Reservoirs, Fick law, and the Darken effect

We study the stationary measures of Ginzburg–Landau (GL) stochastic processes, which describe the magnetization flux induced by the interaction with reservoirs. To privilege simplicity to generality, we restrict to quadratic Hamiltonians where almost explicit formulas can be derived. We discuss the case where reservoirs are represented by boundary generators (mathematical reservoirs) and compare with more physical reservoirs made by large-infinite systems. We prove the validity of the Fick law away from the boundaries. We also obtain in the context of the GL models a mathematical proof of the Darken effect, which shows uphill diffusion of carbon in a specimen partly doped with the addition of Si.