Interplay of reservoirs in a bidirectional system

Motivated by the interplay of multiple species in several real world transport processes, we propose a bidirectional totally asymmetric simple exclusion process with two finite particle reservoirs regulating the inflow of oppositely directed particles corresponding to two different species. The system's stationary characteristics, such as densities, currents, etc., are investigated using a theoretical framework based on mean-field approximation and are supported by extensive Monte Carlo simulations. The impact of individual species populations, quantified by filling factor, has been comprehensively analyzed considering both equal and unequal conditions. For the equal case, the system exhibits the spontaneous symmetry-breaking phenomena and admits both symmetric as well as asymmetric phases. Moreover, the phase diagram exhibits a different asymmetric phase and displays a nonmonotonic variation in the number of phases with respect to the filling factor. For unequal filling factors, the phase schema can display at most five phases including a phase that shows maximal current for one of the species.

Figure 1

Schematic diagram for a bidirectional transport model comprised of a lattice connected to two reservoirs, each accommodating particles of a single species only. Blue and red circles denote two oppositely directed particles traveling from left to right and right to left, respectively. The entrance rates of the two particle species are given by ${\alpha }^{+}$ and ${\alpha }^{-}$, which are controlled by the occupancy of the corresponding reservoirs. The exit rates for both the particles is $\beta$. Two particles of distinct kinds are permitted to swap their positions with a rate $s$, if they encounter each other on neighboring sites.

Figure 2

Stationary phase diagrams for (a) $\mu =0.3,$ (b) $\mu =0.9,$ (c) $\mu =1.1,$ and (d) $\mu =100.$ Symmetric phases are represented by white regions, while the colored regions denote asymmetric phases. Red symbols correspond to Monte Carlo simulation results. The L-L phase remains confined to a curve and acts as a boundary separating H-L and LD-LD phases, displaying a first order phase transition taking average current as the order parameter. The size of the system is taken to be 1500.

Figure 3

Particle density histogram for (a) L-L and (b) H-L phases with the parameters $(\alpha ,\beta ,\mu )=(1,0.1,0.3)$ and (2,0.1,1.5), respectively. Insets show the two-dimensional contour plot.

Figure 4

Plot of currents (a) $(J_{+}+J_{-})/2, |J_{+}-J_{-}|, J_{+}$, and $J_{-}$ and (b) $(J_{+}+J_{-})/2$ vs $\beta$ for $\alpha =3$ and $\mu =1.1$. Inset in (b) displays a discontinuity in $(J_{+}+J_{-})/2$ near $\beta \approx 0.326$. Solid lines represent theoretical results and symbols refer to Monte Carlo simulations.

Figure 5

(a), (b) Position and height of the shock with respect to $\beta$ for $\mu =1.1$ and $\alpha =0.1,1,3$. (c) Density profiles for fixed $\alpha =1$ and $\beta =0.1,0.2,0.25$ with $\mu =1.1$. (d), (e) Position and height of the shock with respect to $\alpha$ for $\mu =1.1$ and $\beta =0.1,0.2,0.3$. (f) Density profiles for fixed $\beta =0.2$ and $\alpha =0.5,1,2$ with $\mu =1.1$. (g) Effective entrance rates with respect to $\mu$ for $\alpha =2$ and $\beta =0.1$. (h) Density profiles for $\alpha =2$ and $\beta =0.1$ for $\mu =0.5,0.8,1.1$. (i) Position of the shock with respect to $\mu$ for $\alpha =2,20$ and $\beta =0.1$. Inset shows the change in position of the shock with respect to $\mu$ for $\alpha =20$ and $\beta =0.1$. In all figures, solid lines represent theoretical results and symbols correspond to Monte Carlo simulations. The size of the lattice is taken to be 3000.

Figure 6

Effect of finite lattice length $L$ on the S-L profile for fixed $(\alpha ,\beta ,\mu )=(1,0.2,1.1)$ and varied values of $L$.

Figure 7

Region width $\left(\mathrm{\Delta }\right)$ of low-low density (L-L) phase with respect to $\beta$ for fixed $\alpha =0.5$ and $\mu =0.5$. The solid curve is a guide to the eye with best-fit polynomial for the discrete simulation data (shown by red markers) until $L\approx 5000$.

