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

TL;DR

This paper explains universal scaling in noisy active matter systems using perturbation theory and Mathieu equation analysis, revealing a cascade of exceptional points and their relation to dynamical phase transitions.

Contribution

It introduces a novel theoretical framework connecting active matter scaling laws to Mathieu equation properties and exceptional points, enhancing understanding of active matter dynamics.

Findings

Universal scaling relations explained by Mathieu equation analysis

Identification of a cascade of exceptional points affecting scaling

Prediction of symmetry-dependent scaling behaviors

Abstract

We show that recent numerical findings of universal scaling relations in systems of noisy, aligning self-propelled particles by K\"ursten [K\"ursten, 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. We also predict that these scaling relations depend on the symmetry of the alignment interactions and discuss the relevance of this structure in the free propagation for self-alignment and cohesion-type interactions.