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

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.