Gates-Lebowitz-Penrose自由能泛函的液滴极小化
Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional
摘要：
研究了Gates-Lebowitz-Penrose自由能泛函${\mathcal F}_{\rm GLP}(m)$的约束极小元的结构，该泛函是密度场$m(x)$的非局部泛函，$x\in {\mathcal T}_L$是边长为$L$的$d$维环面.在低温下，${\mathcal F}_{\rm GLP}$不是凸的，并且具有两个不同的全局极小值，对应于两个平衡态。这里，我们将平均密度$L^{-d}\int_{{\cal T}_L}m(x)\dd x$约束为两个平衡态中的密度之间的固定值$n$，但接近于低密度平衡值。在这种情况下，取决于$n$和$L$的值，高密度相的“液滴”可以或可以不形成在低密度相的背景中。我们确定了液滴形成的临界密度和液滴的性质，作为$n$和$L$的函数。自由能和大偏差功能的粒子模型与远程Kac电位之间的关系，在某些情况下证明，并预计在一般情况下是真实的，然后提供的吉布斯措施的典型微观配置的结构的Kac电位的范围是足够大时的信息。
英文摘要：
We study the structure of the constrained minimizers of the Gates-Lebowitz-Penrose free-energy functional ${\mathcal F}_{\rm GLP}(m)$, non-local functional of a density field $m(x)$, $x\in {\mathcal T}_L$, a $d$-dimensional torus of side length $L$. At low temperatures, ${\mathcal F}_{\rm GLP}$ is not convex, and has two distinct global minimizers, corresponding to two equilibrium states. Here we constrain the average density $L^{-d}\int_{{\cal T}_L}m(x)\dd x$ to be a fixed value $n$ between the densities in the two equilibrium states, but close to the low density equilibrium value. In this case, a "droplet" of the high density phase may or may not form in a background of the low density phase, depending on the values $n$ and $L$. We determine the critical density for droplet formation, and the nature of the droplet, as a function of $n$ and $L$. The relation between the free energy and the large deviations functional for a particle model with long-range Kac potentials, proven in some cases, and expected to be true in general, then provides information on the structure of typical microscopic configurations of the Gibbs measure when the range of the Kac potential is large enough.
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