Current with “wrong” sign and phase transitions

Abstract

We prove that under certain conditions, phase separation is enough to sustain a regime in which current flows along the concentration gradient, a phenomenon which is known in the literature as uphill diffusion. The model we consider here is a version of that proposed by Giacomin and Lebowitz [J. Stat. Phys. 87(1), 37–61 (1997)], which is the continuous mesoscopic limit of a 1d discrete Ising chain with a Kac potential. The magnetization profile lies in the interval [−ε−1, ε−1], ε > 0, staying in contact at the boundaries with infinite reservoirs of fixed magnetization ±μ, μ∈(m*β,1), where m*β=1−1/β, β > 1 representing the inverse temperature. At last, an external field of Heaviside-type of intensity κ > 0 is introduced. According to the axiomatic nonequilibrium theory, we derive from the mesoscopic free energy functional the corresponding stationary equation and prove the existence of a solution, which is antisymmetric with respect to the origin and discontinuous in x = 0, provided ε is small enough. When μ is metastable, the current is positive and bounded from below by a positive constant independent of κ, this meaning that both phase transition and external field contributes to uphill diffusion, which is a regime that actually survives when the external bias is removed.