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

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. CONTENTS

Effective equilibrium states in mixtures of active particles driven by colored noise

“…, which in the onecomponent system is equivalent to the first definition [44], does not yield the same result as βF eff α in the general case of a mixture with nonvanishing probability currents. We thus derive by equating the currents in Eq.…”

“…The nature of this term can be understood by the following example. In the present model we can define the passive (Brownian) particle (label 2) in two ways [44]: we always require that E (2) 2 = 1, i.e., τ (2) = 0 (likewise, in the one-component UCNA, the bare potential of a passive particle can only be recovered when the persistence time is set to zero [44]). Obviously, the persistence time τ (1) of the active species has to remain finite.…”

“…Whereas, the effective potential for a single species is highly accurate [44,45] (1) and D (1) a (as labeled) with τ (2) = 0.1 − τ (1) and D (2) a = 9.6 − D (1) a , such thatτ = 0.05 andD a = 4.8 (the results are invariant when exchanging the particles, i.e., the labels 1 and 2). Here we consider only particles with at least one pair of equal parameters (τ (2) = τ (1) or D (2) a = D ( most likely come along with the following caveats (as detailed in Appendix B): (i) the exact effective potentials can be written in a form similar to Eq.…”

“…Here we consider only particles with at least one pair of equal parameters (τ (2) = τ (1) or D (2) a = D ( most likely come along with the following caveats (as detailed in Appendix B): (i) the exact effective potentials can be written in a form similar to Eq. (36), i.e., the argument of the logarithm follows from the determinant of the effective diffusion tensor [44], only if there is no thermal noise and we assume D (1) a = D (2) a or τ (1) = τ (2) ; (ii) we have no rigorous proof that the underlying assumption J 1 = J 2 of equal probability currents holds for d > 1; (iii) already for two particles of the same species, empirical corrections of the effective potential are required and the quantitative agreement with computer simulation becomes worse with increasing dimensionality. However, we stress that most relevant cases (to be discussed later) are consistent with the assumptions under point (i).…”

“…This procedure yields an approximate Smoluchowski equation, which, in the steady state, admits an analytic solution for the configurational probability distribution [45] and closed formulas for active pressure and interfacial tension [46][47][48]. The former allows us to define effective interaction potentials, which can be directly used to determine density profiles [44,49,50] and rate equations [51] of individual ideal particles and, when implemented in equilibrium liquid-state theory, the structure and phase behavior of interacting systems [11,52,53]. The described effective equilibrium approach is most accurate in one spatial dimension and for small persistence time, which can be explicitly verified by studying exactly solvable models [48].…”