The exact phase diagrams for a class of left-permeable asymmetric exclusion processes

Exclusion processes are well-known prototypical models in non-equilibrium statistical physics. The open asymmetric simple exclusion process (ASEP) involves a single species of particles hopping with excluded volume interaction along a one-dimensional finite lattice connected to reservoirs. Though the dynamical rules of evolution are simple, ASEP exhibits complex behaviour that is intriguing from the point of view of both physics and mathematics. For example, ASEP manifests rich physical phenomena like boundary-induced phase transitions and formation of shock-waves. Besides, ASEP is an integrable model. Integrability renders ASEP amenable to exact analysis using mathematical techniques such as matrix product ansatz (due to Derrida, Evans, Hakim and Pasquier) and Bethe ansatz. ASEP with multiple species is a natural generalization of single-species ASEP. Multispecies ASEPs are interesting also because they have applications in traffic flow, cell motility and biological systems. However, multispecies ASEP is a highly non-trivial problem. In our work, we study an integrable two-species partially asymmetric exclusion process called the left-permeable ASEP or LPASEP and its multispecies generalization, called the mLPASEP. In both these models, the left boundary is permeable to all species but the right boundary is impermeable to one of them. For the LPASEP, we construct a matrix product solution for the stationary state and thereby compute the exact stationary phase diagram for densities and currents. The phase diagram has three phases with a structure similar to the single-species ASEP. We obtain further structure in the phase diagram as we show the existence of subphases by computing boundary densities of each species. These exact results are supported by numerical simulation. We define mLPASEP for an arbitrary number of species. Then using projections onto the LPASEP, we derive densities and currents as well as the phase diagram in the steady state. One observes the phenomenon of dynamical expulsion of one or more species in most of the phases. We explain this phenomenon and the density profiles in each phase using interacting shocks. All the results are corroborated by extensive numerical simulations.