Characterization of a transition in the transport dynamics of a diffusive sandpile by means of recurrence quantification analysis

Recurrence quantification analysis (RQA) is used to characterize a dynamical transition that takes place in the diffusive sandpile. The transition happens when a combination of the drive strength, diffusivity, and overturning size exceeds a critical value. Above the transition, the self-similar transport dynamics associated to the classical (nondiffusive) sandpile is replaced by new transport dynamics dominated by near system-size, quasiperiodic avalanche events. The deterministic content of transport dynamics, as quantified by RQA, turns out to be quite different in both phases. The time series analyzed with RQA in this work correspond to local sand fluxes at different radial locations across the diffusive sandpile.

Figure 1

(a) Sketch of the one-dimensional sandpile in real space explaining the corresponding automaton rules. (b) Sketch of the sandpile automaton rules in gradient space. The parameters for this sandpile are $Z_c=20$ and $N_f=5$. Red cells are unstable; the sand that has been moved to a location is shown in orange; the void left by the move, in light blue.

Figure 2

Scheme with the possible values of $h$ for the last four cells of the sandpile (case $D_0/P_0=0$). $h_u,h_a$, and $h_l$ stand for the upper, average, and lower values of the height, respectively. $h\left(L\right)=0$ is the boundary condition for the bottom cell. Since $h\left(L\right)=0$, the upper limit on the height of the previous cell is $h_u(L-1)=Z_c$. Hence, $h(L-1)\le h_u(L-1)$. But $h(L-1)\ge h_l(L-1)=Z_c-N_f$ to satisfy the steady state condition. Thus, its mean value is $h_a(L-1)=[h_u(L-1)+h_l(L-1)]/2=Z_c-N_f/2$. The same principle can then be applied inwards.

Figure 3

Probability of overturning events per iteration as a function of the cell position for different values of the diffusivity ratio $D_0/P_0$. Filled symbols stand for the analytical estimation by means of Eq. (7). Open symbols stand for the simulations from the diffusive sandpile by using the following parameters: $L=200,Z_c=100,N_f=10$, and $P_0=0.01$.

Figure 4

Typical avalanche activity in the diffusive sandpile for parameter values that put its dynamics below (a–c) or above (b–d) the transition discussed in the main text. Unstable sites are represented with black dots and stable ones are represented with white dots (blanks). Panels (c)–(d) are zooms of the regions marked between horizontal, dashed lines in panels (a)–(b), respectively. Both simulations have identical parameters except that $N_f=5\to \kappa =6.25$ in (a)$-$(c) and $N_f=15\to \kappa =56.25$ in (b)$-$(d).

Figure 5

Time records of $\mathrm{\Gamma }\left(t\right)$ for different regimes: (a) below the transition, inside the SOC region ($x>x_t$); (b) above the transition, inside the SOC region ($x>x_t$); (c) in the vicinity of the crossover location ($x\simeq x_t$); and (d) within the diffusive region ($x<x_t$).

Figure 6

Recurrence plots obtained from local fluxes at points where transport dynamics are different inside the diffusive sandpile: (a) within the diffusion-dominated region ($x<x_t$) and (b) within the avalanche region ($x>x_t$), but with transport dominated by large quasiperiodic avalanches ($\kappa >{\kappa }_c$). The parameters used are $\text{RR}=5\%, m=4$, and $\tau =20\mathrm{\Delta }$.

Figure 7

Fraction of false nearest neighbors as a function of $m$ for different values of the diffusion parameter $D_0/P_0$.

Figure 8

Temporal averaged determinism of the flux temporal series as a function of the position for varying values of the parameter $\kappa$. The parameters used in the simulations were $L=400,P_0=5\times {10}^{-4},N_f=30$, and $Z_c=200$.

Figure 9

Averaged determinism of the flux series as a function of ${log}_{10}\left(\kappa \right)$ for several values of $N_f$.

Figure 10

Evolution of an avalanche in $\left|Z\right|$ space for a sandpile with $N_f=5$ and $Z_c=20$ (see text for explanation). Mean slope profile is around $Z_c-N_f/2=17.5$, as shown with a red dashed line.

Figure 11

Evolution of an avalanche that reaches the center of the pile in $\left|Z\right|$ space for a sandpile with the same parameters as in Fig. 10 (see text for explanation).

Figure 12

Evolution of an avalanche that reaches the edge of the pile in $\left|Z\right|$ space for a sandpile with the same parameters as in Figs. 10 and 11 (see text for explanation).

Figure 13

Histogram of starting (left) and stopping (right) points for avalanches collected for several values of the parameter $N_f$. The parameters in common were $L=400,D_0/P_0=0.25$, and $Z_c=200$. Total duration of each run is $32\times {10}^6$ iterations.