论文标题
统一的自由能能功能和扩散的应用理论
A Unified Theory of Free Energy Functionals and Applications to Diffusion
论文作者
论文摘要
金茨堡 - 兰道型的自由能量功能位于广泛的连续性动力学模型的核心，例如Cahn-Hilliard和Swift-Hohenberg方程。尽管广泛使用了这种模型，但自由能功能中体现的假设通常是有道理的，或者导致物理上不透明的参数。在这里，我们介绍了一种数学上严格的途径，用于构建自由能功能，该途径概括了金茨堡 - 兰道梯度扩展的限制。我们表明，新的形式主义通过建立标准统一了现有的自由描述，该标准遵循该标准，广义自由能将其减少为基于梯度的表示。因此，我们得出了Cahn-Hilliard模型中梯度能量参数的精确物理解释，作为相互作用长度尺度和自由能曲率的乘积。使用模型的自由能函数和硅果合金系统证明了我们方法的实际影响。
Free energy functionals of Ginzburg-Landau type lie at the heart of a broad class of continuum dynamical models, such as the Cahn-Hilliard and Swift-Hohenberg equations. Despite the wide use of such models, the assumptions embodied in the free energy functionals are frequently either poorly justified or lead to physically opaque parameters. Here, we introduce a mathematically rigorous pathway for constructing free energy functionals that generalizes beyond the constraints of Ginzburg-Landau gradient expansions. We show that the new formalism unifies existing free energetic descriptions under a single umbrella by establishing the criteria under which the generalized free energy reduces to gradient-based representations. Consequently, we derive a precise physical interpretation of the gradient energy parameter in the Cahn-Hilliard model as the product of an interaction length scale and the free energy curvature. The practical impact of our approach is demonstrated using both a model free energy function and the silicon-germanium alloy system.
