Disorder Effects and Critical Behavior in Statistical Mechanics and in Quantum Systems

通常的经验是，在一个尺度上发现的物理结构，是具有更多自由度的更精细的复杂系统的集体模态的表现。微观上由粒子、布朗运动等组成的流体。值得注意的是，通常粗糙尺度的物理学是通过简单的自治定律来表达的，而在更精细的描述中却找不到对称性。拟议的研究将在统计力学和场论的一些模型的背景下，从数学上分析这种“尺度极限”的出现。特别注意将致力于在关键系统中看到的大规模特征的数学分析。另一个研究主题将涉及到淬火无序的影响，它经常改变振动模式的特征，导电性能和相变的性质。研究的目的是达到物理上的理解和数学上令人满意的方法来分析这些领域的重要现象。＊＊＊

9512729 Aizenman It is common experience that physical structure found on one scale, is a manifestation of collective modes of a much finer complex system with many more degrees of freedom. Fluids which microscopically consist of particles, Brownian motion, etc. Remarkably, often the physics of the coarser scale is expressed through simple autonomous laws, and symmetries not found in the finer description. The proposed research will analyze mathematically the emergence of such "scaling limits", in the context of a number of models of statistical mechanics, and field theory. Particular attention will be devoted to the mathematical analysis of the large scale features seen in critical systems. Another topic of study will concern the effects of quenched disorder, which quite often modifies the characteristics of vibration modes, electrical conduction properties, and the nature of phase transitions. The study aims at reaching both physical understanding and mathematically satisfactory means of analyzing significant phenomenon these fields. ***