Hydrodynamics of simple active liquids: theemergence of velocity correlations

We derive the hydrodynamics for a system ofNactive, spherical, underdamped particles,interacting through conservative forces. At the microscopic level, we represent the evolution of theparticles in terms of the Kramers equation for the probability density distribution of theirpositions, velocities, and orientations, while at a mesoscopic level we switch to a coarse-graineddescription introducing an appropriate set of hydrodynamic fields given by the lower-ordermoments of the distribution. In addition to the usual density and polarization fields, thehydrodynamics developed in this paper takes into account the velocity and kinetic temperaturefields, which are crucial to understanding new aspects of the behavior of active liquids. Byimposing a suitable closure of the hydrodynamic moment equations and truncation of theBorn–Bogolubov–Green–Kirkwood–Yvon hierarchy, we obtain a closed set of mesoscopic balanceequations. At this stage, we focus our interest onthe small deviations of the hydrodynamic fieldsfrom their averages and apply the methods of the theory of linear hydrodynamic fluctuations. Ourtreatment sheds light on the peculiar properties of isotropic active liquids and their emergentdynamical collective phenomena, such as the spontaneous alignment of the particle velocities. Wepredict the existence within the liquid phase of spatial equal-time Ornstein–Zernike-like velocitycorrelations both for the longitudinal and the transverse modes. At variance with active solids, inactive liquids, the correlation length of the transverse velocity fluctuations is sensibly shorter thanthe length of the longitudinal fluctuations. In particular, the latter depends on the sound speedand increases with the persistence time, while the former displays a weaker dependence on theseparameters. Finally, within the same framework, we derive the dynamical structure factors and theintermediate scattering functions and discuss how the velocity ordering persists in time. We findthat the velocity decorrelates on a time-scale much longer than the one characteristic of passivefluids