Exclusion model with finite reservoirs and particle transfer mechanisms.

Motivated by the complexity of finite-resource transport systems in biological and physical contexts, we study a totally asymmetric simple exclusion process on a lattice coupled to two finite reservoirs: The left reservoir supplies particles to the first site depending on its occupancy, while the right reservoir collects particles leaving the last site. Particles hop unidirectionally from left to right along the lattice under hardcore exclusion, and, in addition, particles can transfer from the right reservoir back to the left, enabling closed-loop circulation. Using mean-field approach, we analytically derive all phase boundaries, density profiles, and shock positions, which are validated through extensive Monte Carlo simulations. The model exhibits a rich variety of phases, with five qualitatively distinct phase diagrams emerging as key parameters vary. Among these, three phase diagrams contain the same set of phases but differ in their topological structure. The shock position displays both monotonic and nonmonotonic dependence on the exit rate β, reflecting the intricate influence of finite resources and the coupling between entry and exit dynamics. A phenomenon, termed the back-and-forth transition, arises from the interplay between the interreservoir particle transfer and the total number of particles.