Counterflow-induced clustering: Exact results

We analyze the cluster formation in a nonergodic stochastic system as a result of counterflow, with the aid of an exactly solvable model. To illustrate the clustering, a two species asymmetric simple exclusion process with impurities on a periodic lattice is considered, where the impurity can activate flips between the two nonconserved species. Exact analytical results, supported by Monte Carlo simulations, show two distinct phases, free-flowing phase and clustering phase. The clustering phase is characterized by constant density and vanishing current of the nonconserved species, whereas the free-flowing phase is identified with nonmonotonic density and nonmonotonic finite current of the same. The n-point spatial correlation between n consecutive vacancies grows with increasing n in the clustering phase, indicating the formation of two macroscopic clusters in this phase, one of the vacancies and the other consisting of all the particles. We define a rearrangement parameter that permutes the ordering of particles in the initial configuration, keeping all the input parameters fixed. This rearrangement parameter reveals the significant effect of nonergodicity on the onset of clustering. For a special choice of the microscopic dynamics, we connect the present model to a system of run-and-tumble particles used to model active matter, where the two species having opposite net bias manifest the two possible run directions of the run-and-tumble particles, and the impurities act as tumbling reagents that enable the tumbling process.