Classical Realizability and Quantum Representability: Truncated Moment Problems in Statistical Physics and Quantum Chemistry

复杂的系统，如由分子制成的液体、由原子制成的大分子、由草制成的草坪等，都不可能完全描述。事实上，这样的描述甚至是不可取的，因为一个人会被无法解释的信息淹没。通常，系统的几个特征，如密度分布、物体间距离的相对频率，是非常重要的。处理这类复杂系统的一个有效方法是专注于这些特性的特性。在这种方法中，以某种特别的方式推导出描述特性的方程系统。那么问题是：这些方程的解仍然与最初考虑的复杂系统兼容吗？换句话说，复杂系统的状态是否存在，这些状态会导致这些特征？例如，如果有效方程预测粒子密度为负，那么答案将是否定的。对于更复杂的特征或特征集合，人们不能指望它们之间的关系如此明显，这种关系通常是以不平等的形式存在的。可实现性和可表现性问题就是识别这些条件，并确定哪些假定的特征实际上可以通过底层系统的状态来实现，可实现性和可表征性在不同的领域反复出现，因此它们似乎是一个非常有前途的复杂系统观点。现在也是解决这些问题的时候了，因为最近人们对这些问题感兴趣，就像在统计力学的许多不同领域，如堵塞、随机堆积、高维最优堆积、非均质材料以及量子化学。由于缺乏对这些问题的基本数学结构的理解，进展受到阻碍，这两个问题都可以解释为高维截断矩问题。即使是二维的情况也已经知道是非常困难的。理想情况下，人们可以得到一种方法，它允许人们从宏观测量中获得微观相互作用，人们可以基于截断矩理论中的一般方法，给出关于假定的可实现相关函数的所有不等式的理论描述。不幸的是，这种描述是如此间接，以至于只有几种情况是明确知道的。用明确的方式表达进一步的条件是一个非常困难的问题。除了它的实际重要性之外，最后一个问题还提供了项目和纯数学领域之间的重要联系。

Complex systems, like liquids made out of molecules, large molecules made out of atoms, lawn made out of grass, etc. are impossible to describe fully. In fact, such a description it is not even desirable, as one would be overwhelmed by information impossible to interpret. Typically, a few characteristics of the system, like density profiles, relative frequencies of inter-object distances, are of great importance. An effective way of treating such complex systems is to concentrate on the properties of these characteristics. In such an approach a system of equations describing the characteristics is derived in some ad hoc manner. The question is then: are the solutions of these equations still compatible with the originally considered complex system? In other words, do states of the complex system exist, which would give rise to these characteristics? As an example, if the effective equations predicted a negative density of particles, then the answer would be 'no'. For more complicated characteristics or collections of characteristics, one cannot expect the relations between them, which are usually in the form of inequalities, to be so obvious. The realizability and representability problems are to identify these conditions and to determine which putative characteristics can in fact be realized by a state of the underlying system.Realizability and representability arise repeatedly in different areas, thus they seem to be a very promising viewpoint on complex systems. It is also timely to attack these problems, due to a recent interest in these problems as in many different areas of statistical mechanics, like jamming, random packing, optimal packing in high dimensions, and heterogeneous materials, as well as in quantum chemistry. Progress is hindered by a lack of understanding of the underlying mathematical structure of these problems, both of which can be interpreted as high-dimensional truncated moment problems. Even the two dimensional case is already known to be very difficult. Ideally, one would obtain an approach which permits one to derive the microscopic interactions from macroscopic measurements.One can give a theoretical description of all inequalities for putative correlation functions characterizing realizability based on a general approach coming from the theory of truncated moment problems. This description is unfortunately so indirect that only a few conditions are known explicitly. It is a very hard problem to express further conditions in an explicit manner. Beside its practical importance this last question provides an important connection between the project and areas of pure mathematics.