Lattice model of reduced jamming by a barrier

We study an asymmetric simple exclusion process in a strip in the presence of a solid impenetrable barrier. We focus on the effect of the barrier on the residence time of the particles, namely, the typical time needed by the particles to cross the whole strip. We explore the conditions for reduced jamming when varying the environment (different drifts, reservoir densities, horizontal diffusion walks, etc.). In particular, we discover an interesting nonmonotonic behavior of the residence time as a function of the barrier length. Besides recovering by means of both the lattice dynamics and the mean-field model well-known aspects like the faster-is-slower effect and the intermittence of the flow, we propose also a birth-and-death process and a reduced one-dimensional (1D) model with variable barrier permeability to capture the behavior of the residence time with respect to the parameters.

Figure 1

Schematic representation of the model in the presence of the barrier. Solid squares represent the particles at rest modeling the barrier.

Figure 2

Simulated 2D density profiles for the lattice model in the presence of the barrier: comparison between low and high bottom boundary densities. On the left ${\rho }_{\mathrm{d}}=0.9$ and on the right ${\rho }_{\mathrm{d}}=0.0$, three values of the barrier width are compared: $W=85$ (top), $W=90$ (middle), and $W=95$ (bottom). The corresponding simulated mean residence time is equal to 81359.8 (b), 101390 (d), 146403 (f) for ${\rho }_{\mathrm{d}}=0.0$; and 146678 (a), 119865 (c), and 162350 (e) for ${\rho }_{\mathrm{d}}=0.9$. The other model parameters used in the simulation are $L_1=100, L_2=400, h=0.5, \delta =0.05, {\rho }_{\mathrm{u}}=1, O_2=3$.

Figure 3

Comparison between the density profile obtained by averaging the 2D lattice simulation and the analytical solution of the 1D mean-field equation. Parameters: $L_1=100, L_2=400, h=0.5, \delta =0, W=70, O_2=3, {\rho }_{\mathrm{u}}=1$, and ${\rho }_{\mathrm{d}}$ as listed in the inset. For the 1D model the tortuosity coefficient has been chosen equal to 0.45 for all the values of ${\rho }_{\mathrm{d}}$. Thick lines correspond to Monte Carlo data for the lattice model and thin lines correspond to the analytical solution of the 1D model.

Figure 4

Density profile obtained by averaging the 2D lattice simulation: comparison for different values of $W$. Parameters: $L_1=100, L_2=400, h=0.5, \delta =0, {\rho }_{\mathrm{u}}=1, {\rho }_{\mathrm{d}}=0$ (a) and ${\rho }_{\mathrm{d}}=0.9$ (b), $O_2=3$, and $W$ as listed in the inset. In the inset we have also listed the Monte Carlo–averaged residence time data discussed in Sec. 3a3.

Figure 5

The birth-and-death and mean-field approximations to the actual measured mean residence time (labeled LA). Parameters: $L_1=100, L_2=400, \delta =0, {\rho }_{\mathrm{u}}=1, {\rho }_{\mathrm{d}}=0, O_2=3$, and $h=0.5$.

Figure 6

The birth-and-death and mean-field approximations to the actual measured mean residence time (labeled LA). Parameters: $L_1=100, L_2=400, \delta =0, {\rho }_{\mathrm{u}}=1, {\rho }_{\mathrm{d}}=0.9, O_2=3$, and $h=0.5$.

Figure 7

Simulated mean residence time for the 2D lattice model as a function of ${\rho }_{\mathrm{d}}$ for the different values of $W$ listed in the inset. Parameters: $L_1=100, L_2=400, h=0.5, \delta =0.1, {\rho }_{\mathrm{u}}=1$, and $O_2=3$.

Figure 8

Diagram of the residence time behavior with respect to the bottom boundary density ${\rho }_{\mathrm{d}}$ and the barrier width $W$. Here we plot the sign of the numerical derivative of $R$ with regard to $W$. Yellow part: the residence time increases with increase in $W$. Blue part: the residence time decreases with increase in $W$. Other parameters: $L_1=400, L_2=100, \delta =0.1, h=0.5$.

Figure 9

Simulated mean residence time for the 2D lattice model as a function of the barrier width $W$ for the different values of $\delta$ listed in the inset. Parameters: $L_1=100, L_2=400, h=0.5, {\rho }_{\mathrm{u}}=1, {\rho }_{\mathrm{d}}=0.9$, and $O_2=3$.

Figure 10

Density profile obtained by averaging the 2D lattice simulation: comparison for different values of $W$. The lattice size is $100\times 400, h=0.5, \delta =0.1$ (top) and $\delta =0.01$ (bottom), ${\rho }_{\mathrm{u}}=1, {\rho }_{\mathrm{d}}=0$ (left), ${\rho }_{\mathrm{d}}=0.4$ (center), and ${\rho }_{\mathrm{d}}=0.9$ (right), and $O_2=3$. The Monte Carlo–averaged residence time measured in the different cases has been reported in the inset.

Figure 11

The Monte Carlo estimate (solid symbols) and the mean-field prediction (open symbols) of the residence time are plotted for $\delta =0.1$, lattice size $100\times 400, h=0.5, {\rho }_{\mathrm{u}}=1, O_2=3, {\rho }_{\mathrm{d}}=0$ (circles), and ${\rho }_{\mathrm{d}}=0.9$ (squares).

Figure 12

Plot of the parameters $m_1$ (solid symbols) and $m_2$ (open symbols) defined in (3.16) for $\delta =0.1$, lattice size $100\times 400, h=0.5, {\rho }_{\mathrm{u}}=1, O_2=3, {\rho }_{\mathrm{d}}=0$ (circles), and ${\rho }_{\mathrm{d}}=0.9$ (squares).