Mathematical aspects of phase separation under long-range interactions

许多物理系统表现出复杂的模式。例如，系统可以显示两个不同相位的空间区域，其中它们之间的界面类似于复杂表面。通常，观测到的结构可以用变分原理来解释或模拟，在变分原理中，可以计算任何容许结构的自由能，观测到的结构是具有最低能量的结构。由此产生的变分问题至少可以说是具有挑战性的。如果一个人希望分析他们，而不是首先采用假设severley限制可能的几何复杂性，一个需要先进的mathematical analysis.This建议属于数学和物理范式的能量驱动模式的形成与竞争的短期和长期的相互作用。物理范式是由二嵌段共聚物，具有显着的能力，自组装成许多迷人的结构，其界面类似于三倍周期性的表面，其平均曲率是constant.The目标的建议是双重的：首先，使用变分问题（数学范式），我们将解决一些核心问题的变分有关规模和几何。其次，在变分问题的一般框架内，我们将讨论基本的物理应用;二嵌段共聚物熔体中的相分离。所涉及的技术范围从严格的分析和几何到统计物理和数值模拟。

Many physical systems exhibit complex patterns. For example, a system may display spatial regions of  two different phases with the interface between them resembling a complex surface. Often, the observed structure can be explained  or simulated by a variational principle in which a free energyof any admissible structure can be computed, with the observed structure being the one with the lowest energy. The resulting variational problems are challenging to say the least. If one wishes to analysis them without first adopting assumptions which severley limit possible geometric complexities, one needs advanced mathematical analysis.This proposal pertains to both mathematical and physical paradigms for energy-driven pattern  formation associated with  competing short and long-range interactions. The physical paradigm is provided by diblock copolymers which have a remarkable ability for self-assembly into many fascinating structures; the interfaces of which resemble triply-periodic surfaces whose mean curvature is constant.The goal of this proposal is two-fold: First, using the variational problem (the mathematical paradigm), we will address some central questions in the calculus of variations concerning scale and geometry. Second, within the general framework of variational problems, we will address the underlying physical application; phase separation in diblock copolymers melts.The techniques expoited range from rigorous analysis and geometry to statistical physics and numerical simulations.