Exclusion processes with avalanches

In an exclusion process with avalanches, when a particle hops to a neighboring empty site which is adjacent to an island the particle on the other end of the island immediately hops, and if it joins another island this triggers another hop. There are no restrictions on the length of the islands and the duration of the avalanche. This process is well defined in the low-density region $\rho <\frac{1}{2}$. We describe the nature of steady states (on a ring) and determine all correlation functions. For the asymmetric version of the process, we compute the steady state current, and we describe shock and rarefaction waves which arise in the evolution of the step-function initial profile. For the symmetric version, we determine the diffusion coefficient and examine the evolution of a tagged particle.

Figure 1

A hopping event on a ring with seven particles and five empty sites. In this example, a particle (shown as an empty disk) hops to the vacant site on the right. This completes the hopping event in the case of the SEP, all other particles (filled disks) remain in their sites. In the case of the EPA, the initial hop triggers an avalanche with three induced hops (shown is the final configuration). The initial configuration has four islands. After the hopping event there are three islands for the SEP and four islands for the EPA: In the latter case, the total number of islands cannot decrease.

Figure 2

Density profiles: $\rho$ versus the scaled spatial coordinate $\xi =x/L$. Shown are simulation results for the system with $L={10}^3$ and $T={10}^8$ for three different densities on the left. Also shown are theoretical predictions Eq. (42).

Figure 3

The amplitude of self-diffusion $\mathcal{D}\left(\rho \right)$ in one dimension as a function of density $\rho$. Shown are simulation results on the ring of length $L={10}^4$ for $T={10}^4$; the averaging was taken over ${10}^3$ configurations for each $\rho$. Also shown for reference is the theoretical prediction Eq. (40).