Moment Problems with a View towards Applications in Statistical Physics
Infinite dimensional moment problems naturally arise in diverse applied areas dealing with the analysis of complex systems like statistical mechanics, fluid dynamics, quantum chemistry, spatial ecology, stochastic geometry, image recognition, etc. However, a concrete progress in such problems is hindered by the lack of a general understanding of the mathematical structure behind them. The primary aim of this project is therefore to establish new advances in the theory of the infinite dimensional moment problem and explore their effective impact on the analysis of complex systems in statistical physics.
Examples of complex systems are many-body systems such as liquid composed of molecules, a molecule composed of atoms, etc. Since such a system consists of a huge number of identical components, the essence of its investigation is to evaluate selected characteristics (usually correlation functions), which encode the most relevant properties of the system. These characteristics are indeed the only ones that give a reasonable picture of the qualitative behaviour of the system. It is therefore fundamental to investigate whether a given candidate correlation function actually represents the correlation function of some random distribution. This problem is well-known as realizability problem and has recently received considerable attention in statistical physics as well as several other areas.
The fundamental mathematical challenge is to derive a general method to solve any realizability problem independently of the specific physical system on which it is posed. This kind of structural investigation is the core of this project and has its starting point in the key-observation that realizability problems can be interpreted as moment problems in infinite dimensional settings. Our aim is to identify new treatable necessary and sufficient realizability conditions, keeping as a guide in our search their practical usefulness in problems belonging statistical physics. The innovative approach we propose is to exploit the two-way interaction existing between moment theory and statistical physics. Indeed, motivated by our previous results, we are convinced that questions arising in applications which are naturally connected to the moment problem can actually shed some light on the unsolved points in moment theory and, at the same time, new theoretical advances in the general moment methods can serve the progress in problems in statistical physics.