Classical Gasses with Singular Densities

We investigate classical continuous systems characterized by singular velocity distributions, where the corresponding Radon measures are defined over the entire space with infinite mass. These singular distributions are used to model particle velocities in systems where traditional velocity distributions do not apply. As a result, the particle positions in such systems no longer conform to conventional configurations in physical space. This necessitates the development of novel analytical tools to understand the underlying models. To address this, we introduce a new conceptual framework that redefines particle configurations in phase space, where each particle is represented by its spatial position and a velocity vector. The key idea is the construction of the Plato space, which is designed to represent idealized particle configurations where the total velocity remains bounded within any compact subset of phase space. This space serves as a crucial bridge to the space of vector-valued discrete Radon measures, where each measure captures the velocity distribution over the entire system. Given the inherent complexity of analyzing infinite-dimensional spaces, we tackle the problem by reformulating it onto a finite-dimensional configuration space. This is achieved by decomposing the infinite space into smaller, more manageable components. A central tool in this reformulation is the K-transform, which is pivotal in enabling harmonic analysis of the space. The K-transform allows us to represent the system in terms of components that are more amenable to analysis, thus simplifying the study of the system's dynamics. Furthermore, we extend previous results in the study of correlation functions by developing correlation measures tailored for these vector-valued Radon measures. These generalized functions provide deeper insights into the correlations between particle positions and velocities, expanding the range of analysis to systems with singular velocity distributions. Through this approach, we develop a robust mathematical framework that sheds light on the structure and dynamics of complex particle systems, especially those characterized by singular velocity distributions. Our results offer a new perspective on systems with non-traditional velocity distributions, advancing the theory and methodology of particle systems in both classical and modern contexts.