On Semigroups of Large Particle Systems and their Scaling Asymptotic Behavior

TL;DR

This paper studies the semigroups of operators for large particle systems, establishing their connection to cumulants of Liouville semigroups and analyzing their mean field asymptotics leading to Vlasov-type equations.

Contribution

It introduces a novel framework linking semigroups of hierarchies to cumulants and derives the Vlasov-type dual hierarchy for mean field limits.

Findings

Semigroups are generated by cumulants of Liouville semigroups.

The mean field limit leads to a Vlasov-type dual hierarchy.

Relationships between the hierarchy and kinetic equations with initial correlations.

Abstract

We consider semigroups of operators for hierarchies of evolution equations of large particle systems, namely, of the dual BBGKY hierarchy for marginal observables and the BBGKY hierarchy for marginal distribution functions. We establish that the generating operators of the expansions for one-parametric families of operators of these hierarchies are the corresponding order cumulants (semi-invariants) of semigroups for the Liouville equations. We also apply constructed semigroups to the description of the kinetic evolution of interacting stochastic Markovian processes, modeling the microscopic evolution of soft active matter. For this purpose we consider the mean field asymptotic behavior of the semigroup generated by the dual BBGKY hierarchy for marginal observables. The constructed scaling limit is governed by the set of recurrence evolution equations, namely, by the Vlasov-type dual hierarchy. Moreover, the relationships of this hierarchy of evolution equations with the Vlasov-type kinetic equation with initial correlations are established.