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Stability analysis for active Brownian particle models

Michele Coti Zelati, Lucas Ertzbischoff, David Gerard-Varet

TL;DR

This work provides a rigorous linear stability analysis for kinetic active Brownian particle models with density-dependent speed, identifying a sharp threshold based on the flux derivative ∂_ρ(ρ v) that governs stability versus instability of the homogeneous state. The authors reduce the problem to a one-dimensional angular equation, derive an explicit non-diffusive dispersion relation, and prove a Landau-damping–like decay in the stable regime, with non-integrable time decay. They extend the analysis to small angular diffusion using a diffusive dispersion relation and Volterra-type equations, obtaining precise decay rates and showing persistence of instability for small ν in the unstable regime. A variant of the main equation is analyzed in parallel, yielding analogous stability thresholds and dispersion relations, thereby broadening the applicability of the results to related active-matter models.

Abstract

We carry out a comprehensive linear stability analysis of active Brownian particle systems around a constant homogeneous state. These scalar models, being important prototypes for the continuous description of active matter, are Fokker-Planck type equations in position-orientation and are known to exhibit motility-induced phase separation. We fully characterize the linear stability and instability regimes, with an explicit threshold depending on the effective speed of the particles. In this way, we rigorously confirm a conjecture on phase separation originating in the physics and applied literature. Our sharp and quantitative (in)stability results are valid both in the non-diffusive case and in the case of small angular diffusion. In the stable non-diffusive regime, we uncover a mixing mechanism reminiscent of Landau damping for the Vlasov equation, albeit with significantly weaker decay. This decay is non-integrable in time and gives rise to substantial mathematical difficulties; in particular, it prevents the use of classical perturbative arguments to treat the case of small angular diffusion.

Stability analysis for active Brownian particle models

TL;DR

This work provides a rigorous linear stability analysis for kinetic active Brownian particle models with density-dependent speed, identifying a sharp threshold based on the flux derivative ∂_ρ(ρ v) that governs stability versus instability of the homogeneous state. The authors reduce the problem to a one-dimensional angular equation, derive an explicit non-diffusive dispersion relation, and prove a Landau-damping–like decay in the stable regime, with non-integrable time decay. They extend the analysis to small angular diffusion using a diffusive dispersion relation and Volterra-type equations, obtaining precise decay rates and showing persistence of instability for small ν in the unstable regime. A variant of the main equation is analyzed in parallel, yielding analogous stability thresholds and dispersion relations, thereby broadening the applicability of the results to related active-matter models.

Abstract

We carry out a comprehensive linear stability analysis of active Brownian particle systems around a constant homogeneous state. These scalar models, being important prototypes for the continuous description of active matter, are Fokker-Planck type equations in position-orientation and are known to exhibit motility-induced phase separation. We fully characterize the linear stability and instability regimes, with an explicit threshold depending on the effective speed of the particles. In this way, we rigorously confirm a conjecture on phase separation originating in the physics and applied literature. Our sharp and quantitative (in)stability results are valid both in the non-diffusive case and in the case of small angular diffusion. In the stable non-diffusive regime, we uncover a mixing mechanism reminiscent of Landau damping for the Vlasov equation, albeit with significantly weaker decay. This decay is non-integrable in time and gives rise to substantial mathematical difficulties; in particular, it prevents the use of classical perturbative arguments to treat the case of small angular diffusion.

Paper Structure

This paper contains 23 sections, 27 theorems, 339 equations, 2 figures.

Table of Contents

  1. Introduction, motivations and main statements
  2. Active Brownian particle model
  3. Main statements
  4. Related works
  5. Strategy of proof
  6. Reduction
  7. Main steps of the proof
  8. Spectral study without diffusion
  9. Mixing estimates in the non-diffusive case
  10. Instability in presence of rotational diffusion
  11. Linear stability and decay estimates with rotational diffusion
  12. Decay estimates for transport--diffusion equations
  13. Stability condition for the Volterra equation
  14. Final estimates
  15. A variant of the main equation \ref{['eq:fulleq-INTRO']}
  16. ...and 8 more sections

Key Result

Theorem 1.2

Assume that assumption-velocity holds, and let $\phi \in (0,1)$ satisfy the instability condition Then the following hold:

Figures (2)

  • Figure 1: Contours $Q_{r_0, R}$ and $\mathscr{C}_\varepsilon$
  • Figure 2: Splitting of $\lbrace -\delta < \mathrm{Re} <0 \rbrace$.

Theorems & Definitions (50)

  • Remark 1.1
  • Theorem 1.2: Linear instability
  • Remark 1.3
  • Theorem 1.4: Linear stability
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1: Linear instability - reduced
  • Theorem 2.2: Linear stability - reduced
  • Definition 3.1
  • Proposition 3.2
  • ...and 40 more