APPLICATIONS OF ANDERSON LOCALIZATION TO DYNAMICAL SYSTEMS AND EVOLUTIONARY PDE

该提案有两个部分。第一部分研究光滑映射的正熵问题。概念如下。拿一块材料，比如说一个球形的面团。 改变它的形状，比如说把它变成一个圆盘。 然后把它收起来，滚成一个球。重复这个过程多次。 自然地，材料的成分，比如盐和其他物质，在每立方英寸的材料中均匀分布。这就是所谓的“混合”。在物理学中，如果混合的速率很高，可以用一个叫做熵的量来测量。 熵方程相当复杂。熵的精确计算是一个很难的数学问题。 一个好消息是，熵可以与所讨论的变换的一些几何特征相关。例如，在一个方向上，材料越强，它就越“薄”，这是有道理的。因此，在收集回材料后，“混合”进行得更快。 只有对于特殊的变换，熵才被计算出来。对于大多数自然的例子来说，这个问题是完全开放的。1969年，苏联物理学家B.奇里科夫发现了一个简单的正方形地图，它是粒子力学、加速器和等离子体物理学的重要模型。这张地图后来被命名为奇里科夫地图，也是标准地图。 地图取决于它的主要参数，衡量其总强度。奇里科夫进行了数值模拟，令人信服地表明，如果强度不低，那么正方形的主要部分表现出快速混合，或者正如数学家所说的“熵是正的”。 标准地图熵的数学确定性评价是一个非常困难的问题。 在目前的提案中，我们计划开发一种全新的标准地图方法。地图自然产生一个属于完全不同物理学的系统，量子力学。有一种猜想认为熵的计算隐藏在这个量子力学系统的某些特殊性质中。 1958年，P.安德森发现了量子力学中的全新现象。这种现象后来被称为安德森本地化。这一发现是1977年诺贝尔奖的主要引文之一，该奖项授予了安德森、莫特和货车弗莱克。我们计划利用这一现象来解决熵问题。 著名的KdV方程是以D.Korteweg和G.德弗里斯，谁提出了它作为一个模型的传播某些特殊的波。 需要知道初始输入来解方程。近似40年前，开发了用于空间输入周期的求解方法。 几年前，该领域的主要专家之一P.Deift提出了一个问题，即开发一种用于初始输入的方法，该方法不是周期性的，而是具有不同周期的周期集合的混合物。在提案的第二部分，我们计划进一步发展我们在过去8年中在这个问题上取得的进展。

The proposal has two parts. The first part addresses positive entropy problem for smooth maps. The concept is as follows. Take a piece of material, say a ball--shaped piece of dough. Transform its shape, say by flattering into a disk. Then gather it back roll and shape back into a ball. Repeat the process a number of times. Naturally, the components of the material, say salt and others, get evenly distributed per each cubic inch of the material. That is called "mixing". In physics if the rate of the mixing is high it can be measured by a quantity called entropy. The entropy equation is rather involved. An accurate evaluation of the entropy is a hard mathematical problem. A good news is that the entropy can be related to some geometrical characteristics of the transformation in question. For instance, it makes sense that the stronger one "stratches" the piece of the material in one direction, the "thinner" it gets in the complimentary direction. So, after gathering back the material, the "mixing" goes faster. Only for special transformations the entropy has been evaluated. For most natural examples the problem is wide open. In 1969 Soviet physicist B.Chirikov discovered a simple map of the square which serves as an important model in mechanics of particles, accelerators, plasma physics. The map was later named Chirikov map and also standard map. The map depends on its major parameter which measures its total intensity. Chirikov produced numerical simulations which show convincingly that if the intensity is not low, then the main portion of the square exhibits fast mixing, or as mathematicians put it "the entropy is positive". To evaluate the standard map entropy with mathematical certainty proved to be a very hard problem. In the current proposal we plan to develop a completely new method for standard map. The map naturally produces a system which belongs to completely different physics,-quantum mechanics. There is a conjecture that the entropy evaluation is hidden in certain special propertie of this quantum mechanical system. In 1958 P.Anderson discovered absolutely new phenomenon in quantum mechanics. The phenomenon was later called Anderson localization. The discovery was one of the major citations for 1977 Nobel prize awarded to Anderson jointly with Mott and Van Vleck. We plan to use this phenomenon for entropy problem. The famous KdV equation was named after D.Korteweg and G. de Vries, who proposed it as a model for propagation of certain special waves. One needs to know the initial input to solve the equation. The solution method was developed for input periodic in space almost 40 years ago. Several years ago P.Deift , who is one of the leading experts in this field, stated a problem to develop a method for initial input which is not periodic but a mixture of a collection of periodic with different periods. In the second part of the proposal we plan to develop further the progress we made in this problem in last 8 years.