Cluster expansions of particle system state with topological nearest-neighbor interaction
V. I. Gerasimenko, I. V. Gapyak
TL;DR
The paper addresses the evolution of states in many-particle systems with topological nearest-neighbor interactions by developing a cumulant- and cluster-expansion framework. It builds a non-perturbative solution to the Liouville hierarchy for cumulants and connects this to the BBGKY hierarchy, while also deriving expansions for reduced distribution and correlation functions. Key contributions include explicit cumulant and cluster-expansion representations, existence/uniqueness results under density-like conditions ($\alpha>2$), and demonstrations of equivalence between different solution forms for the BBGKY hierarchy. The framework clarifies how correlations evolve in infinite-particle systems with topological interactions and provides rigorous tools for reduced observables and reduced correlation functions in complex systems.
Abstract
The article presents the concept of a cumulant representation for distribution functions describing the states of many-particle systems with topological nearest-neighbor interaction. A solution to the Cauchy problem for the hierarchy of nonlinear evolution equations for the cumulants of distribution functions of such systems is constructed. The connection between the constructed solution and the series expansion structure for a solution to the Cauchy problem of the BBGKY hierarchy has been established. Furthermore, the expansion structure for a solution to the Cauchy problem of the hierarchy of evolution equations for reduced observables of topologically interacting particles is established.