Jamming and condensation in one-dimensional driven flow

We revisit the slow-bond (SB) problem of the one-dimensional (1D) totally asymmetric simple exclusion process (TASEP) with modified hopping rates. In the original SB problem, it turns out that a local defect is always relevant to the system as jamming, so that phase separation occurs in the 1D TASEP. However, crossover scaling behaviors are also observed as finite-size effects. In order to check if the SB can be irrelevant to the system with particle interaction, we employ the condensation concept in the zero-range process. The hopping rate in the modified TASEP depends on the interaction parameter and the distance up to the nearest particle in the moving direction, besides the SB factor. In particular, we focus on the interplay of jamming and condensation in the current-density relation of 1D driven flow. Based on mean-field calculations, we present the fundamental diagram and the phase diagram of the modified SB problem, which are numerically checked. Finally, we discuss how the condensation of holes suppresses the jamming of particles and vice versa, where the partially condensed phase is the most interesting, compared to that in the original SB problem.

Figure 1

The modified TASEP is schematically illustrated, where the numbers above the arrows indicate hopping rates, and the site indices are shown at the bottom. The hopping is forbidden due to hard-core repulsion, which is shown as a red cross, and the hopping at the SB is highlighted as the different color (red) arrow.

Figure 2

(a) In the $b-r$ plane, the phase diagram of the modified SB problem is drawn as a heat map, in terms of the density difference $\mathrm{\Delta }={\rho }_{{}_{+}}-{\rho }_{{}_{-}}$. MF phase boundaries are drawn for the NS-S from Eq. (12) (red, solid line) up to $0<b<3$, and the C-PC from Eq. (13) (red, solid line) for $b>3$, and the S-PC from Eq. (15) (dashed line). The NS-C boundary is obtained from Eq. (16) (dashed line). Around $b=0$, the NS-S-NS boundaries are drawn from Eq. (18) (green, solid and dotted lines). The shaded region presents with $J/{\rho }_{{}_{-}}>0.999$, which denotes condensation of holes in numerical measurement. For examples, we show the NS phase at (b) $(b=2.75,r=0.5)$ with $\mathrm{\Delta }=0$ and the PC phase at (c) $(2.75,0.2)$ with $\mathrm{\Delta }>0$. Numerical data are obtained for $L=2^{16}$ at $T\gg 2L^{3/2}$, averaging over ${10}^8$ samples with ${10}^4$ different configurations and ${10}^4$ different times.

Figure 3

Snapshots of spatiotemporal patterns of $L=2^{10}$ (horizontal length of each box) and $T=768$ (vertical length of each box) are plotted at every consecutive $\mathrm{\Delta }t=4\text{MC}$ time step in the steady states $(t\gg 2L^{3/2})$ for various cases. Dots represent particles, and the SB is highlighted for the case of $r<1$ as the vertical line in the middle of the pattern with a different color (red). In each pattern, time elapses from top to bottom, and the direction of particle hopping is to the right. Each column represents the different type of particle interaction: repulsive ($b=-0.75$), neutral ($b=0$), attractive ($b=2.75$), and strongly attractive ($b=4.00$), respectively. Each row is classified by the different SB factor: $r=1.0,0.5,\text{and}0.2$, respectively. For the PC phase, a typical configuration is presented as the two rightmost in the bottom panel.

Figure 4

Density profiles are shown as $b$ varies from repulsive to attractive interactions, $b\in \{-0.75,0.00,2.75,4.00\}$, where we set $r=0.2$ in (a) and $r=0.5$ in (b). Numerical data are obtained in the system of $L=2^{12}$ with the SB that is located at the bond of $(\frac{L}{2},\frac{L}{2}+1)$ (the middle of the system) and averaging over ${10}^8$ samples with ${10}^4$ configurations and ${10}^4$ different times, in the steady-state limit $(t\gg 2L^{3/2})$.

Figure 5

For various $b\in \{-0.75, -0.5, -0.25$, 0, 0.25, 0.5, 1, 2, 3, $4\}$ (from blue to red) that are drawn as different colors, (a) the fundamental diagram by MF approximations is presented with numerical results. Numerically obtained high or low-density ${\rho }_{{}_{\pm }}$ (or $\rho =1/2$ in the NS phase) is plotted with different symbols $(+/\circ )$ for various $r$. Solid lines are drawn by Eq. (8) as $b$ varies, and the dashed line represents the ordinary TASEP ($b=0$). The forbidden-density regions (see the text for detailed discussion) are shown with shaded patterns. (b) Possible high and low densities are plotted as $r$ varies: the case of $b=-0.75$ and the case of $b=2.00$. The dashed guidelines are shown: ($b=-0.75,r=0.25$) with $J=J_1$ and ($b=2.00,r=0.40$) with $J=J_2$, respectively. It is noted that numerical data are overlapped as $r$ gets larger.

Figure 6

The average current $J(b;r)$ and related quantities are shown as a function of $b$. Each line represents the different values of $r$. The color scale from bright to dark is from $r=1$ to $r=0.1$: (a) the ZRP-type current $J$ with $J_{{}_{\max }}={\rho }_{{}_0}=1/2$, (b) the conventional TASEP current $\tilde{J}(\equiv J/u_{{}_{\max }})$ with the maximal current ${\tilde{J}}_{{}_{\mathrm{mc}}}=1/4$, and (c) the relative-current difference between the current without the SB and that with the SB. Symbols represent different system sizes: $L=2^{10}(\diamond ),2^{12}(\times ),2^{14}(\square ),2^{16}(+)$. In (c), the splitting lines in the small value of current difference are caused by finite size effects at the NS-S boundary (see Sec. 4 for detailed discussion).

Figure 7

The distribution of interparticle distance $P(\ell ;L)$ is plotted for $L=2^{10}(\diamond ),2^{12}(\times ),2^{14}(\square ),2^{16}(+)$ at three different phases: (a) The NS phase at $(b=0.75,r=0.80)$ shows $P(\ell ;L)\sim exp(-\ell /{\ell }_{{}_{\mathrm{NS}}})$, where ${\ell }_{{}_{\mathrm{NS}}}$ is independent of $L$; (b) the S phase at (1.5, 0.15) shows $P(\ell ;L)\sim L^{-1/2}exp(-\ell /L^{1/4})$; (c) the PC phase at (6.00, 0.10) shows $P(\ell ;L)L\sim f_{{}_{\mathrm{PC}}}(x_1,x_2)$, where $f_{{}_{\mathrm{PC},1}}\left(x_1\right)=x_1^{-4}$ for $x_1<1, f_{{}_{\mathrm{pc},1-2}}\left(x\right)=\text{constant}$ for $x_1<x<x_2$, and $f_{pc,2}\left(x_2\right)\sim exp(-x_2)$ for $x_2>1$ with $x_1\equiv \ell /L^{1/4}$ and $x_2\equiv \ell /L^{1/2}$. In particular, scaling collapses are tested in the inset of (b) and (c).

Figure 8

The ZRP-type mapping of Fig. 1 is illustrated, where the site index with a number represents the particle index in the modified TASEP, and particles at each site represent links between consecutive particles in the modified TASEP. The slow bond becomes a slow particle as drawn in the different colored (red) circle.

Figure 9

An example of the current-density relation.

Figure 10

Interparticle distance distributions for $L=2^{16}$.