Figure 1

Numerical results for the instantaneous velocity $v\left(t\right)$ at time $t$ averaged over many avalanches of duration $T$ and rescaled to collapse to the form of Eq. (34) with $n=1.$ Results for three different values of the dimensionless driving rate $\tilde{c}=c/D$ are shown. (In the numerics, the driving rate $c$ is varied while the fluctuation strength is fixed at $D=1/2$.) The parabolic shapes with $\tilde{c}$-dependent height predicted by Eq. (35) for $n=1$ match well with the numerics.Reuse & Permissions

Figure 2

Scaling collapses of numerical results of $P\left(v_m\right|T),$ the distribution of maximum velocities for avalanches of duration $T$, onto the form given by Eq. (54) performed for several values of the dimensionless driving rate $\tilde{c}=c/D.$ (In the numerics, the driving rate $c$ is varied while the fluctuation strength is fixed at $D=1/2$.) The corresponding scaling functions $F\left(x\right)$ given by Eq. (55) fit very well with the numerics.Reuse & Permissions

Figure 3

(a) Scaling collapse of $P\left(v_m\right|S),$ the distribution of maximum velocities for avalanches of size $S,$ onto the form of Eq. (66). The rescaled distributions approach the analytical scaling function from Eq. (67) for sufficiently large avalanche sizes. (b) Scaling collapse for the overall maximum velocity distribution $P\left(v_m\right)$ for quasistatic driving ($c=0$) and several different values of $k$ (see the caption of Table  for the physical definition) to the form of Eq. (78). The scaling function plotted is $F\left(x\right)=C{x}^2{\sinh }^{-2}\left(x\right)$, where the constant $C=1.44$ is adjusted to fit near the tail and is proportional to that given in Eq. (78) with $D=1/2$. Adjustment of this nonuniversal constant factor is required since the part of the distribution near the origin that deviates from scaling alters the overall normalization. The inset shows the same maximum velocity distributions before rescaling.Reuse & Permissions

Figure 4

(a) Numerical scaling collapse of the avalanche duration distribution $P\left(T\right)$ for different values of $k$ (see the caption to Table  for the definition) and the driving rate $c\equiv \langle v\rangle /k$ to the form of Eq. (72). The scaling functions plotted are $F_{\tilde{c}}\left(x\right)=C\left(\tilde{c}\right)e^x{x}^{2-\tilde{c}}{(e^x-1)}^{\tilde{c}-2}$, which are proportional to the ones given in Eq. (72). The nonuniversal constants are fit by eye as $C\left(0\right)=3.0$ and $C\left(0.2\right)=3.375.$ The inset shows the same distributions before rescaling with the curves for $\tilde{c}=0.2$ offset for visibility. (b) Collapse of the size distributions for different values of $k$ to the form of Eq. (73). The theoretical curve plotted is $F\left(x\right)=C\exp (-x/2)$, where $C=0.6$, which is, up to normalization, what is given in Eq. (73) with $D=1/2.$ The inset shows the size distributions before rescaling.Reuse & Permissions