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

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.

💡 Research Summary

In this work the authors investigate a novel variant of the totally asymmetric simple exclusion process (TASEP) in which two one‑dimensional driven lanes (denoted T₁ and T₂) run antiparallel to each other and are each coupled at both ends to a finite‑capacity particle reservoir (R₁ and R₂). The total number of particles N₀ in the combined system (both lanes plus the two reservoirs) is conserved, and the overall resource availability is quantified by the filling factor µ = N₀/(2L), where L is the length of each lane (µ ranges from 0 for an empty system to 2 for a fully occupied one).

A key feature of the model is that the entry and exit rates of particles into the lanes are not fixed constants but depend on the instantaneous occupation of the reservoirs. Specifically, the effective entry rates are αᵉᶠᶠ = α f(N) with f(N)=N/L, while the effective exit rates are βᵉᶠᶠ = β g(N) with g(N)=1−N/L. The authors focus on the symmetric case α₁=α₂≡α and β₁=β₂≡β, which reduces the control parameter space to three variables (α, β, µ).

Using a mean‑field (MF) description, the authors write continuum equations for the average densities ρ₁(x) and ρ₂(x) along each lane. In the steady state the particle current J = ρ(1−ρ) is spatially constant, and the flux balance between the reservoirs forces J₁ = J₂ = J. This condition implies either ρ₁ = ρ₂ or ρ₁ + ρ₂ = 1. Because the symmetric boundary conditions enforce identical effective entry and exit rates for the two lanes, the latter possibility is excluded, and the system must satisfy ρ₁ = ρ₂ throughout the bulk. Consequently only four bulk phases are possible: low‑density (LD‑LD), high‑density (HD‑HD), maximal‑current (MC‑MC), and a domain‑wall (DW‑DW) phase in which a shock separates coexisting LD and HD regions on each lane.

In the conventional open TASEP a delocalized domain wall (DDW) appears only on the line α = β < ½, i.e., at a single point in the (α,β) plane. Remarkably, in the present finite‑resource model the DDW occupies an extended region of the (α,β) plane for a given µ. The authors show analytically that when both α and β are below ½ and sufficiently close to each other, the self‑consistent equations for the effective rates admit solutions with a mobile shock whose position is not fixed by the dynamics. This “delocalized” wall can wander over the whole lane, leading to O(L) fluctuations of the particle number in each lane even in the thermodynamic limit.

Phase boundaries are obtained by equating the currents of neighboring phases. The LD↔MC and HD↔MC transitions are continuous (second‑order) because the bulk density changes smoothly, whereas the LD↔HD transition is discontinuous (first‑order) due to a jump in the bulk density. The filling factor µ shifts the location of these boundaries: increasing µ (more resources) pushes the system toward higher‑density phases, while decreasing µ favors low‑density behavior. In the extreme limits µ→0 and µ→2 the system collapses to pure LD‑LD or HD‑HD states, respectively.

To validate the MF predictions, extensive Monte‑Carlo simulations were performed for large L (up to several thousand sites). The simulations measured stationary density profiles, currents, and the probability distribution of the shock position. The numerical results confirm the MF phase diagram, the existence of an extended DDW region, and the scaling of fluctuations: particle‑number variance in each lane grows linearly with L in the DDW phase, whereas reservoir‑population fluctuations decay as 1/L, indicating that the reservoirs become effectively deterministic in the large‑L limit.

The paper discusses the broader implications of these findings. In biological contexts, the model captures situations where a finite pool of ribosomes or molecular motors interacts with multiple filaments, leading to large stochastic fluctuations in translation or transport rates when resources are limited. In vehicular traffic, a finite fleet of cars circulating on a network of roads can exhibit similar delocalized jams over a range of entry/exit rates. The key conceptual advance is the demonstration that coupling multiple driven lanes to finite, dynamically regulated reservoirs can generate a qualitatively new non‑equilibrium phase structure—most notably an extended region of delocalized domain walls—absent in previously studied TASEP variants.

In summary, the authors provide (i) a clear definition of a symmetric two‑lane TASEP with finite reservoirs, (ii) a mean‑field analysis that predicts four bulk phases with equal densities in both lanes, (iii) the discovery of an extended delocalized‑domain‑wall region in the (α,β,µ) parameter space, (iv) analytical expressions for phase boundaries and the nature of the transitions, and (v) Monte‑Carlo evidence supporting the theoretical predictions. These results enrich our understanding of driven diffusive systems under resource constraints and open avenues for exploring similar mechanisms in biological transport, traffic flow, and other nonequilibrium processes with conserved particle number.