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Usually, in a non-equilibrium setting, a current brings mass from the highestdensity regions to the lowest density ones. Although rare, the oppositephenomenon (known as "uphill diffusion") has also been observed inmulticomponent systems, where it appears as an artificial effect of theinteraction among components. We show here that uphill diffusion can be asubstantial effect, i.e. it may occur even in single component systems as aconsequence of some external work. To this aim we consider the 2D ferromagneticIsing model in contact with two reservoirs that fix, at the left and the rightboundaries, magnetizations of the same magnitude but of opposite signs. Weprovide numerical evidence that a class of non-equilibrium steady states existsin which, by tuning the reservoir magnetizations, the current in the systemchanges from "downhill" to "uphill". Moreover, we also show that, in suchnon-equilibrium set-up, the current vanishes precisely when the reservoirmagnetizations equal the magnetization of the corresponding equilibriumdynamics, thus establishing a novel relation between equilibrium andnon-equilibrium properties.

Non-equilibrium 2D Ising model with stationary uphill diffusion