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We study the stationary measures of Ginzburg-Landau (GL) stochastic processeswhich describe the magnetization flux induced by the interaction withreservoirs. To privilege simplicity to generality we restrict to quadraticHamiltonians where almost explicit formulas can be derived. We discuss the casewhere reservoirs are represented by boundary generators (mathematicalreservoirs) and compare with more physical reservoirs made by large-infinitesystems.  We prove the validity of the Fick law away from the boundaries.  %and the existence of boundary layers for the mathematical reservoirs. Wealso obtain in the context of the GL models a mathematical proof of the Darkeneffect which shows uphill diffusion of carbon in specimen partly doped with theaddition of Si.

Reservoirs, Fick law and the Darken effect