Environmental Averaging

TL;DR

This paper introduces a unified framework for analyzing self-organized alignment models based on environmental averaging, covering mean-field limits, hydrodynamic regimes, and relaxation properties, with new criteria and estimates.

Contribution

It formalizes environmental averaging, unifies various alignment models, and provides new analytical results including relaxation criteria and spectral gap estimates.

Findings

Asymptotic alignment criteria based on connectivity.

Spectral gap estimates independent of density bounds.

Unconditional relaxation for Fokker-Planck-Alignment models.

Abstract

Many classical examples of models of self-organized dynamics, including the Cucker-Smale, Motsch-Tadmor, multi-species, and several others, include an alignment force that is based upon density-weighted averaging protocol. Those protocols can be viewed as special cases of `environmental averaging'. In this paper we formalize this concept and introduce a unified framework for systematic analysis of alignment models. A series of studies are presented including the mean-field limit in deterministic and stochastic settings, hydrodynamic limits in the monokinetic and Maxwellian regimes, hypocoercivity and global relaxation for dissipative kinetic models, several general alignment results based on chain connectivity and spectral gap analysis. These studies cover many of the known results and reveal new ones, which include asymptotic alignment criteria based on connectivity conditions, new estimates on the spectral gap of the alignment force that do not rely on the upper bound of the macroscopic density, uniform gain of positivity for solutions of the Fokker-Planck-Alignment model based on smooth environmental averaging. As a consequence, we establish unconditional relaxation result for global solutions to the Fokker-Planck-Alignment model, which presents a substantial improvement over previously known perturbative results.