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Mixing, Enhanced Dissipation and Phase Transition in the Kinetic Vicsek Model

Mengyang Gu, Siming He

TL;DR

This work develops a rigorous kinetic framework for the Vicsek model of cell polarization, deriving the mesoscopic PDE $\partial_t f + v\mathbf{p}(\theta)\cdot \nabla_{\mathbf{x}} f + \kappa \partial_\theta(f L[f]) = \nu \partial_\theta^2 f$ from agent-based dynamics. It identifies nonlinear enhanced dissipation and mixing as the core mechanisms driving rapid spatial homogenization, enabling efficient information exchange and reducing the inhomogeneous dynamics to a homogeneous effective description on long times. By analyzing the effective homogeneous dynamics via a free-energy functional and linear-stability criteria, it characterizes phase-transition-like behavior depending on the ratio $\kappa/\nu$, including stability thresholds and Fisher-information-based convergence. These results provide a rigorous justification of phase separation in the spatially inhomogeneous Vicsek model and link to Fisher-von Mises-type equilibria, with implications for pattern formation in tissue engineering and cellular alignment.

Abstract

In this paper, we study the kinetic Vicsek model, which serves as a starting point for describing the polarization phenomena observed in the experiments of fibroblasts moving on liquid crystalline substrates. The long-time behavior of the kinetic equation is analyzed, revealing that, within specific parameter regimes, the mixing and enhanced dissipation phenomena stabilize the dynamics and ensure effective information communication among agents. Consequently, the solution exhibits features similar to those of a spatially-homogeneous system. As a result, we confirm the phase transition observed in the agent-based Vicsek model at the kinetic level.

Mixing, Enhanced Dissipation and Phase Transition in the Kinetic Vicsek Model

TL;DR

This work develops a rigorous kinetic framework for the Vicsek model of cell polarization, deriving the mesoscopic PDE from agent-based dynamics. It identifies nonlinear enhanced dissipation and mixing as the core mechanisms driving rapid spatial homogenization, enabling efficient information exchange and reducing the inhomogeneous dynamics to a homogeneous effective description on long times. By analyzing the effective homogeneous dynamics via a free-energy functional and linear-stability criteria, it characterizes phase-transition-like behavior depending on the ratio , including stability thresholds and Fisher-information-based convergence. These results provide a rigorous justification of phase separation in the spatially inhomogeneous Vicsek model and link to Fisher-von Mises-type equilibria, with implications for pattern formation in tissue engineering and cellular alignment.

Abstract

In this paper, we study the kinetic Vicsek model, which serves as a starting point for describing the polarization phenomena observed in the experiments of fibroblasts moving on liquid crystalline substrates. The long-time behavior of the kinetic equation is analyzed, revealing that, within specific parameter regimes, the mixing and enhanced dissipation phenomena stabilize the dynamics and ensure effective information communication among agents. Consequently, the solution exhibits features similar to those of a spatially-homogeneous system. As a result, we confirm the phase transition observed in the agent-based Vicsek model at the kinetic level.
Paper Structure (8 sections, 10 theorems, 160 equations)

This paper contains 8 sections, 10 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Nonlinear Enhanced Dissipation and Mixing
  3. Analysis of the Effective Dynamics
  4. Linear Enhanced Dissipation and Inviscid Damping
  5. Connection between Different Formulations
  6. The Sobolev Estimates of the Solution
  7. Energy Minimizers of the Spatially Homogeneous System
  8. Derivation of the Kinetic Model

Key Result

Theorem 1.1

Consider solutions to equation eq:bsc subject to initial condition $0< f_0\in C^\infty({\mathbb T}^3)$. Assume that the speed profile $v(\cdot)\in C^\infty(\mathbb{R}_+)$ takes values in $(1/2,1]$ and that the $C^\infty$ smooth influence functions $\Phi$ and $\Psi$ satisfy the structural conditions Here, $C_2$ is a universal constant. b) Mixing: If one assumes that the agent speed $v(t)\equiv 1$

Theorems & Definitions (28)

  • Theorem 1.1: Spatial Homogenization
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Corollary 1.1
  • Theorem 1.3
  • Remark 1.4
  • proof : Proof of Theorem \ref{['thm_1']} a)
  • proof : Proof of Theorem \ref{['thm_1']} b)
  • ...and 18 more