Nonequilibrium steady states in a closed inhomogeneous asymmetric exclusion process with generic particle nonconservation

We study the totally asymmetric exclusion process (TASEP) on a nonuniform one-dimensional ring consisting of two segments having unequal hopping rates, or defects. We allow weak particle nonconservation via Langmuir kinetics (LK), which are parametrized by generic unequal attachment and detachment rates. For an extended defect, in the thermodynamic limit the system generically displays inhomogeneous density profiles in the steady state—the faster segment is either in a phase with spatially varying density having no density discontinuity, or a phase with a discontinuous density changes. Nonequilibrium phase transitions between the above phases are controlled by the inhomogeneity and LK. The slower segment displays only macroscopically uniform bulk density profiles in the steady states, reminiscent of the maximal current phase of TASEP but with a bulk density generally different from half. With a point defect, there are spatially uniform low- and high-density phases as well, in addition to the inhomogeneous density profiles observed for an extended defect. In all the cases, it is argued that the mean particle density in the steady state is controlled only by the ratio of the LK attachment and detachment rates.

Figure 1

TASEP and LK dynamics on a 1D ring with unequal hopping rates in CHI and CHII; the LK detachment and attachment rates are $\omega$ and $\omega K$, respectively; see the text.

Figure 2

Plots of the density profiles in CDP ($K=0.5,p=0.2,\mathrm{\Omega }=4.5,l=1/2,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown. MCS results for $n_{\text{II}}\left(x\right)$ are shown with magenta triangles; $x_{\alpha }$ and $x_{\beta }$ are extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right), j_{\text{IA}}\left(x\right)$, and $j_0\left(x\right)$ are plotted; $x_{\alpha }$ and $x_{\beta }$ are extracted (see the text), which match well with their corresponding MCS results.

Figure 3

Plots of the density profiles in SP ($K=0.5,p=0.2,\mathrm{\Omega }=2.25,l=1/2,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown. MCS results for $n_{\text{II}}\left(x\right)$ are shown with magenta triangles; $x_w$ is extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right)$ and $j_{\text{IA}}\left(x\right)$ are plotted; $x_w$ has been extracted (see the text), which match well with their corresponding MCS results.

Figure 4

Plots of the density profiles in the borderline case ($K=0.5,p=0.2,\mathrm{\Omega }=3.75,l=1/2,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown. MCS results for $n_{\text{II}}\left(x\right)$ are shown with magenta triangles; $x_{\alpha }$ and $x_{\beta }$ are extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right), j_{\text{IA}}\left(x\right)$, and $j_0\left(x\right)$ are plotted; $x_{\alpha }$ and $x_{\beta }$ are extracted (see the text), which match well with their corresponding MCS results.

Figure 5

Phase diagram in the $\mathrm{\Omega }-p$ plane for extended defects, with $K=0.5$: mean-field result (red continuous line) and MCS results (black circles) are shown, which agree well. Here $l=1/2$ and $N=1000$.

Figure 6

Phase diagram in the $\mathrm{\Omega }-p$ plane for extended defects, with $K=0.2$: mean-field result (red continuous line) and MCS results (black circles) are shown, which agree well. Note that while Fig. 6 is quantitatively different from Fig. 5, the phase diagrams are qualitatively similar as they separate the same phases from each other. Here $l=1/2$ and $N=1000$.

Figure 7

Plots of the density profiles in SP ($K=0.2,p=0.2,\mathrm{\Omega }=4.5, l=1/2,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown. MCS results for $n_{\text{II}}\left(x\right)$ are shown with magenta triangles; $x_{\alpha }$ and $x_{\beta }$ are extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right), j_{\text{IA}}\left(x\right)$, and $j_0\left(x\right)$ are plotted; $x_{\alpha }$ and $x_{\beta }$ are extracted (see the text), which match well with their corresponding MCS results.

Figure 8

Plots of the density profiles in the LD region for a point defect located at $x=1$ ($K=0.5,p=0.6,\mathrm{\Omega }=1.0,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown. $n_0$ (green dotted line) is the average density of the system. We have extended the $x$ axis up to $x=1.1$ to resolve the peak at $x=1$ (the location of the point defect) properly.

Figure 9

Plots of the density profiles in the CDP ($K=0.5,p=0.05,\mathrm{\Omega }=2.0,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown; $x_w$ is extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right)$ and $j_{\text{IA}}\left(x\right)$ are plotted; $x_w$ has been extracted (see the text). Qualitative agreement with their corresponding MCS results is evident.

Figure 10

Plots of the density profiles in the SP phase ($K=0.5,p=0.2,\mathrm{\Omega }=0.75,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown; $x_w$ is extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right)$ and $j_{\text{IA}}\left(x\right)$ are plotted; $x_w$ has been extracted (see the text). Qualitative agreement with their corresponding MCS results is evident.

Figure 11

Plots of the density profiles in the borderline case for a point defect located at $x=1$ ($K=0.5,p=0.2,\mathrm{\Omega }=1.25,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown; $x_{\alpha }$ and $x_{\beta }$ are extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right), j_{\text{IA}}\left(x\right)$, and $j_0\left(x\right)$ are plotted; $x_{\alpha }$ and $x_{\beta }$ are extracted (see the text), which match well with their corresponding MCS results.

Figure 12

Phase diagram in the $\mathrm{\Omega }-p$ plane for a point defect located at $x=1$, with $K=0.5$: mean-field results (red solid line) and the MCS results are shown. The green dotted horizontal line, separating the LD phase from other phase regions, is obtained from MFT and satisfies the equation $p=K$. The corresponding MCS data points are represented by the blue triangles. Here $N=1000$.

Figure 13

Phase diagram in the $\mathrm{\Omega }-p$ plane for a point defect located at $x=1$, with $K=0.2$: mean-field results (red solid line) and the MCS results are shown. The green dotted horizontal line, separating the LD phase from other phase regions, is obtained from MFT and satisfies the equation $p=K$. The corresponding MCS data points are represented by the blue triangles. Note that a lower value of $K$ corresponds to a lower attachment rate, and hence the LD profile occupies a bigger region in the phase diagram than it does for a higher value of $K$ (see Fig. 12). Here $N=1000$.

Figure 14

Plots of the density profiles in the SP for a point defect located at $x=1$ ($K=0.2,p=0.05,\mathrm{\Omega }=2.0,N=2000$). Mean-field solutions $n_{\text{IB}}\left(x\right)$ (black dotted line), $n_{\text{IA}}\left(x\right)$ (blue solid line), and the corresponding MCS results $n_{\text{I}}\left(x\right)$ (red squares) are shown; $x_{\alpha }$ and $x_{\beta }$ are extracted from MCS data (see the text). $n_0$ (green dotted line) is the average density of the system. Inset: mean-field values of the currents $j_{\text{IB}}\left(x\right), j_{\text{IA}}\left(x\right)$, and $j_0\left(x\right)$ are plotted; $x_{\alpha }$ and $x_{\beta }$ are extracted (see the text), which match well with their corresponding MCS results.