Critical behavior for random initial conditions in the one-dimensional fixed-energy Manna sandpile model

A fixed-energy Manna sandpile model undergoes an absorbing phase transition at a critical ${\rho }_c$, where an order parameter $\varphi \left(t\right)$ decays as $t^{-\alpha }$ in time $t$. As the prototype of the Manna class, the model has been extensively studied in one dimension. However, the previous estimates of ${\rho }_c$ and some critical exponents are different, depending on the types of initial conditions; random, natural, and regular conditions. The estimates of ${\rho }_c$ for the random and the regular conditions are the lower and the upper bound among currently known estimates, respectively. In this work, for the random conditions, ${\rho }_c$ and $\alpha$ are measured by taking into account finite-size (FS) effects. At the previous estimate of ${\rho }_c$, simulation results show that the temporal decay of $\varphi \left(t\right)$ is strongly affected by the FS effects up to much larger system size ($\sim {10}^6$). For the sizes for which $\varphi \left(t\right)$ is independent up to $t=2\times {10}^7$, we estimate ${\rho }_c=0.8925\left(1\right)$ and $\alpha =0.110\left(5\right)$, which clearly differ from the previous results for the random conditions, ${\rho }_c=0.89199\left(5\right)$ and $\alpha =0.141\left(24\right)$. Instead, the present ${\rho }_c$ agrees with ${\rho }_c=0.89255\left(2\right)$ of the regular conditions. In addition, the present $\alpha$ is substantially distinguishable from the results of the other types of initial conditions, $\alpha =0.159\left(3\right)$ and $0.146\left(2\right)$ for the natural and the regular conditions, respectively, which supports the claim of the initial condition dependence of dynamical exponents in the Manna class.

Figure 1

Plots of $\varphi (L,t)$ and $P(L,t)$ at criticality. (a) CP at $p_c=0.232674$. Two solid lines are $\varphi (L,t)$ for $L=512$ (black) and 4096 (red) from bottom. Dashed and dotted lines are $P\left(t\right)$ of $L=512$ and 4096. The vertical line indicates $t_P$. (b) FE Manna sandpile model at ${\rho }_c=0.892$. Two solid lines are $\varphi (L,t)$ for $5\times {10}^4$ (red) and $L=2000$ (black) from bottom. Dashed and dotted lines are $P\left(t\right)$ of $L=2000$ and $5\times {10}^4$. The vertical line indicates the expected ${\tau }_{\ell }$. In each panel, each $P\left(t\right)$ is multiplied by $0.5$ in (a) and $0.15$ in (b).

Figure 2

Plots of $\varphi (L,t)$ at ${\rho }_c=0.89199$ of various $L\mathrm{s}$. Main plot shows $\varphi$ of $L={10}^6$ (red line), ${10}^5$ (blue line), and $5\times {10}^4$ (black line) from top to bottom. Inset shows $\varphi$ of $L={10}^6$ ($◯$), $2\times {10}^6$ ($+$), and $3\times {10}^6$ ($\times$).

Figure 3

(a) $\varphi \left(t\right)$ of $L=2\times {10}^6$ and various $\rho$ larger than the known ${\rho }_c=0.89199$. For eight lines from top, $\rho$ of each line decreases by ${10}^{-4}$ from $0.8928$ to $0.8921$. The bottom line indicates $\varphi \left(t\right)$ of $\rho =0.89199$. (b) $\alpha \left(t\right)$ of each $\varphi$ in (a) with the same order of $\rho$ from top to bottom. In each panel, the fourth line (red color) from top indicates the data of ${\rho }_c=0.8925$.

Figure 4

The scaling plot of $\varphi t^{\alpha }$ vs. $t$ for $L=2\times {10}^6$. Main plot shows the scaling plot with $\alpha =0.11$ for several $\rho$'s. Each line corresponds to the plot of $\rho =0.8927, 0.8926, 0.8925$ (red), $0.8924$, and $0.8923$ from top to bottom. Inset shows the scaling plot for $\rho =0.8925$ with three $\alpha$ values, $0.115, 0.110$, and $0.105$ from top to bottom as indicated in the legend.