# Spectral Insights into Active Matter: Exceptional Points and the Mathieu Equation

## TL;DR

This paper explains how certain behaviors in active matter systems can be understood using mathematical tools like the Mathieu equation and perturbation theory.

## Contribution

The paper provides a theoretical explanation for universal scaling relations in active matter using perturbation theory and the Mathieu equation.

## Key findings

- A cascade of exceptional points leads to non-trivial fractional scaling exponents in active matter systems.
- Scaling relations depend on the symmetry of alignment interactions in self-propelled particle systems.

## Abstract

We show that recent numerical findings of universal scaling relations in systems of noisy, aligning self-propelled particles by Rüdiger Kürstencan 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.