Asymmetric exclusion process with global hopping

We study a one-dimensional totally asymmetric simple exclusion process with one special site from which particles fly to any empty site (not just to the neighboring site). The system attains a nontrivial stationary state with a density profile varying over the spatial extent of the system. The density profile undergoes a nonequilibrium phase transition when the average density passes through the critical value $1-{\left[4(1-\ln 2)\right]}^{-1}=0.185277...,$ viz., in addition to the discontinuity in the vicinity of the special site, a shock wave is formed in the bulk of the system when the density exceeds the critical density.

Figure 1

The TASEP with global hopping from one site.

Figure 2

Smooth profiles. The top curve is the critical profile, i.e., $v=v_c$, and the lower curve is a subcritical profile with $v=0.95$. They are realized by Monte Carlo simulations. The symbols $\square$ and $\circ$ correspond to simulation results with parameters $(v,\beta )=(v_c,0.6)$ and $(v,\beta )=(0.95,0.6)$, respectively. We have set $L={10}^5$ sites and took an average over time steps ${10}^{11}\le T\le 2\times {10}^{11}$.

Figure 3

The right-hand side of Eq. (16) (top), the density just before the shock $r$ (middle), and the position of the shock $s$ (bottom) are determined by $R$. The fraction of the vacant sites is chosen as the following four values: $v=0.1$ (dotted), 0.5 (bold), $v_c$, and 0.9 (dashed).

Figure 4

The density profiles [Eqs. (11) and (13)] with a shock arising at $v=0.6$ and $R=0.9,0.7,0.5$. They are realized by simulations with $\beta =0.1(\square ),0.3(\circ ),0.7\left(▵\right)$, respectively, where $L={10}^5$ and the averages over time steps ${10}^{11}\le T\le 2\times {10}^{11}$ were taken.

Figure 5

The phase diagram of the model. The full lines represent the boundaries between the three phases of the model: a phase with a smooth density profile and two domains where a shock appears. The two shock phases differ at the right boundary: When $R\ne {\rho }_L,$ a boundary layer is formed; when $R={\rho }_L=1-\beta$, the solution of the inviscid hydrodynamic equation provides an excellent description of the density profile of the entire system.

Figure 6

A smooth profile and a shock profile are solutions to Eq. (6) with $v=0.85$ (solid lines). For the shock profile, we have set the parameter $R=0.75$. The symbols $\circ$ correspond to the simulation result with $(v,\beta )=(0.85,0.25)$, where $L={10}^5$ and the average over time steps ${10}^{11}\le T\le 2\times {10}^{11}$ was taken. We observe from the simulation that the shock profile is selected with $R=1-\beta$.

Figure 7

Comparison with a special case of the PFF model. The parameters of our model are chosen as $(\beta ,v)=(0.6,0.95)$ ($\circ$, smooth profile) and $(\beta ,v)=(0.6,0.3)$ ($\square$, shock profile), corresponding to the PFF model with $(\beta ,{\Omega }_a)=(0.6,0.09645\cdots )$ ($\times$) and $(0.6,0.8)$ ($+$), respectively. In both models we have set $L={10}^3$ (and we took an average over time steps $5\times {10}^8\le T\le 5\times {10}^9$), but we observe a difference around the shock position.