标题： 探索耦合Cahn-Hilliard方程的自由能全局景观 显示英文标题

标题： Exploring global landscape of free energy for the coupled Cahn-Hilliard equations

摘要： 描述无限维自由能的复杂景观通常是一个具有挑战性的问题。 这种困难源于存在许多最小值点，从而导致大量鞍点的存在。 这些因素使得预测所需构型的位置或预测从初始条件到最终状态的轨迹和路径变得困难。 相比之下，实验观察表明，在受控条件下，即使在噪声存在的情况下，也可以可重复地获得特定形态，并且产率很高。 本研究探讨了阐明自由能景观全局结构的可能性，并在不依赖全面穷举方法的情况下实现对轨道的控制以达到期望的最小值点。 此外，它试图从数学上解释某些实验设置在实现高产率结果方面的有效性。 聚焦于溶剂中两种聚合物的相分离，我们进行了一维分析，揭示了全局自由能景观和与松弛参数相关的轨迹行为。 开发了两种关键方法：一种是类似于分岔跟踪的鞍点搜索方法，旨在全面识别所有鞍点。 另一种策略是调整每个变量的时间导数之前的松弛参数，与实验设置相一致。 这种方法能够控制轨迹行为以达到期望的结构，克服了最速下降方法的局限性。 通过调节这些松弛参数，可以抑制由于不可避免的波动引起的轨迹行为的不确定性。 这些方法共同提供了一个数学框架，与实验中的高产率现象相呼应，有助于更深入地理解其潜在机制。 显示英文摘要

摘要： Describing the complex landscape of infinite-dimensional free energy is generally a challenging problem. This difficulty arises from the existence of numerous minimizers and, consequently, a vast number of saddle points. These factors make it challenging to predict the location of desired configurations or to forecast the trajectories and pathways leading from an initial condition to the final state. In contrast, experimental observations demonstrate that specific morphologies can be reproducibly obtained in high yield under controlled conditions, even amidst noise. This study investigates the possibility of elucidating the global structure of the free energy landscape and enabling the control of orbits toward desired minimizers without relying on exhaustive brute-force methods. Furthermore, it seeks to mathematically explain the efficacy of certain experimental setups in achieving high-yield outcomes. Focusing on the phase separation of two polymers in a solvent, we conduct a one-dimensional analysis that reveals the global free energy landscape and relaxation-parameter-dependent trajectory behaviors. Two key methodologies are developed: one is a saddle point search method, akin to bifurcation tracking. This method aims to comprehensively identify all saddle points. The other is a strategy that adjusts the relaxation parameters preceding each variable's time derivative, aligning with experimental setups. This approach enables control over trajectory behaviors toward desired structures, overcoming the limitations of steepest descent methods. By tuning these relaxation parameters, uncertainties in trajectory behavior due to inevitable fluctuations can be suppressed. These methodologies collectively offer a mathematical framework that mirrors experimental high-yield phenomena, facilitating a deeper understanding of the underlying mechanisms.