Current with "wrong" sign and phase transitions

TL;DR

This paper proves that phase separation can sustain uphill diffusion in a mesoscopic Ising model with long-range interactions, external fields, and boundary reservoirs, revealing new conditions for non-equilibrium current flow.

Contribution

It establishes the existence of stationary solutions exhibiting uphill diffusion due to phase transitions and external fields in a mesoscopic Ising model.

Findings

Uphill diffusion persists under certain phase separation conditions.

Stationary solutions are antisymmetric and discontinuous at the origin.

External fields and phase transitions both contribute to uphill diffusion.

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 \textit{uphill diffusion}. The model we consider here is a version of that proposed in [G. B. Giacomin, J. L. Lebowitz, Phase segregation dynamics in particle system with long range interactions, Journal of Statistical Physics 87(1) (1997): 37-61], which is the continuous mesoscopic limit of a $1d$ discrete Ising chain with a Kac potential. The magnetization profile lies in the interval $\left[-\varepsilon^{-1},\varepsilon^{-1}\right]$, $\varepsilon>0$, staying in contact at the boundaries with infinite reservoirs of fixed magnetization $\pm\mu$, $\mu\in(m^*\left(\beta\right),1)$, where $m^*\left(\beta\right)=\sqrt{1-1/\beta}$, $\beta>1$ representing the inverse temperature. At last, an external field of Heaviside-type of intensity $\kappa>0$ is introduced. According to the axiomatic non-equilibrium 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 $\varepsilon$ is small enough. When $\mu$ is metastable, the current is positive and bounded from below by a positive constant independent of $\kappa$, this meaning that both phase transition as well as external field contributes to uphill diffusion, which is a regime that actually survives when the external bias is removed.