Exact steady states in the asymmetric simple exclusion process beyond one dimension

Abstract

The asymmetric simple exclusion process (ASEP) is a paradigmatic nonequilibrium many-body system that describes the asymmetric random walk of particles with exclusion interactions in a lattice. Although the ASEP is recognized as an exactly solvable model, most of the exact results obtained so far are limited to one-dimensional systems. Here, we construct the exact steady states of the ASEP with closed and periodic boundary conditions in arbitrary dimensions. This is achieved through the concept of transition decomposition, which enables the treatment of the multi-dimensional ASEP as a composite of the one-dimensional ASEPs.

Figures (3)

• Figure 1: ASEP under various boundary conditions. The 1D ASEP with (a) periodic boundary conditions and (b) closed boundary conditions. The 2D ASEP with (c) periodic$\times$closed boundary conditions (multi-lane ASEP) and (d) closed$\times$closed boundary conditions.
• Figure 2: Example of the transition of states from a configuration $n=\{ (1,1), (2,2), (2,3) \}$ in the 2D ($2\times3$) ASEP with $\ell=1$. All configurations that can transition from a configuration $n$ in the 2D ASEP can be regarded as the transition of the five 1D ASEPs (a)-(e).
• Figure 3: (a) 2D ASEP with $\ell=1$. We set $(L_1,L_2)=(3,8)$, $p_2(r_1)=0.3$ for $r_1=1,2$ and $p_2(r_1)=1.0$ for $r_1=3$, and $q_2(r_2)=0$. (b) Relation between the quasi-one-dimensional current $j$ and the density $\rho$ for various hopping rates $(p_1,q_1)$ in the 2D ASEP.