Phase diagram of generalized totally asymmetric simple exclusion process on an open chain: Liggett-like boundary conditions

The totally asymmetric simple exclusion process with generalized update is a version of the discrete time totally asymmetric exclusion process with an additional interparticle interaction that controls the degree of particle clustering. Though the model was shown to be integrable on the ring and on the infinite lattice, on the open chain it was studied mainly numerically, while no analytic results existed even for its phase diagram. In this paper, we introduce boundary conditions associated with the infinite translation invariant stationary states of the model, which allow us to obtain the exact phase diagram analytically. We discuss the phase diagram in detail and confirm the analytic predictions by extensive numerical simulations.

Figure 1

Two instances of gTASEP configuration updates having probabilities (a) $\alpha p^2(1-\mu )\beta$ and (b) $\tilde{\alpha }{p}^2{\mu }^3\beta$. A particle jumps from a lattice site to the next site on the right with probability $p$ or stays with probability $(1-p)$ if the target site was empty before the update. The probabilities change to $\mu$ and $(1-\mu )$, respectively, if the target site has emptied at the same step. Similarly, in (a), a particle enters to the leftmost site that was empty before the update with probability $\alpha$, and in (b), the probability is $\tilde{\alpha }$ since the site was occupied. The particle in the rightmost site leaves the lattice with probability $\beta$.

Figure 2

Typical plots of $\tilde{\alpha }$ vs $\alpha$ in (a) attracting, $p<\mu$, and (b) repulsive, $p>\mu$, regimes for two definitions of $\tilde{\alpha }$: formula (6) (solid curves) and formula (7) (dashed curves).

Figure 3

Typical phase diagram of TASEP-like system with three phases: HD, LD, and MC. The HD-LD phase coexistence line is straight for known exactly solvable TASEPs. The stationary state trivializes on the dashed line.

Figure 4

An example of the current density plot for $p=0.8,\mu =0.9.$ The points $c_{+},c_{-}$, and $c_t$ show a pair of left and right reservoir densities corresponding to a point at the phase coexistence line and the density of the MC phase, respectively.

Figure 5

The values of ${\alpha }_t$ and ${\beta }_t$ at the triple point as functions of $p$ corresponding to $\mu =0.9$. The graphs cross at two points, corresponding to the PU and BSU cases. The down and up triangles, $▿,▵,$ are their values measured in simulations. The cross symbols $+$ are the values of ${\beta }_t$ in gTASEP with the left BC (7), which coincide with the ones of the Liggett's BCs within numerical precision.

Figure 6

The values of ${\alpha }_t$ and ${\beta }_t$ at the triple point as functions of $\mu$ corresponding to $p=0.6$ and $p=0.9$. The graphs cross at two points, corresponding to the PU and BSU cases. The down and up triangles, $▿,▵,$ are their values measured in simulations. The cross symbols $+$ are the values of ${\beta }_t$ in gTASEP with the left BC (7), which coincide with the ones of the Liggett's BCs within numerical precision.

Figure 7

The phase diagrams of gTASEP in the $\alpha -\beta$ plane for (a) $p=0.6$ and (b) $p=0.9$ with various values of $\mu$. The dashed curves are the lines (32), where the stationary state trivializes. The triangle symbols $▴$ are the simulation results.

Figure 8

Manifestation of HD-MC transitions. Numerical results for different values of $\beta$ at fixed $p=0.6,\mu =0.9,\alpha =0.9>{\alpha }_t=0.3228$. (a) Density profiles $c\left(x\right),x\in (0,1]$ along the system; (b) dependence of the current $j$ and the bulk density $c_b$ on $\beta$ at fixed $\alpha ,p,\mu$.

Figure 9

Manifestation of HD-LD transitions. Numerical results for different values of $\alpha \in (0.01,0.07)$ at fixed $p=0.6,\mu =0.99,\beta =0.2<{\beta }_t=0.49296$. (a) Density profiles for different values of $\alpha$. The profile at the coexistence line is perfectly straight. The change of the density profiles shape is used to determine the coexistence line coordinates. (b) Change of the bulk density $c_b$ at fixed $\beta$ as $\alpha$ crosses its value ${\alpha }_c$ at the HD-LD coexistence line. The transition is a bit “smoothed” since the system is finite.

Figure 10

Comparison of the phase diagrams of gTASEP with Liggett-like BCs (solid lines) and those with BC (7) (dashed lines) at three different values of $p,\mu$. Black triangles $▴$ are the simulations results. We show the triple point and yet another point illustrating that the HD-LD phase coexistence line of gTASEP with BC (7) is straight. One can see that the triple point of gTASEP with BC (7) is to the left of that with Liggett-like BCs when $p>\mu$ and is to the right when $\mu >p$.