1}{2} (\sqrt{17}-1)$. We obtain the probability distributions in any $d$ for the maximal radius of an avalanche and the minimal distance of approach to a given point, as well as the probability of not hitting a cone in $d=2$. These results equivalently apply to the continuum limit of some branching diffusions, as detailed in a companion paper.
