We construct a solution for the $1d$ integro-differential stationary equation derived from a finite-volume version of the mesoscopic model 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)]. This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials. The microscopic system is in contact with reservoirs of fixed magnetization and infinite volume, so that their density is not affected by any exchange with the bulk in the original Kawasaki dynamics. At the mesoscopic level, this condition is mimicked by the adoption Dirichlet boundary conditions. We derive the stationary equation of the model starting from the Lebowitz-Penrose free energy functional defined on the interval $[-\varepsilon^{-1},\varepsilon^{-1}]$, $\varepsilon>0$. For $\varepsilon$ small, we prove that below the critical temperature there exists a solution that carries positive current provided boundary values are opposite in sign and lie in the metastable region. Such profile is no longer monotone, connecting the two phases through an antisymmetric interface localized around the origin. This represents 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. However uniqueness is lacking, and we have a clue that the stationary solution obtained is not unique, as suggested by numerical simulations.

中文翻译：

---

长程相互作用磁系统中的固定电流

我们为从 [GB Giacomin, JL Lebowitz, “具有长程相互作用的粒子系统中的相分离动力学”，Journal of Statistical 中提出的细观模型的有限体积版本派生的 $1d$ 积分微分平稳方程构建了一个解物理学 87(1) (1997)]。这是 Ising 自旋链通过 Kac 电位在远距离相互作用的连续极限。微观系统与固定磁化强度和无限体积的储层接触，因此它们的密度不受与原始川崎动力学中的体积的任何交换的影响。在细观层面，这种条件通过采用 Dirichlet 边界条件来模拟。我们从定义在区间 $[-\varepsilon^{-1},\varepsilon^{-1}]$, $\varepsilon>0$ 上的 Lebowitz-Penrose 自由能泛函开始推导出模型的平稳方程。对于 $\varepsilon$ 小，我们证明在临界温度以下存在一个带有正电流的解，前提是边界值符号相反并且位于亚稳态区域。这种轮廓不再是单调的，通过位于原点周围的反对称界面连接两相。这代表了在经历相变的单组分系统中沿着浓度梯度存在扩散的分析证明，这种现象通常被称为上坡扩散。然而缺乏唯一性，我们有线索得到的平稳解不是唯一的，