Stationary densities in a weakly nonconserving asymmetric exclusion processes with finite resources

TL;DR

This work analyzes a TASEP on a finite lattice that exchanges particles with a finite reservoir implementing Langmuir kinetics in the bulk. By deriving a mean-field continuum equation and solving for stationary density profiles, the authors map phase diagrams in the control-parameter space $(\alpha,\beta,\Omega)$, revealing three phases: LD-HD, LD-MC-HD, and MC, with certain homogeneous phases absent due to reservoir-induced boundary coupling $\alpha_{\mathrm{eff}}=\beta_{\mathrm{eff}}=\dfrac{\alpha\beta}{\alpha+\beta}$. The study identifies domain-wall localization at $x=1/2$ in the LD-HD regime, and a three-phase LD-MC-HD region whose MC width grows with $\Omega$, ultimately expanding to cover the bulk as $\Omega$ increases. Monte Carlo simulations corroborate the mean-field predictions, demonstrating a strong agreement and highlighting how finite resources qualitatively alter nonequilibrium phase behavior compared to conventional open TASEP with Langmuir kinetics. The results offer a physical basis for finite-resource transport phenomena in biological and vehicular contexts and point to avenues for extensions, including unequal Langmuir rates and alternative reservoir dynamics.

Abstract

Asymmetric exclusion process (TASEP) along a one-dimensional (1D) open channel sets the paradigm for 1D driven models and nonequilibrium phase transitions in open 1D models. Inspired by the phenomenologies of an open TASEP with Langmuir kinetics (Lk) and with finite resources, we study the stationary densities and phase transitions in a TASEP with Lk connected to a particle reservoir at its both ends. We calculate the stationary density profiles and the phase transitions. The resulting phase diagrams in the plane of the control parameters are significantly different from their counterparts in an open TASEP with Lk. In particular, some of the phases admissible in the open TASEP with Lk model are no longer possible. Intriguingly, our model that is closely related to a TASEP coupled with Lk on a ring with a point defect, admits more phases than the latter. Phenomenological implications of our results are discussed.

Figures (7)

• Figure 1: Schematic diagram of the model. A one-dimensional lattice $T$ having $L$ sites ($j = 1, 2, \ldots, L$) executing TASEP dynamics is coupled to a particle reservoir $R$ at both boundaries ($j=1,\,L$). Each site can hold at most one particle, while the reservoir population is bounded by $L$. Particles enter at site $j=1$ with rate $\alpha_{\text{eff}}$ and exit from site $j=L$ with rate $\beta_{\text{eff}}$, as defined in Eq. \ref{['effective-entry-and-exit-rates']}, both depending on the instantaneous reservoir occupation $N_R$. Particles at sites $1 \le j < L$ hop to the next site $j+1$ if it is empty, with the hopping rate set to $1$ to fix the time scale. Additionally, within the bulk ($1 < j < L$), particles may attach to an empty site or detach from an occupied site with rates $\omega_A$ and $\omega_D$, respectively, as specified in Eq. \ref{['attachment-detachment-rates']}.
• Figure 2: 2D phase diagrams of our model are presented. The top panel illustrates the phase diagrams in the $(\alpha, \beta)$-plane for three different values of $\Omega$: (a) $\Omega = 0.3$, (b) $\Omega = 0.5$, and (c) $\Omega = 1$. The bottom panel shows the phase diagrams in the $(\alpha, \Omega)$-plane for three distinct values of $\beta$: (d) $\beta = 0.3$, (e) $\beta = 1$, and (f) $\beta = 500$. The black solid lines represent the phase boundaries obtained from the mean-field analysis, while the colored points correspond to the phase boundaries determined through Monte Carlo simulations. Both results show excellent agreement with each other.
• Figure 3: Phase diagram in the $\alpha$-$\beta$-$\Omega$-plane exhibits three distinct phases: LD-HD (green), LD-MC-HD (orange), and MC (cyan).
• Figure 4: Plots of the width $\Delta$ of the MC phase segment within the LD-MC-HD phase are shown as functions of (a) $\Omega$ for fixed $\alpha=\beta=0.95$, and (b) $\alpha$ for fixed $\beta=0.75$ and $\Omega=0.7$. The mean-field theory (MFT) predictions (black solid lines) show excellent agreement with the corresponding Monte Carlo simulation (MCS) results (colored symbols). Evidently, $\Delta$ increases monotonically with $\Omega$ for fixed $\alpha$ and $\beta$, and also increases with $\alpha$ for fixed $\beta$ and $\Omega$, in accordance with Eq. \ref{['mc-phase-width']}.
• Figure 5: (a) Density and (b) current profiles for different values of $\alpha$ with fixed $\beta = 2.5$ and $\Omega = 0.3$. For $\alpha = 0.1,\,0.25,\,0.4$, the system is in the LD-HD phase with density given by Eq. \ref{['xa-greater-xb']} and a domain wall located at $x_{w} = 1/2$ (Eq. \ref{['dw-pos']}). As $\alpha$ increases, the domain-wall height (Eq. \ref{['dw-height']}) decreases continuously and vanishes at $\alpha = 0.4$, producing a fully linear profile. For $\alpha = 0.5,\,0.6$, the system enters the LD-MC-HD phase described by Eq. \ref{['xa-less-xb']}, where the width of the maximal-current region (Eq. \ref{['mc-phase-width']}) grows with increasing $\alpha$. At $\alpha = 1$, the system reaches the MC phase with bulk density $1/2$ and boundary layers of vanishing width in the thermodynamic limit. The density changes smoothly across LD-HD/LD-MC-HD and LD-MC-HD/MC boundaries, indicating second-order transitions. The corresponding current profiles in Fig. \ref{['den-curr-profile1']}(b) also vary smoothly with $x$. Simulations are performed for system size $L = 1000$, with averages taken over $10^{6}$ Monte Carlo steps. The Monte Carlo results (colored points) show excellent agreement with the mean-field predictions (black solid lines).