Reservoirs, Fick law and the Darken effect

TL;DR

This paper analyzes the stationary measures of Ginzburg-Landau stochastic processes with quadratic Hamiltonians, proving Fick's law validity and mathematically demonstrating the Darken effect of uphill diffusion in specific models.

Contribution

It provides explicit formulas for stationary measures, compares different reservoir models, and offers a mathematical proof of the Darken effect in Ginzburg-Landau models.

Findings

Fick's law holds away from boundaries in the models.

Explicit formulas for stationary measures are derived.

Mathematical proof of the Darken effect in the context of GL models.

Abstract

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. %and the existence of boundary layers for the mathematical reservoirs. We also obtain in the context of the GL models a mathematical proof of the Darken effect which shows uphill diffusion of carbon in specimen partly doped with the addition of Si.