Effects of saturation in the transient process of a dye laser. III. The case of colored noise with large and small correlation time

Statistical mechanics of active Ornstein-Uhlenbeck particles

We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.

Velocity distribution in active particles systems

We derive an analytic expression for the distribution of velocities of multiple interacting active particles which we test by numerical simulations. In clear contrast with equilibrium we find that the velocities are coupled to positions. Our model shows that, even for two particles only, the individual velocities display a variance depending on the interparticle separation and the emergence of correlations between the velocities of the particles. When considering systems composed of many particles we find an analytic expression connecting the overall velocity variance to density, at the mean-field level, and to the pair distribution function valid in the limit of small noise correlation times. Finally we discuss the intriguing analogies and main differences between our effective free energy functional and the theoretical scenario proposed so far for phase-separating active particles.

Towards a statistical mechanical theory of active fluids

We present a stochastic description of a model of N mutually interacting active particles in the presence of external fields and characterize its steady state behavior in the absence of currents. To reproduce the effects of the experimentally observed persistence of the trajectories of the active particles we consider a Gaussian force having a non-vanishing correlation time τ, whose finiteness is a measure of the activity of the system. With these ingredients we show that it is possible to develop a statistical mechanical approach similar to the one employed in the study of equilibrium liquids and to obtain the explicit form of the many-particle distribution function by means of the multidimensional unified colored noise approximation. Such a distribution plays a role analogous to the Gibbs distribution in equilibrium statistical mechanics and provides complete information about the microscopic state of the system. From here we develop a method to determine the one- and two-particle distribution functions in the spirit of the Born-Green-Yvon (BGY) equations of equilibrium statistical mechanics. The resulting equations which contain extra-correlations induced by the activity allow us to determine the stationary density profiles in the presence of external fields, the pair correlations and the pressure of active fluids. In the low density regime we obtained the effective pair potential ϕ(r) acting between two isolated particles separated by a distance, r, showing the existence of an effective attraction between them induced by activity. Based on these results, in the second half of the paper we propose a mean field theory as an approach simpler than the BGY hierarchy and use it to derive a van der Waals expression of the equation of state.

Multidimensional stationary probability distribution for interacting active particles

We derive the stationary probability distribution for a non-equilibrium system composed by an arbitrary number of degrees of freedom that are subject to Gaussian colored noise and a conservative potential. This is based on a multidimensional version of the Unified Colored Noise Approximation. By comparing theory with numerical simulations we demonstrate that the theoretical probability density quantitatively describes the accumulation of active particles around repulsive obstacles. In particular, for two particles with repulsive interactions, the probability of close contact decreases when one of the two particle is pinned. Moreover, in the case of isotropic confining potentials, the radial density profile shows a non trivial scaling with radius. Finally we show that the theory well approximates the “pressure” generated by the active particles allowing to derive an equation of state for a system of non-interacting colored noise-driven particles.