Flocking and concentration behavior for the stochastic Cucker-Smale system in a harmonic field

TL;DR

This paper studies how a harmonic potential influences flocking and concentration in a stochastic Cucker-Smale system, establishing conditions for these behaviors and analyzing the mean-field limit.

Contribution

It introduces a stochastic Lyapunov functional to derive conditions for flocking and concentration, and provides a rigorous mean-field limit analysis.

Findings

Almost sure velocity flocking and spatial concentration are achieved under certain conditions.

Numerical verification supports the theoretical results.

A rigorous mean-field limit estimate from stochastic to kinetic models is established.

Abstract

We consider the Cucker-Smale system with multiplicative noise in a harmonic potential field and investigate the effect of harmonic potential field. In the presence of external potential force, the system is expected to emerge into almost surely velocity flocking and spatial concentration, due to the alignment mechanism, confining harmonic potential field and multiplicative noise. By constructing a stochastic Lyapunov functional, we derive a sufficient condition for the almost surely flocking and concentration behavior for the stochastic particle model, and verify it numerically. Then, we discuss the flocking and concentration behavior for the mean field Vlasov-type kinetic model. Moreover, a rigorous analysis of the uniform mean-field limit estimate for the limit process from the stochastic model to the kinetic one is provided.