Hydrodynamics of simple active liquids: the emergence of velocity correlations

TL;DR

This paper develops a hydrodynamic theory for active liquids, revealing how velocity correlations and collective behaviors emerge from microscopic particle interactions and fluctuations.

Contribution

It introduces a comprehensive hydrodynamic framework including velocity and temperature fields, and predicts velocity correlation phenomena in active liquids.

Findings

Velocity correlations exhibit Ornstein-Zernike-like behavior.

Transverse velocity correlation length is shorter than longitudinal.

Longitudinal correlations depend on sound speed and persistence time.

Abstract

We derive the Hydrodynamics for a system of N active, spherical, underdamped particles, interacting through conservative forces. At the microscopic level, we represent the evolution of the particles in terms of the Kramers equation for the probability density distribution of their positions, velocities, and orientations, while at a mesoscopic level we switch to a coarse-grained description introducing an appropriate set of hydrodynamic fields given by the lower-order moments of the distribution. In addition to the usual density and polarization fields, the Hydrodynamics developed in this paper takes into account the velocity and kinetic temperature fields, which are crucial to understanding new aspects of the behavior of active liquids. By imposing a suitable closure of the hydrodynamic moment equations and truncation of the Born-Bogolubov-Green-Kirkwood-Yvon hierarchy, we obtain a closed set of mesoscopic balance equations. At this stage, we focus our interest on the small deviations of the hydrodynamic fields from their averages and apply the methods of the theory of linear hydrodynamic fluctuations. Our treatment sheds light on the peculiar properties of isotropic active liquids and their emergent dynamical collective phenomena, such as the spontaneous alignment of the particle velocities. We predict the existence within the liquid phase of spatial equal-time Ornstein-Zernike-like velocity correlations both for the longitudinal and the transverse modes. At variance with active solids, in active liquids, the correlation length of the transverse velocity fluctuations is sensibly shorter than the length of the longitudinal fluctuations. In particular, the latter depends on the sound speed and increases with the persistence time, while the former displays a weaker dependence on these parameters. Finally, within the same framework, we derive the dynamical structure factors ...