Multi-species generalization of the totally asymmetric simple exclusion process

TL;DR

This paper extends the TASEP model to multiple species with hierarchical dynamics, deriving coupled PDEs for hydrodynamics, exploring boundary effects, and developing integrability tools for finite-time distributions, revealing complex phenomena and general principles.

Contribution

It introduces a multi-species TASEP with hierarchical dynamics, derives its hydrodynamic PDEs, and develops integrability methods for finite-time distributions, advancing understanding of multi-species driven systems.

Findings

Coupled PDEs describe the hydrodynamic limit for two-species TASEP.

Boundary-induced phase diagram principles are generalized for multiple species.

Finite-time particle distribution formulas are developed using integrability tools.

Abstract

Exclusion processes in one dimension first appeared in the 70s and have since dragged much attention from communities in different domains: stochastic processes, out-of-equilibriums statistical physics, and more recently integrable systems. While the state of the art for a single species totally asymmetric simple exclusion process (TASEP) can be described, from different aspects as mature, much less is known when multiple interacting species are present. Using tools from integrable systems and hydrodynamics in the first place and stochastic processes in the second place, this work attempts to study the behavior of a novel version of the model with different species of particles having hierarchical dynamics that depend on arbitrary parameters. While Burger's equation famously represents the hydrodynamic limit of TASEP with a single species, we present a counterpart coupled system of PDE representing the hydrodynamic limit for a model with two species. The solutions of these PDEs display a rich phenomenology of solutions best characterized through the underlying normal modes. This system with two species can be used as a toy model for studying driven diffusive systems with open boundaries. Using heuristics, we present results suggesting a general principle governing the boundary-induced phase diagram of systems with multiple coupled driven conserved quantities, generalizing thus the extremal current principle known for the case of a single driven quantity. Using integrability tools, we develop a formalism allowing the computation of the finite-time probability distribution of particle positions on the 1D lattice, generalizing therefore known results for TASEP and other multi-species models. We finally study the behavior and the impact of a single second-class impurity initially located at the interface separating two regions of different densities of first-class particles.