Spectral insights into active matter: Exceptional Points and the Mathieu equation

TL;DR

The paper explains the universal scaling observed in noisy, aligning active matter by mapping the free one-particle Fokker-Planck operator to the Mathieu equation with an imaginary parameter $q=2k\,\mathrm{Pe}$. Through perturbation theory and Mathieu-function theory, it identifies a cascade of exceptional points that governs the spectral structure, yielding exact exponents: in the small-$q$ limit, $\alpha_{\text{low}}=2$ and $\beta_{\text{low}}=1$; in the large-$q$ limit, $\alpha_{\text{high}}=1/2$ and $\beta_{\text{high}}=-1/8$, with a universal high-activity projection $\sim q^{-1/8}$. These spectral insights, extended to mean-field interactions, predict the scaling of the critical coupling $\Gamma_c \sim q^{\gamma}$ with symmetry-dependent exponents $\gamma$, capturing the two regimes observed in simulations for polar and higher-symmetry alignments. The work reveals a dynamical phase-transition-like mechanism rooted in non-Hermitian spectral topology, offering a rigorous foundation for non-equilibrium scaling laws in active matter and suggesting broad applicability to mode selection and density-order coupling. Overall, it connects perturbative analysis, exceptional-point cascades, and Mathieu-function asymptotics to provide exact, universal scaling predictions for noisy active systems.

Abstract

We show that recent numerical findings of universal scaling relations in systems of noisy, aligning self-propelled particles by Kürsten [Kürsten, arXiv:2402.18711v2 [cond-mat.soft] (2025)] can robustly be explained by perturbation theory and known results for the Mathieu equation with purely imaginary parameter. In particular, we highlight the significance of a cascade of exceptional points that leads to non-trivial fractional scaling exponents in the singular-perturbation limit of high activity. Crucially, these features are rooted in the Fokker-Planck operator corresponding to free self-propulsion. This can be viewed as a dynamical phase transition in the dynamics of noisy active matter.

Figures (3)

• Figure 1: Projection of the numerically found lowest eigenmode onto the lowest three Fourier modes as a function of the activity parameter $q$ and for various numbers $Q$ of considered modes, see text for detail. The axes use double-logarithmic scaling. We are also providing the scaling behaviors $q^1$ and $q^{-\frac{1}{8}}$ (black dashed lines) found in earlier numerical works kuerstennumerical as guides to the eye. The diagonalization of the system eq. \ref{['eq:eigenvaluematrix']} was done using the Eigen library eigenweb.
• Figure 2: Large: Real part of numerically determined eigenvalues for $Q=200$ shown in log-log format as a function of the control parameter $q$. The power-law relations of eq. \ref{['eq:lambda']} are indicated by dashed black lines. For visual clarity, we display every other eigenvalue differently, highlighting the coalescence of pairs of eigenvalues in a cascade of exceptional points. Inset: Number of EPs seen up to $q$. The number grows as $q^{\frac{1}{2}}$ which we indicate via a dashed black line. The diagonalization of the system eq. \ref{['eq:eigenvaluematrix']} was done using the Eigen library eigenweb.
• Figure 3: Numerical data for the system of coupled exceptional points in eq. \ref{['eq:simple']} obtained via direct diagonalization using the computer algebra system MathematicaMathematica. Here, we use $c=0.1$. Top: Eigenvalues, the real part is shown in blue and the imaginary part in orange. The spectrum clearly has two exceptional points (slightly shifted due to the perturbative coupling). Bottom: Numerical evidence for $\varepsilon^{-\frac{1}{8}}$ scaling (as indicated by the dashed black line) in the projection of the eigenvectors to the $(0,0,0,1)$ mode. The different vectors are represented by differing colors with two curves coinciding (red line, corresponding to the upper two eigenvalues).