Effects of junctional correlations in the totally asymmetric simple exclusion process on random regular networks

We investigate the totally asymmetric simple exclusion process on closed and directed random regular networks, which is a simple model of active transport in the one-dimensional segments coupled by junctions. By a pair mean-field theory and detailed numerical analyses, it is found that the correlations at junctions induce two notable deviations from the simple mean-field theory, which neglects these correlations: (1) the narrower range of particle density for phase coexistence and (2) the algebraic decay of density profile with exponent $1/2$ even outside the maximal-current phase. We show that these anomalies are attributable to the effective slow bonds formed by the network junctions.

Figure 1

TASEP on the CDRR networks and its approximate descriptions. (a) Each junction (node) has $c$ outgoing and $c$ incoming links, where each link is by itself a chain of $L$ inner sites. The dynamics is identical to the 1D TASEP except that a particle at a junction randomly chooses one of the $c$ adjacent sites in the outgoing links as its next destination. (b) The DMF theory assumes that each link is an open 1D system whose entry rate $\tilde{\alpha }$ and exit rate $\tilde{\beta }$ are given by the occupancy ${\rho }_u$ of each junction. (c) The DPMF theory assumes that only the pairs consisting of a junction and its adjacent site are correlated. The entry (exit) rate of a link is given by the occupancy ${\rho }_1$ (${\rho }_L$) of the first (last) inner site of the link.

Figure 2

Fundamental diagram of the density-current relation for the CDRR networks with $c=2$ (red/gray) and $c=3$ (black). The DMF predictions (dashed lines), the DPMF predictions (solid lines), and the simulation results obtained at $L=100$ ($\times$), 200 ($◯$), 400 ($△$), 800 ($\square$) are shown. (Inset) Covariance (${\sigma }_{ab}\equiv \langle {\tau }_a{\tau }_b\rangle -\langle {\tau }_a\rangle \langle {\tau }_b\rangle$) between a junction and its adjacent site in an incoming link (upper two curves) and in an outgoing link (lower two curves) at $L=800$, where the lines are guides to the eye. The CDRR networks with $N={10}^3$ are used.

Figure 3

Deviations of the simulation results from the DPMF (main plot) and the DMF (inset) predictions. The CDRR networks with $N={10}^3$ and $c=2$ are used. The lines are guides to the eye.

Figure 4

Variances of the mean occupancies ${\rho }_u$ among difference junctions (main plot) and the mean currents $J_{uv}$ among different links (inset) as functions of the observation time $T$. A CDRR network with $N={10}^3$, $c=2$, and $L={10}^2$ is used. The lines are guides to the eye.

Figure 5

(a) Density profile of particles in the low-density phase. For $\rho =0.34$, the particle density converges to the bulk value $\tilde{\rho }$ (b) algebraically near the entrance and (c) exponentially near the exit. (d) Density profile of particles in the high-density phase. For $\rho =0.66$, the particle density converges to the bulk value $\tilde{\rho }$ (e) algebraically near the exit and (f) exponentially near the entrance. The CDRR networks with $N={10}^3$ and $c=2$ are used.