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Environmental Averaging

Roman Shvydkoy

TL;DR

It is proved that solutions to the Fokker-Planck model based on a smooth environmental averaging gain Gaussian tails uniformly in time and, consequently, relax unconditionally to the global Maxwellian.

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.

Environmental Averaging

TL;DR

It is proved that solutions to the Fokker-Planck model based on a smooth environmental averaging gain Gaussian tails uniformly in time and, consequently, relax unconditionally to the global Maxwellian.

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.
Paper Structure (40 sections, 58 theorems, 790 equations, 3 tables)

This paper contains 40 sections, 58 theorems, 790 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Basic concept and examples
  3. Examples
  4. Classes of models and their properties
  5. Mapping properties. Jensen inequality
  6. Reproducing kernel
  7. Conservative models and contractivity
  8. Symmetric models
  9. Galilean invariance
  10. Ball-positivity
  11. Thickness, regularity, and well-posedness of microscopic systems
  12. Locality of communication
  13. Thickness
  14. Regularity of $\mathcal{M}$ and continuous dependence on $\rho$
  15. Well-posedness of agent-based systems
  16. ...and 25 more sections

Key Result

Theorem 1.1

If $\beta \leqslant 1$, all solutions to e:CSintro align exponentially fast to the mean velocity $\bar{v} = \frac{1}{ \sum_{j=1}^N m_j} \sum_{j=1}^N m_j v_j$, while flock remains bounded where $C,\delta,\bar{D}$ depend only on the initial condition and parameters of the kernel. If $\beta >1$ there are solutions that do not align.

Theorems & Definitions (137)

  • Theorem 1.1: CS2007aCS2007b
  • Theorem 1.2
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6: Topological models
  • Example 2.7: Models with strict Segregation
  • Example 2.8: Smooth Segregation
  • ...and 127 more