Exact Solution of the One-Dimensional Deterministic Fixed-Energy Sandpile

By reason of the strongly nonergodic dynamical behavior, universality properties of deterministic fixed-energy sandpiles are still an open and debated issue. We investigate the one-dimensional model, whose microscopical dynamics can be solved exactly, and provide a deeper understanding of the origin of the nonergodicity. By means of exact arguments, we prove the occurrence of orbits of well-defined periods and their dependence on the conserved energy density. Further statistical estimates of the size of the attraction’s basins of the different periodic orbits lead to a complete characterization of the activity vs energy density phase diagram in the limit of large system’s size.

Figure 1

The activity diagram $\rho (\zeta )$ (a) and the behavior of the period $T$ as a function of the energy density $\zeta$ (b) for a 1D-DFES of size $N=20$. Figures display all results, without averaging, obtained starting from 100 randomly chosen initial configurations at each energy value.

Figure 2

Graphical sketch of the method used to find out the structure of periodic configurations. The case $W=2$ is considered. Top figure shows the correspondence between the binary representation of the spatiotemporal dynamics and the evolution of the real configuration. An example of the procedure used to recover the real values assumed by a given site during the period is reported in the bottom series of draws. Site values can be univocally determined in $W+1$ steps. At each step, the value of the site at a certain time is computed from a number of constraints on its (spatiotemporal) neighbors.

Figure 3

In panel (a), the number ${\mu }_T(N)$ of configurations belonging to the attraction’s basin of orbits of period $T=1$ (crosses), 2 (squares), and $N$ (circles) is plotted as function of $N$. The exponential rate of growth is estimated by curve fitting [dashed line for ${\mu }_1(N)$ and dot-dashed line for ${\mu }_N(N)$]. The inset shows that the ratio ${\mu }_1(N)/{\mu }_N(N)$ decreases with $N$. Panel (b) reports the total number $\Omega (N)$ of configurations at $\zeta =1$ (or $\zeta =2$) as function of the system size $N$ (circles). The fit (dashed line) shows an exponential growth as $\Omega (N)\propto {\alpha }^N$ with $\alpha \simeq 3.5$.