Matrix product solution of a left-permeable two-species asymmetric exclusion process

We study a two-species partially asymmetric exclusion process where the left boundary is permeable for the “slower” species but the right boundary is not. We find a matrix product solution for the stationary state and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero.

Figure 1

Stationary phase diagram of the semipermeable two-species ASEP. The density of species 1, ${\rho }^{\left(1\right)}$, is a fixed parameter in this system.

Figure 2

Phase diagram for lattice average current and density with $\lambda =\gamma /\alpha$.

Figure 3

Density profiles in the HD, LD and MC phases. Plots show densities of species 0 (red dotted line), 1 (green dashed line), and 2 (blue solid line) vs normalized site position $x=i/L$ for $L=500$ in (a) subphase HD1 for $\lambda =1.75,b=3.91,$ and $d=-0.66(\alpha =0.32,\gamma =0.56,\beta =0.35,\delta =0.9,q=0.41)$, (b) subphase HD2 for $\lambda =0.20,b=3.91,$ and $d=-0.66(\alpha =0.81,\gamma =0.16,\beta =0.35,\delta =0.9,q=0.41)$, (c) subphase LD1 for $\lambda =3.18,b=1.25,$ and $d=-0.59(\alpha =0.22,\gamma =0.7,\beta =0.64,\delta =0.47,q=0.41)$, (d) subphase LD2 for $\lambda =1.16,b=0.94,$ and $d=-0.68(\alpha =0.5,\gamma =0.58,\beta =0.95,\delta =0.61,q=0.41)$, (e) MC phase for $\lambda =0.37,b=0.75,$ and $d=-0.59(\alpha =0.62,\gamma =0.23,\beta =0.83,\delta =0.37,q=0.41)$, and (f) MC phase for $\lambda =0.91,b=0.26,$ and $d=-0.45(\alpha =0.45,\gamma =0.41,\beta =0.85,\delta =0.1,q=0.41)$. The subphases LD1, LD2, HD1, and HD2 will be described in Sec. 5b.

Figure 4

Plots of (a) time-averaged densities for $L=500$ and (b) instantaneous density profiles for $L=2500$, coarse-grained over 50 sites, on the coexistence line for $\lambda =0.37,b=0.75,$ and $d=-0.59(\alpha =0.15,\gamma =0.74,\beta =0.28,\delta =0.89,q=0.41)$. Each plot shows densities of species 0 (red dotted line), 1 (green dashed line), and 2 (blue solid line) vs normalized site position $x=i/L$.

Figure 5

Division of the HD phase according to the scaling of ${\rho }_L^{\left(1\right)}$, as given in Eqs. (42, 43, 44).

Figure 6

Division of phases according to the difference between bulk and boundary densities for a fixed value of $d$.