Cucker-Smale flocking particles with multiplicative noises: stochastic mean-field limit and phase transition

TL;DR

This paper studies a stochastic Cucker-Smale flocking model with multiplicative noise, proving a mean-field limit to a stochastic PDE and analyzing phase transitions and long-term behavior of the system.

Contribution

It rigorously establishes the stochastic mean-field limit for the model and analyzes phase transition phenomena and stability of solutions.

Findings

Existence of a phase transition from non-flocking to flocking states as noise strength varies.

Quantitative error estimates for the mean-field approximation using Wasserstein distance.

Exponential convergence of velocities to the mean in the limiting stochastic PDE.

Abstract

In this paper, we consider the Cucker-Smale flocking particles which are subject to the same velocity-dependent noise, which exhibits a phase change phenomenon occurs bringing the system from a "non flocking" to a "flocking" state as the strength of noises decreases. We rigorously show the stochastic mean-field limit from the many-particle Cucker-Smale system with multiplicative noises to the Vlasov-type stochastic partial differential equation as the number of particles goes to infinity. More precisely, we provide a quantitative error estimate between solutions to the stochastic particle system and measure-valued solutions to the expected limiting stochastic partial differential equation by using the Wasserstein distance. For the limiting equation, we construct global-in-time measure-valued solutions and study the stability and large-time behavior showing the convergence of velocities to their mean exponentially fast almost surely.