Figure 1

Spectrum $S\left({\rho }^{\prime }\right)$ computed with $\overline{\rho }$ defined on $[0,T^{*}]$, where $T^{*}=2\times {10}^8$, for $128\times 128$ and $512\times 512$ lattices; highest frequencies are omitted. The blue and black lines give the best fits. The power laws of the type $S\sim {\phi }^{-\gamma }$ imply the dependence $S\sim T^{\gamma }$.

Figure 2

Ensemble-averaged spectrum. Right part labeled (a) contains: (i) yellow and green curves $S_n\left(\rho \right)$ computed with the initial catalog $\rho \left(t\right), t\in [0,2\times {10}^7]$ split into $n=100$ segments of the length of $T^{*}=2\times {10}^5$ each for $N$ equaled to 256 and 1024, respectively; (ii) blue curve $S_n\left({\rho }^{\prime }\right)$ computed with the thinned catalog ${\rho }^{\prime }\left(t\right), t\in [0,8\times {10}^8], T^{*}=8\times {10}^6, n=100, N=1024$ (the latter as for the green curve). Left part (b) sharing the horizontal axis with (a) and having the stretched vertical axis displays same blue curve but shown on partly different interval.

Figure 3

Scheme of the spectrum: four components shown with solid lines.

Figure 4

Normalized spectrum $S^{*}·\phi$ in circles, found with logarithmic bins of the length $\tau =\sqrt{1.2}$ for four lattices. The triangles ${\mathcal{L}}_{\infty }\mathcal{M}\mathcal{H}$ corresponds to the triangles $\mathcal{L}\mathcal{M}\mathcal{H}$ displayed on Fig. 3. The fits ${\phi }^{\mathrm{\Gamma }}$ are found for $\mathcal{M}\mathcal{H}$ and ${\mathcal{L}}^{\prime }\mathcal{M}$ instead of ${\mathcal{L}}_{\infty }\mathcal{M}$. The periods corresponding to the points ${\mathcal{L}}_{\infty }, {\mathcal{L}}^{\prime }, \mathcal{M}$, and $\mathcal{H}$ scale as $\sim N^2, \sim N^{1.5}, \sim N^{1.3}$, and $\sim N^1$, respectively.

Inset, shared period axis: the coefficient of variation of the values of the spectrum $S^{*}$ related to $N=512$.

Figure 5

Mean rate of the stress accumulation computed as the ensemble average of the slopes of the $\rho \left(t\right)$ trends that lasts $\mu$ time moments after the stress level ${\rho }_{*}$ is passed; $N=1024$.

Figure 6

Taken from Fig. 2 ensemble-spectrum average $S_n$ (in blue) computed with $N=1024$ and the thinned out catalog is complemented by two linear fits on the displayed intervals, the graph of another $S_n$ computation (in red) from the same catalog ${\rho }^{\prime }$ but preceded by the multiplication by the Hamming window, and two fits of the latter $S_n$.

Figure 7

The normalized difference $(S_{\nu =24}^{*}-S_{\nu =25}^{*})/S_{\nu =25}^{*}$ between two summed spectra $S^{*}$ computed with $\nu =24$ and $\nu =25$ for $N=512$ and $N=1024$, where $S_{\nu =25}^{*}$ is taken from Fig. 4. Inset contains the right part of the main figure zoomed along the vertical axis.