Emergence of stationary uphill currents in 2D Ising models: the role of reservoirs and boundary conditions

We investigate the dynamics of a 2D Ising model on a square lattice with conservative Kawasaki dynamics in the bulk, coupled with two external reservoirs that pull the dynamics out of equilibrium. Two different mechanisms for the action of the reservoirs are considered. In the first, called ISF, the condition of local equilibrium between reservoir and the lattice is not satisfied. The second mechanism, called detailed balance (DB), implements a DB condition, thus satisfying the local equilibrium property. We provide numerical evidence that, for a suitable choice of the temperature (i.e. below the critical temperature of the equilibrium 2D Ising model) and the reservoir magnetizations, in the long time limit the ISF model undergoes a ferromagnetic phase transition and gives rise to stationary uphill currents, namely positive spins diffuse from the reservoir with lower magnetization to the reservoir with higher magnetization. The same phenomenon does not occur for DB dynamics with properly chosen boundary conditions. Our analysis extends the results reported in Colangeli et al. [Phys. Rev. E 97, 030103(R) (2018)], shedding also light on the effect of temperature and the role of different boundary conditions for this model. These issues may be relevant in a variety of situations (e.g. mesoscopic systems) in which the violation of the local equilibrium condition produces unexpected phenomena that seem to contradict the standard laws of transport.