# Brownian Molecules Formed by Delayed Harmonic Interactions

## TL;DR

This paper investigates how delayed harmonic interactions among Brownian particles lead to complex, molecule-like structures, analyzing their stability, entropy, and transition dynamics through analytical and simulation methods.

## Contribution

It provides the first analytical solutions for small systems and develops a new transition-state theory for understanding stability and transitions in delayed stochastic systems.

## Key findings

- Particles form unstable molecule-like structures with increasing delay and interaction strength.
- Analytical solutions for N ≤ 3 particles are derived.
- A new approximate transition-state theory characterizes structural transitions.

## Abstract

A time-delayed response of individual living organisms to information exchanged within flocks or swarms leads to the emergence of complex collective behaviors. A recent experimental setup by (Khadka et al 2018 Nat. Commun. 9 3864), employing synthetic microswimmers, allows to emulate and study such behavior in a controlled way, in the lab. Motivated by these experiments, we study a system of N Brownian particles interacting via a retarded harmonic interaction. For $N \leq 3$ , we characterize its collective behavior analytically, by solving the pertinent stochastic delay-differential equations, and for $N>3$ by Brownian dynamics simulations. The particles form molecule-like non-equilibrium structures which become unstable with increasing number of particles, delay time, and interaction strength. We evaluate the entropy and information fluxes maintaining these structures and, to quantitatively characterize their stability, develop an approximate time-dependent transition-state theory to characterize transitions between different isomers of the molecules. For completeness, we include a comprehensive discussion of the analytical solution procedure for systems of linear stochastic delay differential equations in finite dimension, and new results for covariance and time-correlation matrices.