Global Solutions to Multi-dimensional Topological Euler Alignment Systems

Lear, Daniel; Reynolds, David N.; Shvydkoy, Roman

doi:10.1007/s40818-021-00116-z

Cited by 8 publications

(4 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A new was of modeling interactions which implement topological, rather than Euclidean measure of distance, has long been advocated by many empirical studies [SB14, NMG14, BCC + 08, CCG + 12]. The fist symmetric topological model was introduced in [ST20b], see also [LRS22,RS20], although it incorporated singular communication. Its smooth variant fits within our framework of environmental averaging.…”

Section: Basic Concept and Examplesmentioning

confidence: 99%

Environmental averaging

Shvydkoy¹

2022

Preprint

View full text Add to dashboard Cite

We introduce a new analytical framework designed to systematically study models of collective behavior based on smooth communication. Many examples including the classical Cucker-Smale, Motsch-Tadmor, multi-species, and several new models fall under the framework and can be viewed as special cases of 'environmental averaging'. By identifying special properties of the averaging protocols we reveal new alignment results that are otherwise obscured by specificity of the model if analyzed individually. In particular, it is shown that the kinetic Cucker-Smale model based on a Bochner-positive communication kernel φ = ψ * ψ aligns provided the density of the flock obeys a lower bound ρ 1 t 1/4 , and no bound from above is required. Similar results are proved for other positive models.A series of studies are presented based on the introduced framework -mean-field limit in deterministic and stochastic settings, hydrodynamic limits in the monokinetic and Maxwellian regimes, global hypocoercivity and relaxation for dissipative kinetic models, several general alignment results based on chain connectivity of the system. In each study we identify a set of natural regularity conditions on the averaging protocol sufficient for the end-result in question. Numerous applications to existing models and new examples are revealed. A detailed well-posedness theory for kinetic Vlasov-alignment and Fokker-Planck-alignment models is developed. It is shown that solutions to the Fokker-Planck model based on a smooth environmental averaging gain Gaussian tails uniformly in time and relax unconditionally. This applies, in particular, to the classical Cucker-Smale model and presents a substantial improvement over previously obtained perturbative results.

show abstract

Section: Basic Concept and Examplesmentioning

confidence: 99%

Environmental averaging

Shvydkoy¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…A notable special case worth mentioning is that of "parallel shear flocks", which was examined in [21,Section 4.1]. In this context, we consider shear velocities represented by ρ = ρ(x 2 , • • • , x d , t), u = u(x 2 , • • • , x d , t) in the system (1.5).…”

Section: The Supercritical Regime: Refined Regularity Criterionmentioning

confidence: 99%

Global well-posedness for the 3D damped micropolar Bénard system with zero thermal conductivity

Tan

2021

Applied Mathematics Letters

View full text Add to dashboard Cite

“…It has drawn researchers from a variety of fields over the past few decades, extending outside biology. Many mathematicians also study this recently, see [19], [45], [29], [30] [40], [41], [14], [49], [33], [18] and the reference therein.Additionally, theoretical physicists in particular look into the similarities between large animal collectives and statistical physics systems like fluids or magnets, where local interactions between a lot of relatively simple components result in the emergence of novel properties at the macroscopic level that can be challenging to link to the characteristics of the individual components. [6,39,28].…”

Section: Introductionmentioning

confidence: 99%

Animal Groupings With Phase Change

Gérard¹

2023

Preprint

View full text Add to dashboard Cite

Fish schools, bird flocks, and insect swarms are a few examples of animal groupings that engage in collective behaviors that are intriguing demonstrations of self-organization in biology. Theoretical debates regarding the potential repercussions of collective effects in such living systems have been conducted using concepts and techniques from statistical physics. In order to maximize their capacity for collective computation, it has been suggested that such collective systems should function close to a phase transition, specifically a (pseudo-)critical point. This paper will begin by reviewing key phase transitions displayed by animal collectives and highlighting the challenges in translating statistical physics concepts to biological systems. After that, we'll go over the present state of the "criticality hypothesis" study, including how to assess the distance from criticality and the functional repercussions for animal groups that operate close to a phase transition. We will emphasis the growing theory that analyzes the advantages of living systems being able to tune to an ideal distance from criticality rather than emphasizing the optimality of being precisely at a critical point. We will conclude by outlining upcoming difficulties in understanding collective behavior at the nexus of physics and biology.

show abstract

Global Solutions to Multi-dimensional Topological Euler Alignment Systems

Cited by 8 publications

References 33 publications

Environmental averaging

Environmental averaging

Global well-posedness for the 3D damped micropolar Bénard system with zero thermal conductivity

Animal Groupings With Phase Change

Contact Info

Product

Resources

About