arXiv math/03043712003-04-24math.PR

Ising和渗流模型中的重整化、大偏差和相分离

Renormalization, large deviations and phase separation in Ising and percolation models

Ágoston Pisztora

发表信息：: Proceedings of the ICM, Beijing 2002, vol. 3, 79--96

摘要：

相分离是一种相当常见的物理现象，例如从潮湿空气（雾，雨）中形成水滴，从各向同性材料（如液体）中分离晶体结构，甚至在打开苏打罐时形成嘶嘶作响的气泡。很久以前（至少在现象学水平上）就认识到，表现出多个平衡相的系统可以用适当的变分原理来描述：相以这样一种方式排列，即与相边界相关的能量最小。典型地，这导致几乎确定性的行为，并且相边界是相当规则的。然而，当从微观的角度来看，该系统由一群具有相对较强的短程相互作用的不规则运动的分子组成，上述宏观描述的简单性看起来不仅仅是奇迹。事实上，当从分子水平开始时，有更多的问题需要提出和理解：我们将看到哪些阶段？为什么只有这些发生？为什么相界是尖锐的？我们应该如何找到（定义）与界面相关的能量？只有这样，我们才能问这样一个问题：为什么系统会使这种能量最小化？只是在过去的十年中，数学上令人满意的理解这一现象已经实现。谈话的主要目标是目前的事态发展，从而集中在与拉斐尔瑟夫联合工作所取得的成果。连接到其他领域的数学比概率论或统计力学将突出，即几何测度论和变分法。

英文摘要：

Phase separation is a fairly common physical phenomenon with examples including the formation of water droplets from humid air (fog, rain), the separation of a crystalline structure from an isotropic material such as a liquid or even the formation of the sizzling gas bubbles when a soda can is opened. It was recognized long ago (at least on a phenomenological level) that systems exhibiting several phases in equilibrium can be described with an appropriate variational principle: the phases arrange themselves in such a way that the energy associated with the phase boundaries is minimal. Typically this leads to an almost deterministic behavior and the phase boundaries are fairly regular. However, when looked at from a microscopic point of view, the system consists of a bunch of erratically moving molecules with relatively strong short-range interaction and the simplicity of the above macroscopic description looks more than miraculous. Indeed, when starting from the molecular level, there are many more questions to be asked and understood: which are the phases which we will see? why do only those occur? why are the phase boundaries sharp? how should we find (define) the energy associated with the interfaces? Only then can we ask the question: why does the system minimize this energy? It is only in the last decade that a mathematically satisfactory understanding of this phenomenon has been achieved. The main goal of the talk is to present the current state of affairs focusing thereby on results obtained in joint works with Raphael Cerf. The connection to fields of mathematics other than probability theory or statistical mechanics will be highlighted; namely, to geometric measure theory and to the calculus of variations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文