Eigen microstates in self-organized criticality

We employ the eigen microstates approach to explore the self-organized criticality (SOC) in two celebrated sandpile models, namely the BTW model and the Manna model. In both models, phase transitions from the absorbing state to the critical state can be understood by the emergence of dominant eigen microstates with significantly increased weights. Spatial eigen microstates of avalanches can be uniformly characterized by a linear system size rescaling. The first temporal eigen microstates reveal scaling relations in both models. Furthermore, by finite-size scaling analysis of the first eigen microstates, we numerically estimate critical exponents, i.e., $\sqrt{{\sigma }_0{w}_1}/{\tilde{v}}_1\propto L^D$ and ${\tilde{v}}_1\propto L^{D(1-{\tau }_s)/2}$. Our findings could provide profound insights into eigen microstates of the universality and phase transition in nonequilibrium complex systems governed by self-organized criticality.

Figure 1

Evolution of the average height of the sandpile from the absorbing state to the critical state for (a) the BTW model and (b) the Manna model with $L=32$. The system reaches the critical state after the time step indicated by the red dashed line. PDFs of the avalanche size in subcritical and critical states for (c) the BTW model and (d) the Manna model. The lines represent fitted lines with slopes of $-1.1\pm 0.07$ in (c) and $-1.17\pm 0.04$ in (d), respectively.

Figure 2

Evolution of the weights: (a) $w_1$ and (b) $w_2$ and $w_3$ of the eigenensemble from the absorbing state to the critical state for the BTW model with $L=32$. (c) and (d) Similar to (a) and (b), but for the Manna model. Each ensemble matrix is calculated based on $M=5\times {10}^5$ from independent $5\times {10}^5$ realizations.

Figure 3

Spatial distributions of the rescaled eigen microstates ${\mathit{u}}_{\mathit{1}}L$ for the BTW model at the critical state with different system sizes: (a) $L=32$, (b) $L=64$, and (c) $L=128$. Correspondingly, (d)–(f) and (g)–(i) present the spatial distributions of ${\mathit{u}}_{\mathit{2}}L$ and ${\mathit{u}}_{\mathit{3}}L$ for the same system sizes, respectively.

Figure 4

Similar to Fig. 3, but for the Manna model.

Figure 5

PDFs of the components of the temporal eigen microstates ${\mathit{v}}_{\mathit{1}}$ for (a) the BTW model and (c) the Manna model at the critical state with various system sizes. Rescaled PDFs for (b) the BTW model and (d) the Manna model using the critical exponents ${\tau }_s=1.27$ and $D=2.75$.

Figure 6

${\mathrm{Log}}_{10}\text{−}{\log }_{10}$ plot of (a) $w_1$, (b) ${\sigma }_0$, (c) ${\tilde{v}}_1$, and (d) $\sqrt{{\sigma }_0{w}_1}/{\tilde{v}}_1$ as functions of the system size $L$ for both the BTW model and the Manna model at the critical state. The fitted lines are presented.

Figure 7

${\mathrm{Log}}_{10}\text{−}{\log }_{10}$ plot of (a) $w_1^a{L}^2$ and (b) ${\tilde{v}}_1^a$ as functions of the system size $L$ for both the BTW model and the Manna model at the critical state. The fitted lines are presented.