Hydrodynamics of Multi-Species Driven Diffusive Systems with Open Boundaries: A Two-Tasep Study

TL;DR

The paper addresses how to determine the open-boundary steady state of multi-species driven diffusive systems by coupling the Riemann problem for the hydrodynamic equations with boundary-rate constraints, using the Two-TASEP as a case study. It generalizes the extremal-current principle to multiple species through a framework that relies on solving the Riemann problem at the origin and a consistent map between boundary and bulk densities, without requiring product measures or integrable boundaries. The key contributions include explicit currents written in terms of two Riemann invariants, the identification of five phases governed by the signs of the characteristic speeds, and a practical algorithm to obtain bulk and boundary densities for arbitrary boundary rates. The approach provides a transparent, largely analytic picture of boundary-driven phase behavior in multi-component systems and suggests directions for extending to more species and other models beyond exclusion processes.

Abstract

In this short note, we review a recently developed method for analysing multi-component driven diffusive systems with open boundaries. The approach generalises the extremal-current principle known for single-component models and is based on solving the Riemann problem for the corresponding hydrodynamic equations. As a case study, we focus on a two-species exclusion process on a lattice (Two-TASEP), where two types of particles move in opposite directions with two arbitrary rates and exchange positions upon encounter with a third rate. Despite its simplicity, this toy model effectively captures the key features of multi-species driven diffusive systems, including phase separation phenomena. This allows us to illustrate the critical role played by the underlying Riemann invariants in determining the system's macroscopic behavior.

Figures (7)

• Figure 1: Example of a multi-species driven diffusive system; molecular transport on an axon, the main nerve fiber of a neuron, featuring different types of molecular motors, each type has a preferred directed of motion and a characteristic rate. When molecules from different types meet, they interact, (by possibly slowing down), figure adapted from hirokawa2010molecular
• Figure 2: Phase diagram of a single-species TASEP, shown in terms of boundary densities on the left and bulk density on the right. Acronyms in black represent the conventional phase names: HD (High Density), LD (Low Density), and MC (Maximal Current). Acronyms in red indicate the new phase names: RI (Right Induced), LI (Left Induced), and BI (Bulk Induced). The symbols ${+, 0, -}$ refer to the characteristic velocities in the bulk.
• Figure 3: The two-TASEP as a traffic flow model, $\bullet$ particles represent cars, $\circ$ particles represent bikes
• Figure 4: Representation of the solutions to the Riemann problem in the $\boldsymbol{z}$ domain. Points $L$ and $R$ correspond to $(z^L_\alpha, z^L_\beta)$ and $(z^R_\alpha, z^R_\beta)$, respectively. The continuous line represents the trajectory of a rarefaction fan, while the dashed line represents a shock solution. The left panel illustrates elementary solutions that respect the respective constraints on the initial conditions, whereas the right panel shows global solution trajectories constructed by stitching together segments of elementary solutions.
• Figure 5: Phase diagram of the Two-TASEP: The signs correspond to the velocities $v_\alpha$ and $v_\beta$ in order. In blue the boundaries of the $z$ domain verifying $z_{\alpha}+z_{\beta} < 1$, $z_{\alpha} < \alpha$ and $z_{\beta} < \beta$. For this figure, $\alpha = 0.8$ and $\beta=0.9$