Figure 8

Stationary phase diagrams for (a) ${\mu }_{+}=1$ and (b) ${\mu }_{+}=20$ with ${\mu }_{-}=0.3$. Solid lines represent theoretical results and symbols correspond to Monte Carlo simulations. The length of the lattice is 1500.

Figure 9

Stationary phase diagrams for (a) ${\mu }_{+}=0.98$, (b) ${\mu }_{+}=1.5$, and (c) ${\mu }_{+}=15$ with ${\mu }_{-}=0.9$. Solid lines represent theoretical results and symbols correspond to Monte Carlo simulations. The length of the lattice is set to 1500.

Figure 10

Position of the shock vs (a) ${\mu }_{+}$ with ${\mu }_{-}=0.3$, (b) $\beta$ with ${\mu }_{+}=1, {\mu }_{-}=0.3$, and (c) $\alpha$ with ${\mu }_{+}=1, {\mu }_{-}=0.3$. The rest of the parameter values are mentioned in the respective figures. Inset in (a) shows the change in $x_w$ with respect to ${\mu }_{+}$, which is almost linear. Inset in (b) is a zoomed figure for smaller values of $\beta$. In all figures, solid lines represent theoretical results and symbols correspond to Monte Carlo simulations. The length of the lattice is set to 1500.

Figure 11

Stationary phase diagrams for (a) ${\mu }_{-}=0.3$, (b) ${\mu }_{-}=0.5$, and (c) ${\mu }_{-}=0.54$ with ${\mu }_{+}=0.55$. Solid lines represent theoretical results and symbols correspond to Monte Carlo simulations. The length of the lattice is taken to be 1500.

Figure 12

(a) Effective entrance rate ${\alpha }_{\mathrm{eff}}^{-}$ with respect to ${\mu }_{-}$ for ${\mu }_{+}=0.55, \alpha =15,\text{and}\beta =0.8$. (b) Particle density ${\rho }_{-}$ with respect to $\alpha$ for distinct values of $\beta$. (c) Density profiles of the negative species for different values of $\beta , \alpha =3, {\mu }_{+}=1$, and ${\mu }_{-}=0.3$. The rest of the parameter values are mentioned in the respective figures. In all figures, solid lines represent theoretical results and symbols correspond to Monte Carlo simulations. The length of the lattice is taken to be 1500.

Figure 13

Different dynamic regions based on the filling factors ${\mu }_{+}$ and ${\mu }_{-}$. Eleven distinct regions numbered I to XI have phase regimes that are qualitatively different. Table 2 provides a tabular description of the phase regimes that can exist in each of these different regions. In the table, a phase that does not exist is indicated by empty entries.

Figure 14

Density profiles for (a) H-L, (b) L-L, (c) S-L phases, (d) LD-LD and (e) MC-MC with $(\alpha ,\beta ,\mu )=(2,0.2,10),(2,0.315,1.1),(1,0.1,0.3),(1,0.5,0.9),\text{and}(3.5,0.8,1.1)$ with symmetric filling factors, and (f) M-M phase with $(\alpha ,\beta ,{\mu }_{+},{\mu }_{-})=(4.5,0.8,15,0.9)$. The lattice length is taken to be 1500. Red and blue solid lines are mean-field results for $+$ and $-$ particles, respectively, while filled markers correspond to Monte Carlo simulations.

Figure 15

(a) Density profile of the two particle species for $\alpha =0.7,\beta =1, {\mu }_1={\mu }_2=1.1$, and $s=0.8$ obtained using mean-field theory. Dashed lines correspond to solution obtained with initial condition ${\rho }_{+}={\rho }_{-}=0.4$, dotted line is plotted for initial condition ${\rho }_{+}={\rho }_{-}=0.7$, and symbols correspond to Monte Carlo simulations. (b) Density profile of the two particle species for $\alpha =0.5,\beta =0.2, {\mu }_1={\mu }_2=1.1$, and $s=0.8$. Solid lines represent mean-field results and symbols correspond to Monte Carlo simulations.