Evolution Equations for Cumulants of Distribution Functions of Particle Systems with Topological Interaction

The article formulates the concept of a cumulant representation for distribution functions that describe the state of many-particle systems with topological interaction, i.e., using the interaction potential determined by the proximity rank of particles. Cumulants of probability distribution functions are interpreted as correlations of particle states and are defined as solutions of the corresponding cluster expansions of probability distri­bution functions. We emphasize that the correlations that arise during the evolution of a system of particles with topological interaction naturally differ from the structure of correlations of many-particle systems, the state of which is traditionally described by symmetric probability distribution functions. The article establishes a hierarchy of evolution nonlinear equations for correlation functions (hierarchy of recursive Liouville equations). In the space sequences of integrated functions, a non-perturbative solution of the Cauchy problem for a hierarchy of such nonlinear evolution equations is constructed. Typical properties of the expansion of such a solution, which are generated by the properties of its generating operators, namely, cumulants of groups of operators of Liouville equations, are investigated. Based on the dynamics of correlations, the structure of series expansions is established, which determines reduced distribution functions, as well as reduced correlation functions, which, in particular, made it possible to substantiate the structure of generating operators of the solution of the Cauchy problem for the hierarchy of BBGKY equations (Bogolyubov–Born–Green–Kirkwood–Yvon). It is proved that the structure of expansions for correlation functions, the generating operators of which are cumulants of groups of operators of the corresponding order of Liouville equations, induces the cumulant structure of series expansions for reduced distribution functions and reduced correlation functions. Thus, the dynamics of state correlations generate the dynamics of many-particle systems with topological interaction.