Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system

Abstract

We study a stochastic Cucker-Smale flocking system in which particles interact with the environment through white noise. We provide the definition of flocking for the stochastic system, and show that when the communication rate is constant, the system exhibits a flocking behavior independent of the initial configurations. For the case of a radially symmetric communication rate with a positive lower bound, we show that the relative fluctuations of the particle velocity around the mean velocity have a uniformly bounded variance in time. We conclude with numerical simulations that validate our analytical results.

Langevin Dynamics with Variable Coefficients and Nonconservative Forces: From Stationary States to Numerical Methods

“…An SDE model was presented there without analysis. A subsequent paper [5] provided numerical evidence for flocking states. In these papers, the SDE approach consists of a Langevin-type model with coordinate-dependent friction and additive (typically uniform) white noise, that is they take the form…”

“…When we say that µ is an invariant measure of the SDE (5) or that the latter preserves the probability measure µ, we mean that the probability density ρ is a stationary solution of the Fokker-Planck equation associated to the SDE, i.e.,…”

“…Second, we do not assume a conservative force field. Such generalized forms of Langevin dynamics can be used to model diffusion (or thermal) gradients by particle simulation [1][2][3], as well as a variety of models for flocking [4][5][6], protein folding [7] and bacterial suspensions [8]. Problems with nonconservative forces are considered often in the physics literature, but the precise characterization of the convergence to stationary conditions is rarely discussed.…”

How Do Cultural Classes Emerge from Assimilation and Distinction? An Extension of the Cucker-Smale Flocking Model

“…The robustness of the Cucker-Smale model to these noises is being lively studied in mathematics. For example, the Cucker-Smale system is proved to keep flocking in the presence of constant-strength noises (Ha, Lee, & Levy, 2009) and multiplicative noises (Ahn & Ha, 2010 …”

MULTIPLE SOLUTIONS TO DISCRETE BOUNDARY VALUE PROBLEMS FOR THE p-LAPLACIAN WITH POTENTIAL TERMS ON FINITE GRAPHS

“…The reason why we consider the discrete p-Laplacian on weighted graphs is that it is one of the best methods to represent linear or nonlinear phenomena with network structures (for examples, see [6,8,9,17,18,19]). …”