Distribution of joint local and total size and of extension for avalanches in the Brownian force model

The Brownian force model is a mean-field model for local velocities during avalanches in elastic interfaces of internal space dimension $d$, driven in a random medium. It is exactly solvable via a nonlinear differential equation. We study avalanches following a kick, i.e., a step in the driving force. We first recall the calculation of the distributions of the global size (total swept area) and of the local jump size for an arbitrary kick amplitude. We extend this calculation to the joint density of local and global sizes within a single avalanche in the limit of an infinitesimal kick. When the interface is driven by a single point, we find new exponents ${\tau }_0=5/3$ and $\tau =7/4$, depending on whether the force or the displacement is imposed. We show that the extension of a “single avalanche” along one internal direction (i.e., the total length in $d=1$) is finite, and we calculate its distribution following either a local or a global kick. In all cases, it exhibits a divergence $P\left(\ell \right)\sim {\ell }^{-3}$ at small $\ell$. Most of our results are tested in a numerical simulation in dimension $d=1$.

Figure 1

An avalanche in $d=1$.

Figure 2

An avalanche in $d=2$; the transverse direction is orthogonal to the plane of the figure, and the colored zone corresponds to the support of the avalanche.

Figure 3

Green histogram : global avalanche-size distribution from a direct numerical simulation of a discretized version of Eq. (1) with the following parameters: $N=1024,m=0.01,df=m^2\delta w=1$, and $dt=0.05$. Red line: theoretical result given in Eq. (16). For details about the simulation, see Appendix pp8.

Figure 4

Green histogram: Local avalanche-size distribution from a direct numerical simulation of a discretized version of (1) with parameters $N=1024,m=0.01,df=m^2\delta w=1$, and $dt=0.05$. Red line: the theoretical result given in Eq. (21). For details about the simulation, see Appendix pp8.

Figure 5

Distribution of $\alpha$, defined in Eq. (26), from numerical simulations. This is compared to the theoretical prediction (33). Keeping only large-size avalanches, this converges (without any adjustable parameter) to the $\delta w=0^{+}$ result. Numerical parameters used here are $N=1024,m=0.02,\delta w=10,dt=0.01$, different from the one used in Figs. 3 and 4 as we want to be close to the $\delta w=0^{+}$ limit.

Figure 6

Decay amplitude $R\left(k\right)$ as a function of the aspect ratio $k$ involved in the joint density of $\ell$ and $k$, and defined in Eqs. (60) and (61).

Figure 7

The distribution of extensions $\rho \left(\ell \right)$, as obtained from the elliptic integrals (F7) and (F8) (black line). The (straight) green dotted line is the small-$\ell$ asymptotics (69), whereas the (curved) red dotted line is the large-$\ell$ asymptotics (71). The numerical simulation (green histogram) is cut at small scale due to discretization effects. Numerical parameters are $N=2^{10},m=0.05,dw=100$, and $dt=0.01$.

Figure 8

Representation of the potential energy $V\left(y\right)$ as a function of $y$, and lines of constant total energy, with $E=0$ in red, $E>0$ in blue, and $E<0$ in green.

Figure 9

Phase-space diagram, i.e., trajectories represented with $y^{\prime }$ as a function of $y$. The case $E=0$ is in red, $E>0$ is in blue, and $E<0$ is in green. We can see that the properties of the solution (periodicity, divergences, etc.) depend strongly on the value of $E$.

Figure 10

Solutions with energy 0 of Eq. (B1); left: $y^{+}\left(x\right)$, right: $y^{-}\left(x\right)$.

Figure 11

Graphical representation of the construction of solutions of the instanton equation for $\lambda >0$ (blue) and $\lambda <0$ (green). The dotted part of the curve represents the discontinuity in the derivative. The red line represents the $E=0$ solution of (B1), the only one needed to solve the instanton equation with one local source.

Figure 12

The generating function $Z\left(\lambda \right)=6(1-z)$ is represented here with some indications of the link with the construction of the instanton solution; the green and blue dots correspond to the solutions represented in Fig. 11.

Figure 13

Instanton solutions involved in the computation of $Z\left(r\right)$ for $r=1$: in blue, ${\tilde{u}}_1\left(x\right)$, in red, ${\tilde{u}}_{\infty }\left(x\right)$, and in purple, ${\tilde{u}}_{\infty }(x-1)$.