Bulk and surface transitions in asymmetric simple exclusion process: Impact on boundary layers

In this paper, we study the boundary-induced phase transitions in a particle nonconserving asymmetric simple exclusion process with open boundaries. Using a boundary layer analysis on the mean field version of the model, we show that the key signatures of various bulk phase transitions are present in the boundary layers of the density profiles. In addition, we also find surface transitions in the low- and high-density phases. The surface transition in the low-density phase provides a complete description of the nonequilibrium critical point found in this system.

Figure 1

Phase diagram for $K={\omega }_a∕{\omega }_d=1$ and $\Omega ={\omega }_{d}N<0.5$. $N$ is the number of lattice points. The coexistence of a maximal-current and a low- or a high-density phase is represented by $LM$ or $HM$, respectively. The coexistence of low-density, maximal-current, and high-density phases is similarly represented by $LMH$. Approach to different phase boundaries is indicated by the paths with arrows. Divergences, as the phase boundaries are approached along these paths, are discussed in Sec. 3A, 3C, 4. “$l$” and “$r$” indicate the presence of boundary layers at left and right boundaries, respectively. Dashed lines represent the surface transition lines.

Figure 2

Phase diagram for $K\ne 1$ and $\Omega ={\omega }_{d}N<0.5$. $N$ represents the number of lattice points. Dashed lines represent the surface transition lines. “bl” implies boundary layer. “l” and “r” indicate the presence of boundary layers at left and right boundaries, respectively. The filled black circle represents the critical point.

Figure 3

Density profile with positive slope at the right boundary for $K=1$ with $\alpha =0.1$, $\gamma =0.44$, $ϵ=0.0035$, and $\Omega =0.2$.

Figure 4

Density profile with negative slope at the right boundary for $K=1$ with $\alpha =0.2$, $\gamma =0.3$, $ϵ=0.0035$, and $\Omega =0.2$.

Figure 5

Dotted line represents the surface transition line in the low-density phase for $K=3$ and $\Omega =0.1$. The solid line corresponds to the phase boundary between the low-density and the shock phase. The intersection of the two lines is the critical point $({\alpha }_c,{\gamma }_c)$.

Figure 6

Density profile with positive slope at the left boundary for $K=1$ with $\alpha =0.55$, $\gamma =0.8$, $ϵ=0.002$ and $\Omega =0.2$.

Figure 7

Density profile with negative slope at the left boundary for $K=1$ with $\alpha =0.6$, $\gamma =0.76$, $ϵ=0.001$, and $\Omega =0.2$.

Figure 8

Density profile in the low-density maximal current $\left(LM\right)$ phase with $\alpha =0.45$, $\gamma =0.4$, $ϵ=0.002$ and $\Omega =0.2$.

Figure 9

Density profile in the high-density maximal current $\left(HM\right)$ phase with $\alpha =0.55$, $\gamma =0.58$, $ϵ=0.002$, and $\Omega =0.2$.

Figure 10

Contour plots in the $p$-$f$ plane for $\Omega =0.1$. The arrows on different contours indicate the directions of increasing $x^{*}$.