Phase transition and asymptotic behaviour of flocking Cucker-Smale model

TL;DR

This paper analyzes a noisy flocking Cucker-Smale model, identifying a noise threshold for phase transition, classifying stationary solutions, and establishing their stability and asymptotic convergence rates.

Contribution

It introduces a precise noise threshold for phase transition and provides a detailed stability and convergence analysis of stationary solutions in the model.

Findings

Identified the noise parameter threshold for phase transition.

Classified stationary solutions and their stability.

Proved exponential convergence to stable solutions.

Abstract

In this paper, we study a continuous ocking Cucker-Smale model with noise, which has isotropic and polarized stationary solutions depending on the intensity of the noise. The first result establishes the threshold value of the noise parameter which drives the phase transition. This threshold value is used to classify all stationary solutions and their linear stability properties. Using an entropy, these stability properties are extended to the non-linear regime. The second result is concerned with the asymptotic behaviour of the solutions of the evolution problem. In several cases, we prove that stable solutions attract the other solutions with an optimal exponential rate of convergence determined by the spectral gap of the linearized problem around the stable solutions. The spectral gap has to be computed in a norm adapted to the non-local term.