Totally asymmetric exclusion process with long-range hopping

Generalization of the one-dimensional totally asymmetric exclusion process (TASEP) with open boundary conditions in which particles are allowed to jump $l$ sites ahead with the probability $p_l\sim 1∕l^{\sigma +1}$ is studied by Monte Carlo simulations and the domain-wall approach. For $\sigma >1$ the standard TASEP phase diagram is recovered, but the density profiles display additional features when $1<\sigma <2$. At the first-order transition line, the domain wall is localized and phase separation is observed. In the maximum-current phase the profile has an algebraic decay with a $\sigma$-dependent exponent. Within the $\sigma \le 1$ regime, where the transitions are found to be absent, analytical results in the continuum mean-field approximation are derived in the limit $\sigma =-1$.

Figure 1

Fundamental diagram of the TASEP with long-range hopping with $\sigma =1.5$ (solid line) and with nearest-neighbor hopping (dashed line).

Figure 2

Phase diagram for $\sigma >1$ from the results of Monte Carlo simulations consisting of low-density $\left(A\right)$, high-density $\left(B\right)$, and maximum-current $\left(C\right)$ phases.

Figure 3

Dependence of the bulk density $\overline{\rho }$ on $\alpha$ for $\beta =0.3$ and various $\sigma (L=200,t=2\times {10}^6$ MCS/site) compared to the bulk density in the TASEP (dashed lines).

Figure 4

Dependence of current $j_L(\alpha ,\beta ,\sigma )∕{\lambda }_L$ on $\alpha$ for $\beta =0.7$ and various $\sigma (L=200,t=2\times {10}^6$ MCS/site) compared to the current in the TASEP (dashed line).

Figure 5

(Color online) Comparison of the density profiles $(\alpha =\beta =0.2$, $\sigma =1.5$) for various chain sizes $L=10$, $100$, and $1000$ obtained from MC simulations (symbols) and from the domain-wall approach (solid lines).

Figure 6

(Color online) Comparison of the density profiles $(\alpha =\beta =0.2$, $\sigma =3.0$) for various chain sizes $L=10$, $100$, and $1000$ obtained from simulations (symbols) and from the domain-wall approach (solid lines).

Figure 7

Standard deviation of the domain-wall position according to the stationary probability $P_i$ for $\alpha =\beta =0.1$ and various $\sigma$.

Figure 8

(Color online) Deviation $\mathrm{\Delta }\rho (k∕L)$ of the density profile from its bulk value $\overline{\rho }=1∕2$ for $\sigma =1.5$ and various system sizes $L$ $(\alpha =\beta =0.8$), before (a) and after (b) the scaling relation (26) was applied.

Figure 9

(Color online) Deviation $\mathrm{\Delta }\rho (k∕L)$ of the density profile from its bulk value $\overline{\rho }=1∕2$ for various system sizes $L=800$, $1500$, and $5000$ $(\alpha =\beta =0.8$) for $\sigma =1.2$, $1.5$, and $1.8$ (from top to bottom). The profiles $\mathrm{\Delta }\rho (k∕L)$ for the same $\sigma$ are scaled to the profile $\mathrm{\Delta }\rho (k∕L_0)(L_0=5000)$ according to Eq. (26).