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

-A simple theoretical approach is used to investigate active colloids at the free interface and near repulsive substrates. We employ dynamical density functional theory to determine the steady-state density profiles in an effective equilibrium system [Farage et al., Phys. Rev. E, 91 (2015) 042310]. In addition to the known accumulation at surfaces, we predict wetting and drying transitions at a flat repulsive wall and capillary condensation and evaporation in a slit pore. These new phenomena are closely related to the motility-induced phase separation (MIPS) in the bulk.Introduction. -Understanding self-organisation and non-equilibrium phase behaviour in active systems is currently a subject of intense research activity [1][2][3]. Bulk systems of spherically symmetric, repulsive active Brownian particles (ABPs) have been demonstrated to undergo a motility-induced phase separation (MIPS) into coexisting high-and low-density non-equilibrium phases [3][4][5][6][7][8][9]. The phenomenon of MIPS is a consequence of the persistent trajectories of active particles; the particles run into each other and thus tend to cluster [9]. Systems for which the passive interaction has an attractive component exhibit an even richer collective behaviour. In this case, it has been found that increasing activity can lead first to a suppression of passive phase separation [10][11][12], followed by a re-entrant phase separation at high activity [11,12].

The escape rate of a Brownian particle over a potential barrier is accurately described by the Kramers theory. A quantitative theory explicitly taking the activity of Brownian particles into account has been lacking due to the inherently out-of-equilibrium nature of these particles. Using an effective equilibrium approach [Farage et al., Phys. Rev. E 91, 042310 (2015)] we study the escape rate of active particles over a potential barrier and compare our analytical results with data from direct numerical simulation of the colored noise Langevin equation. The effective equilibrium approach generates an effective potential that, when used as input to Kramers rate theory, provides results in excellent agreement with the simulation data. DOI: 10.1103/PhysRevE.95.012115 The escape of a Brownian particle over a potential barrier is a thermally activated process. Kramers theory accurately describes the escape process by taking into account the force acting on a particle due to the confining potential and solvent-induced Brownian motion. Kramers showed that in the limit of vanishing particle flux across the barrier, the escape rate decreases exponentially with increasing barrier height [1]. In contrast to Brownian particles, active particles undergo both Brownian motion and a self-propulsion, which requires a continual consumption of energy from the local environment [2][3][4][5]. Due to self-propulsion, active particles are expected to escape a potential barrier at a higher rate than their passive counterparts. However, a quantitative description of their escape rate, explicitly taking the activity into account has been lacking. Active particles, in general, have coupled orientational and positional degrees of freedom [6,7]. This makes the theoretical treatment of escaping active particles over a potential barrier a difficult problem as shown in Ref. [8], in which the authors explicitly considered the orientational diffusion of self-propelled particles.In this paper we show that a Kramers-like rate expression can be obtained for a closely related model system of active particles in which the velocities are represented by a stochastic variable and the orientations are not considered explicitly. For small activity the steady-state properties obtained from this model exhibit intriguing similarities with an equilibrium system [9] and several sedimentation and trapping problems are analytically tractable on the single-particle level [10]. As a starting point for a theoretical treatment of the nonstationary case we will employ this model in the form of a coarse-grained Langevin equation for the particle position [6,7,11] with activity of particles appearing as a colored-noise term. It is important to note that the colored-noise Langevin equation describes a non-Markovian process and thus cannot yield an exact Fokker-Planck equation.The colored-noise Langevin equation serves as the basis for effective equilibrium approaches that map an active system to a passive equilibrium system with modified interaction potential and an...

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Active particles driven by colored noise can be approximately mapped onto a system that obeys detailed balance. The effective interactions which can be derived for such a system allow to describe the structure and phase behavior of the active fluid by means of an effective free energy. In this paper we explain why the related thermodynamic results for pressure and interfacial tension do not represent the results one would measure mechanically. We derive a dynamical density functional theory, which in the steady state simultaneously validates the use of effective interactions and provides access to mechanical quantities. Our calculations suggest that in the colored-noise model the mechanical pressure in coexisting phases might be unequal and the interfacial tension can become negative. CONTENTS Acknowledgements 12A. Effective free energy for the full Fox approach 12 B. Effective dynamical density functional theory 13 C. Effective route to the active pressure and interfacial tension 14 D. Alternative rescaling of effective mechanical quantities 15 E. Mean-field free energy in the mechanical picture 16References 17