Statistical Mechanics of Active Fluids


Active matter is made of active particles which are able to convert energy
from the environment into directed persistent motion. Run and Tumble and
Active Brownian particle (ABP) models have been initially proposed to interpret
the experimental observations conducted in a laboratory on bacterial suspensions.
These models are very efficient in numerical simulations but do not allow for
much analytical progress. More recently, the Gaussian colored noise (GCN) model,
introduced with the idea of capturing the peculiar aspect of of GCN and ABP models,
i.e. the persistence of the trajectories of the active particles, has gained
a lot of attention. Such a behavior is reproduced by a Gaussian forcing term,
identified with the active force and having a non vanishing correlation time τ,
whose finiteness is a measure of the activity of the system.
Based on these assumptions, we present a description of a model of N mutually
interacting active particles in the presence of external fields and characterize
its steady state behavior. 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 a complete information about the microscopic steady 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 determining 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 obtain the effective pair potential φ acting between two isolated
particles separated by a distance, r, showing the existence an effective attraction.
We give a simple derivation of the governing equation and analyze some of its
recent applications ranging from the study of the swim pressure, its relation to the
mobility, to the state induced by a moving object.