# Effective equilibrium states in the colored-noise model for active matter I. Pairwise forces in the Fox and unified colored noise approximations

## TL;DR

This paper compares the Unified Colored Noise Approximation and Fox approximation for modeling active matter with colored noise, analyzing their effectiveness in predicting pairwise interactions and steady states, and proposing empirical corrections for limitations.

## Contribution

It provides a comparative analysis of UCNA and Fox approximations for active matter, introduces empirical modifications, and explores force-balance conditions at low activity.

## Key findings

- Fox approximation more accurate with white noise
- Limitations for potentials with negative slope or curvature
- Empirical correction improves theoretical predictions

## Abstract

The equations of motion of active systems can be modeled in terms of Ornstein-Uhlenbeck processes (OUPs) with appropriate correlators. For further theoretical studies, these should be approximated to yield a Markovian picture for the dynamics and a simplified steady-state condition. We perform a comparative study of the Unified Colored Noise Approximation (UCNA) and the approximation scheme by Fox recently employed within this context. We review the approximations necessary to define effective interaction potentials in the low-density limit and study the conditions for which these represent the behavior observed in two-body simulations for the OUPs model and Active Brownian particles. The demonstrated limitations of the theory for potentials with a negative slope or curvature can be qualitatively corrected by a new empirical modification. In general, we find that in the presence of translational white noise the Fox approach is more accurate. Finally, we examine an alternative way to define a force-balance condition in the limit of small activity.