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Stability of stationary states for mean field models with multichromatic interaction potentials

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

TL;DR

This work investigates the stability of stationary states in mean-field McKean–Vlasov dynamics on the torus with multichromatic interaction potentials. Using a combination of self-consistency analysis, second-variation calculus, and linearization, the authors derive a critical temperature $\\beta_c = \min_k{-2/a_k}$ that marks the transition from a uniform to nonuniform state, and provide explicit eigenstructure for the linearized operator near the transition. They extend the analysis to peaked states under various multichromatic potentials, employing perturbation theory to obtain leading-order corrections to eigenvalues and to characterize the emergence and stability of multimodal steady states. Theoretical results are supported by extensive numerical simulations of both the PDE and N-particle SDEs, confirming phase transitions, the role of Fourier modes in shaping steady states, and the agreement between mean-field predictions and particle-based dynamics. The findings advance understanding of phase transitions and pattern formation in nonlocal interacting particle systems and have implications for synchronization, active matter, and collective behavior models.

Abstract

We consider weakly interacting diffusions on the torus, for multichromatic interaction potentials. We consider interaction potentials that are not H-stable, leading to phase transitions in the mean field limit. We show that the mean field dynamics can exhibit multipeak stationary states, where the number of peaks is related to the number of nonzero Fourier modes of the interaction. We also consider the effect of a confining potential on the structure of non-uniform steady states. We approach the problem by means of analysis, perturbation theory and numerical simulations for the interacting particle systems and the PDEs.

Stability of stationary states for mean field models with multichromatic interaction potentials

TL;DR

This work investigates the stability of stationary states in mean-field McKean–Vlasov dynamics on the torus with multichromatic interaction potentials. Using a combination of self-consistency analysis, second-variation calculus, and linearization, the authors derive a critical temperature that marks the transition from a uniform to nonuniform state, and provide explicit eigenstructure for the linearized operator near the transition. They extend the analysis to peaked states under various multichromatic potentials, employing perturbation theory to obtain leading-order corrections to eigenvalues and to characterize the emergence and stability of multimodal steady states. Theoretical results are supported by extensive numerical simulations of both the PDE and N-particle SDEs, confirming phase transitions, the role of Fourier modes in shaping steady states, and the agreement between mean-field predictions and particle-based dynamics. The findings advance understanding of phase transitions and pattern formation in nonlocal interacting particle systems and have implications for synchronization, active matter, and collective behavior models.

Abstract

We consider weakly interacting diffusions on the torus, for multichromatic interaction potentials. We consider interaction potentials that are not H-stable, leading to phase transitions in the mean field limit. We show that the mean field dynamics can exhibit multipeak stationary states, where the number of peaks is related to the number of nonzero Fourier modes of the interaction. We also consider the effect of a confining potential on the structure of non-uniform steady states. We approach the problem by means of analysis, perturbation theory and numerical simulations for the interacting particle systems and the PDEs.
Paper Structure (22 sections, 2 theorems, 78 equations, 15 figures, 3 tables)

This paper contains 22 sections, 2 theorems, 78 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Our Contributions
  4. Set-up and self-consistency equations
  5. Phase transitions, stability analysis and the self-consistency equation
  6. Examples
  7. Stability analysis of the uniform state
  8. Second variation of the free energy
  9. Linearisation of the McKean-Vlasov equation
  10. Stability analysis of the peaked states
  11. The Kuramoto model
  12. A higher order harmonic potential
  13. Multichromatic potentials
  14. Numerical experiments
  15. Steady states and intermediate dynamics
  16. ...and 7 more sections

Key Result

Proposition 2.1

The generalised Kuramoto model $W(x)=-\cos(k x)$, for some $k \in \mathbb{N}$, $k \neq 0$ exhibits a continuous transition point at the linear instability threshold ${\beta}_c$. For $\beta>\beta_c$, the equation $\mathcal{F}(\rho,\beta, \kappa)=0$ has only two solutions in $L^2([0,2\pi])$ (up to tra

Figures (15)

  • Figure 1: Probability distributions corresponding to the three different solutions of the $(r_1, r_2)$ equations for the bichromatic system with $\beta = 3$ (left) and $\beta = 10$ (right).
  • Figure 2: First eigenvalue as a function of $\delta$, $W(x) = -\cos(x)$.
  • Figure 3: First eigenvalue as a function of $\eta$, $W(x) = -\cos(x)-\frac{1}{2} \cos(2x)$.
  • Figure 4: PDE dynamics for different values of $\beta$ for the interacting potential $W(x) = - 3\cos(x) - \cos(2x)$ for times $T = 5$ (left) and $T = 10^3$ (right). The critical temperature is $\beta_c = \frac{2}{3}$.
  • Figure 5: PDE dynamics for different values of $\beta$ for the interacting potential $W(x) = - 3\cos(x) - \cos(2x)-\cos(3x)$ for times $T = 5$ (left) and $T = 10^3$ (right). The critical temperature is $\beta_c = \frac{2}{3}$.
  • ...and 10 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Lemma 2.2
  • proof