Dynamics of Aligning Active Matter: Mapping to a Schr\"odinger Equation and Exact Diagonalization

TL;DR

This paper maps the dynamics of aligning active matter to a Schr"odinger equation, uses exact diagonalization for analytical insights, and explores non-reciprocal interactions affecting steady states and entropy production.

Contribution

It introduces a rigorous exact diagonalization approach to analyze the spectrum of aligning active matter and extends the methodology to non-reciprocal, non-Hermitian interactions.

Findings

Exact analytical results for relaxational modes.

Extension to non-reciprocal interactions with non-Hermitian Schr"odinger problems.

Quantification of entropy production in non-reciprocal steady states.

Abstract

There has been recent interest in the relaxational modes of small-scale fully connected systems of aligning self-propelled particles (Spera et al., Phys. Rev. Lett. {\bf 132}: 078301 (2024)). We revisit the classical connection between Fokker-Planck and Schr\"odinger equations to address this by means of exact diagonalization, allowing for rigorous analytical insight into the full spectrum. This allows us to extract exact results which we compare to the existing result from linearized statistical field theory. We derive asymptotically correct analytical results that improve upon the prior approximations. We show that this methodology can fruitfully be extended to the case of non-reciprocal interactions which gives rise to a non-Hermitian Schr\"odinger problem akin to those in open quantum mechanics. While the non-reciprocity can be chosen such as not to alter the stationary distribution, it fundamentally changes the nature of the steady state which we quantify via the entropy production. We discuss the case of low particle numbers as well as the emergence of mean-field dynamics at large numbers.