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Mean-Field Limits of Deterministic and Stochastic Flocking Models with Nonlinear Velocity Alignment

Vinh Nguyen, Roman Shvydkoy, Changhui Tan

TL;DR

This work analyzes mean-field limits for flocking models with nonlinear velocity alignment $A(\bv)=|\bv|^{p-2}{\bv}$ under fat-tailed communication. By deriving a Vlasov-type kinetic equation and establishing a Dobrushin-type stability bound, the authors prove rigorous mean-field limits and quantitative propagation of chaos in both deterministic and stochastic settings, including multiplicative noise that depends on local interaction strength. They obtain explicit rates that depend on the nonlinear exponent $p$ and tail parameter $\alpha$, with sub-exponential growth and logarithmic corrections in critical cases, and extend the classical Cucker–Smale theory to nonlinear alignment. The results provide a unified, rate-driven framework for understanding how large-agent flocking systems converge to mean-field dynamics and how finite-size effects decay, with implications for modeling collective behavior in biology and engineered systems. The stochastic analysis further yields a Fokker-Planck-Alignment limit, showing robust convergence under strength-dependent noise and broadens the applicability to more realistic noisy environments.

Abstract

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol $φ$ and a non-linear coupling of velocities given by the power law $A(\bv) = |\bv|^{p-2}\bv$, $p > 2$. The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the $k$-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.

Mean-Field Limits of Deterministic and Stochastic Flocking Models with Nonlinear Velocity Alignment

TL;DR

This work analyzes mean-field limits for flocking models with nonlinear velocity alignment under fat-tailed communication. By deriving a Vlasov-type kinetic equation and establishing a Dobrushin-type stability bound, the authors prove rigorous mean-field limits and quantitative propagation of chaos in both deterministic and stochastic settings, including multiplicative noise that depends on local interaction strength. They obtain explicit rates that depend on the nonlinear exponent and tail parameter , with sub-exponential growth and logarithmic corrections in critical cases, and extend the classical Cucker–Smale theory to nonlinear alignment. The results provide a unified, rate-driven framework for understanding how large-agent flocking systems converge to mean-field dynamics and how finite-size effects decay, with implications for modeling collective behavior in biology and engineered systems. The stochastic analysis further yields a Fokker-Planck-Alignment limit, showing robust convergence under strength-dependent noise and broadens the applicability to more realistic noisy environments.

Abstract

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol and a non-linear coupling of velocities given by the power law , . The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the -particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.
Paper Structure (15 sections, 12 theorems, 206 equations, 1 table)

This paper contains 15 sections, 12 theorems, 206 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Statement of Main Results
  3. Wasserstein distance
  4. Flocking and alignment with fat-tailed communication protocol
  5. Key auxiliary lemmas
  6. Statement of main results
  7. Mean-Field Limit For Deterministic Model
  8. Flocking and alignment estimates
  9. Stability
  10. Mean-field Limit
  11. Propagation of Chaos for Deterministic Model with Fat-tailed Kernels
  12. Mean-Field Limit for Stochastic Model
  13. Proof of key auxillary lemmas
  14. Proof of Lemma \ref{['lem:ab-aux']}
  15. Proof of Lemma \ref{['lem:ab-aux2']}

Key Result

Theorem 2.2

Suppose $(\mathcal{D}(t),\mathcal{V}(t))$ satisfy the paired inequality SDDI with bounded initial data $(\mathcal{D}_0,\mathcal{V}_0)$, and the communication protocol $\phi$ satisfies fat-tail. Then, for any $t\geqslant0$, we have the following bounds:

Theorems & Definitions (19)

  • Definition 2.1
  • Theorem 2.2: Flocking and alignment estimates black2024asymptotic
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Theorem 2.6: Stability
  • Theorem 2.7: Mean-field limit
  • Theorem 2.8: Propagation of chaos
  • Theorem 2.9
  • Proposition 3.1
  • ...and 9 more