Stationary densities and delocalized domain walls in asymmetric exclusion processes competing for finite pools of resources

TL;DR

This paper analyzes two antiparallel TASEP lanes connected to finite particle reservoirs, focusing on stationary densities and domain-wall behavior under resource conservation. Using mean-field theory complemented by extensive Monte Carlo simulations, the authors show that delocalized domain walls (DDWs) appear in a broad region of parameter space, causing large density fluctuations in the TASEP lanes while reservoir number fluctuations vanish in the thermodynamic limit. All four phases LD-LD, HD-HD, MC-MC, and DW-DW coexist for 1/2<μ<3/2, with the DW-DW phase defined by a pair of DDWs whose relative positions are constrained only by particle-number conservation. The phase boundaries are derived analytically within MF theory and confirmed numerically, revealing a multicritical point where all four phases meet, and highlighting the extended topology of the phase diagram compared to open-TASEP and previously studied finite-resource models. These findings have potential implications for resource-limited transport systems, including ribosome traffic on mRNA, where extended DDW regions imply robust, large-scale fluctuations in particle densities across coupled channels.

Abstract

We explore the stationary densities and domain walls in the steady states of a pair of asymmetric exclusion processes (TASEP) antiparallelly coupled to two particle reservoirs without any spatial extent by using the model in Haldar et al., Phys. Rev. E {\bf 111}, 014154 (2025). We show that the model admits a pair of {\em delocalized} domain walls, which exist for some choices of the model parameters that define the effective entry and exit rates into the TASEP lanes. Surprisingly, in the parameter space spanned by these model parameters, the region corresponding to delocalized domain walls covers an {\em extended} region, in contrast to the delocalized domain walls that appear only along a line in the relevant parameter space of the other known variants of TASEP. This implies large fluctuations in the TASEP particle numbers even in the thermodynamic limit that can be found over a range of the control parameters. The corresponding phase diagrams in the plane of the control parameters have different topology from those for an open TASEP or other models with multiple TASEPs connected to two reservoirs.

Figures (11)

• Figure 1: Schematic model diagram: Two lattices $T_{1}$ and $T_{2}$ with $L$ sites in each are connected to two particle reservoirs $R_{1}$ and $R_{2}$ containing finite number of particles $N_{1}$ and $N_{2}$, respectively. Particles move following the TASEP update rules (1)-(5), see text. Unlike the open TASEP, entry and exit rates are dynamically controlled by instantaneous reservoir populations, see Eqs. (\ref{['eff-rates-T1']})-(\ref{['f-and-g']}).
• Figure 2: Phase diagrams in the $\alpha$-$\beta$ space with a set of representative values of $0 < \mu \le 1$. Depending on the value of $\mu$, which can take values between 0 and 2, two or four phases are simultaneously found. These are LD-LD, HD-HD, MC-MC, and DW-DW and are represented by green, gray, yellow, and blue regions, respectively, according to MFT. These phases are delineated by black solid lines according to MFT, see Eqs. (\ref{['ldld-and-mcmc-boundary']}), (\ref{['hdhd-and-mcmc-boundary']}), (\ref{['ldld-and-dwdw-bundary']}), and (\ref{['hdhd-and-dwdw-bundary']}). MCS confirm the phase boundaries with colored points: red squares (between LD-LD and MC-MC), blue circles (between HD-HD and MC-MC), orange triangles (between LD-LD and DW-DW), and magenta triangles (between HD-HD and DW-DW). System size is $L=1000$ and $2 \times 10^{9}$ Monte Carlo steps are taken. Phase diagrams for $1 < \mu \le 2$ are presented in Fig. \ref{['ph-pd']}, exhibiting the particle-hole symmetry of the model.
• Figure 3: (a) Plot of $\rho_{\text{LD-LD}}$ as a function of $\alpha$ for fixed $\mu=0.3$ [see Eq. (\ref{['rho-for-ld-ld-phase']})]. For large $\alpha$, $\rho_{\text{LD-LD}}$ approaches $\mu$. (b) Plot of $\rho_{\text{HD-HD}}$ as a function of $\beta$ for fixed $\mu=1.7$ [see Eq. (\ref{['rho-for-hd-hd-phase']})]. For large $\beta$, $\rho_{\text{HD-HD}}$ approaches $\mu - 1$. MFT and MCS results show excellent agreement.
• Figure 4: Steady state time-averaged reservoir populations $\langle N_1\rangle,\langle N_2\rangle$ when the TASEP lanes are in their MC-MC phase. Within error bars, $\langle N_1\rangle =\langle N_2\rangle$ is found.
• Figure 5: Steady state density profiles in both lanes $T_{1}$ and $T_{2}$ are presented in the DW-DW phase. These are delocalized domain walls (DDW). System size is $L=1000$ and $2 \times 10^{9}$ Monte Carlo steps are taken. (a) Partially delocalized with average position $x_{0}<1/2$, span $\Delta=2x_{0}$, and parameter values: $\alpha=0.5$, $\beta=0.2$, and $\mu=1$; (b) Partially delocalized: $x_{0}>1/2$, $\Delta=2(1-x_{0})$, and parameter values: $\alpha=0.2$, $\beta=0.5$, and $\mu=1$; and (c) Fully delocalized: $x_{0}=1/2$, $\Delta=1$, parameter values: $\alpha=\beta=0.5$, and $\mu=1$. MFT predictions and MCS outcomes match well with each other.