Stationary currents in long-range interacting magnetic systems
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Stationary currents in long-range interacting magnetic systems Roberto Boccagna1 Received: 19 November 2019 / Accepted: 20 July 2020 / © The Author(s) 2020
Abstract We construct a solution for the 1d integro-differential stationary equation derived from a finite-volume version of the mesoscopic model proposed in Giacomin and Lebowitz (J. Stat. Phys. 87(1), 37–61, 1997). This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials, staying in contact at the two edges with reservoirs of fixed magnetizations. The stationary equation of the model is introduced here starting from the Lebowitz-Penrose free energy functional defined on the interval [−ε−1 , ε−1 ], ε > 0. Below the critical temperature, and for ε small enough, we obtain a solution that is no longer monotone when opposite in sign, metastable boundary conditions are imposed. Moreover, the mesoscopic current flows along the magnetization gradient. This can be considered as an analytic proof of the existence of diffusion along the concentration gradient in one-component systems undergoing a phase transition, a phenomenon generally known as uphill diffusion. In our proof uniqueness is lacking, and we have clues that the stationary solution obtained is not unique, as suggested by numerical simulations. Keywords Uphill diffusion · Kac potentials · Fick’s law · Phase transitions Mathematics Subject Classification (2010) 82C26
1 Introduction The aim of this paper is to study Fick’s law of transport in one-component systems undergoing a second order phase transition. In this context, it represents a step forward towards the establishment of a well posed theory for diffusion along the gradient (uphill diffusion) in the out-of-equilibrium setting [2–4]. Fick’s law relates the flux J
Roberto Boccagna
[email protected] 1
Universit`a dell’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila, Italia
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Math Phys Anal Geom
(2020) 23:30
of a given substance to the gradient of its concentration ρ, which we suppose to be a differentiable function of the position in [0, L]: d ρ, (1) dx at fixed boundary conditions ρ (0) = ρ− , ρ (L) = ρ+ , with ρ− < ρ+ without loss of generality. Here, D > 0 is the diffusion coefficient. According to (1), the flux is always in the direction of decreasing gradient, i.e. from the region at higher concentration to the region at lower concentration. Thus, the solution of (1) connects monotonically ρ− to ρ+ , as represented in a sketchy way in Fig. 1. Indeed, (1) should be modified when considering systems that consist of many components, since diffusion may be also affected by possible microscopic, chemical interactions among different substances. Evidences of unexpected behaviors have already been reported by Nernst [5], Onsager [6] and especially Darken [7–9], who performed a crucial experiment in the late 40’s. His setup consisted of pairs of doped steels (Fe-Si with a different wt. % of silicon, Fe-Si and Fe-Mn or Fe-Si and Fe-Mo) containing a small difference in the carbon co
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