Towards kinetic equations of open systems of active soft matter
V. I. Gerasimenko
TL;DR
The work develops a rigorous, observables-centric kinetic description for open systems of self-propelled entities in active soft matter. By formulating a dual BBGKY hierarchy for reduced observables and constructing a non-perturbative cumulant-based solution, it connects observable dynamics to a reduced-state description via functionals of the state. A key result is a non-Markovian Fokker–Planck generalized kinetic equation for the reduced distribution $F_{1+0}(t,\mathbf{u})$ that incorporates initial system–environment correlations through higher-order functionals $F_{1+s}(t|F_{1}(t))$, with convergence guaranteed under $\alpha<e^{-4}$. This framework facilitates exact (within the initial-data scope) description of correlation propagation and creation, and it provides a principled path from microscopic Markov-jump dynamics to macroscopic, memory-bearing kinetics, with clear avenues for scaling limits and extensions to other interaction structures.
Abstract
The chapter presents some new approaches to describing the collective behavior of complex systems of mathematical biology based on the evolution equations of observables such as open systems. This representation of kinetic evolution has looked to be the most direct and mathematically fully consistent formulation modeling the collective behavior of biological systems since the traditionally used concept of the state in kinetic theory is more subtle and is an implicit characteristic of the populations of living creatures. One of the advantages of the developed approach is the opportunity to construct kinetic equations for open complex systems in scaling approximations, involving initial correlations, in particular, that can characterize the condensed states of active soft matter. An approach is also related to the challenge of a rigorous derivation of the non-Markovian kinetic equations from underlying many-entity dynamics, which makes it possible to describe the memory effects of the collective behavior of living creatures.