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The equations of motion of active systems can be modeled in terms ofOrnstein-Uhlenbeck processes (OUPs) with appropriate correlators. For furthertheoretical studies, these should be approximated to yield a Markovian picturefor the dynamics and a simplified steady-state condition. We perform acomparative study of the Unified Colored Noise Approximation (UCNA) and theapproximation scheme by Fox recently employed within this context. We reviewthe approximations necessary to define effective interaction potentials in thelow-density limit and study the conditions for which these represent thebehavior observed in two-body simulations for the OUPs model and ActiveBrownian particles. The demonstrated limitations of the theory for potentialswith a negative slope or curvature can be qualitatively corrected by a newempirical modification. In general, we find that in the presence oftranslational white noise the Fox approach is more accurate. Finally, weexamine an alternative way to define a force-balance condition in the limit ofsmall activity.

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