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Critical coupling thresholds for tilted Kuramoto-Vicsek models with a confining potential

Benedetta Bertoli, Benjamin D. Goddard, Grigorios A. Pavliotis

Abstract

We study a Kuramoto-Vicsek model of self-propelled particles with periodic boundary conditions subject to a constant angular tilt and a confining potential, and its mean-field (Fokker-Planck) behaviour. In the absence of confinement, the uniform density is stationary and we compute the critical coupling for four normalisation variants of the interaction kernel, showing that the leading instability is always spatially homogeneous. When the confining field is present, the uniform state is no longer stationary. We construct the new steady state perturbatively and apply eigenvalue perturbation theory to derive an explicit formula for the critical coupling as a function of the field strength. The threshold increases quadratically with confinement strength, and the tilt enters through the steady-state correction despite having no effect on the threshold in the absence of confinement. We verify the prediction numerically and derive self-consistency equations for stationary states with general multichromatic potentials.

Critical coupling thresholds for tilted Kuramoto-Vicsek models with a confining potential

Abstract

We study a Kuramoto-Vicsek model of self-propelled particles with periodic boundary conditions subject to a constant angular tilt and a confining potential, and its mean-field (Fokker-Planck) behaviour. In the absence of confinement, the uniform density is stationary and we compute the critical coupling for four normalisation variants of the interaction kernel, showing that the leading instability is always spatially homogeneous. When the confining field is present, the uniform state is no longer stationary. We construct the new steady state perturbatively and apply eigenvalue perturbation theory to derive an explicit formula for the critical coupling as a function of the field strength. The threshold increases quadratically with confinement strength, and the tilt enters through the steady-state correction despite having no effect on the threshold in the absence of confinement. We verify the prediction numerically and derive self-consistency equations for stationary states with general multichromatic potentials.
Paper Structure (24 sections, 4 theorems, 71 equations, 7 figures, 1 table)

This paper contains 24 sections, 4 theorems, 71 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and stationary-state formulation
  3. Particle system and mean-field limit
  4. Additive versus Non-Additive Interactions
  5. Self-consistency formulation for stationary states
  6. Spatially homogeneous reduction ($v_0 \neq 0$)
  7. Full spatial dependence ($v_0 = 0$)
  8. Reference case: $h=0$
  9. Linearisation about the uniform state
  10. Critical thresholds for the four normalisations
  11. Fully normalised interaction
  12. Unnormalised interaction
  13. Partial normalisation in $\theta$
  14. Partial normalisation in $\mathbf{x}$
  15. Justification of spatial homogeneity
  16. ...and 9 more sections

Key Result

Lemma 2.2

For $\Gamma$ sufficiently large, the map $T:\mathcal{P}\to\mathcal{P}$ defined by is a contraction on $(\mathcal{P}, \|\cdot\|_{L^1})$, and therefore admits a unique fixed point.

Figures (7)

  • Figure 1: Final density profiles $\rho(\theta,t_{\max})$ for several values of the alignment strength $\gamma$ and tilt parameter $F$. The transition from the uniform state to a clustered profile occurs at the same $\gamma$ for all $F$, consistent with Proposition \ref{['prop:F-invariance']}.
  • Figure 2: Steady-state density for the unnormalised interaction with $R=0.2$. The critical threshold is $\gamma_c=\Gamma/R=5$.
  • Figure 3: Steady-state density for the fully normalised interaction. The critical threshold is $\gamma_c = 2\Gamma = 2$.
  • Figure 4: Steady-state density for the partially normalised (in $\theta$) interaction with $R = 0.3$. The critical threshold is $\gamma_c = \Gamma/R = 3.33$.
  • Figure 5: Steady-state density for the partially normalised (in $x$) interaction. The critical threshold is $\gamma_c = \Gamma/\pi \approx 0.32$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Remark 2.1: Periodicity of the self-consistency integral
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • Remark 4.1: General normalisation
  • Lemma 4.2
  • proof
  • Proposition 4.3: Critical coupling with confinement