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arXiv · 1810.04639

Current with "wrong" sign and phase transitions

Roberto Boccagna
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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 $\pmμ$, $μ\in(m^*\left(β\right),1)$, where $m^*\left(β\right)=\sqrt{1-1/β}$, $β>1$ representing the inverse temperature. At last, an external field of Heaviside-type of intensity $κ>0$ is introduced. According to the axiomatic 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 $μ$ is metastable, the current is positive and bounded from below by a positive constant independent of $κ$, 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.

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Roberto Boccagna. 2018-10-10. Current with "wrong" sign and phase transitions. https://doi.org/10.1063/1.5065566

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