[Paper Review] Hydrodynamics of simple active liquids: the emergence of velocity correlations

Umberto Marini Bettolo Marconi, Lorenzo Caprini+1 more|arXiv (Cornell University)|Jan 1, 2021

Micro and Nano RoboticsPhysics and Astronomy113 references31 citations

TL;DR

This paper develops a hydrodynamic theory for active liquids composed of underdamped, spherical particles with persistent active forces, deriving a closed set of mesoscopic balance equations for density, polarization, velocity, and kinetic temperature fields. It predicts the emergence of long-ranged, equal-time velocity correlations—both longitudinal and transverse—where the longitudinal correlation length increases with persistence time and sound speed, while transverse correlations decay more rapidly, revealing a key distinction from active solids and highlighting slow relaxation of velocity order in time.

ABSTRACT

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

Motivation & Objective

• To develop a systematic coarse-grained hydrodynamic theory for active liquids composed of underdamped, interacting particles with persistent active forces.
• To identify and describe the emergence of collective velocity correlations in the absence of explicit alignment interactions.
• To understand the dynamical behavior of velocity fluctuations and their relaxation timescales in active liquids compared to passive fluids.
• To derive and analyze the dynamical structure factors and intermediate scattering functions within the hydrodynamic framework.
• To clarify the role of persistence time and sound speed in determining the spatial and temporal characteristics of velocity correlations.

Proposed method

• Formulate a microscopic description using the Kramers-Fokker-Planck (KFP) equation for the N-particle probability distribution function, incorporating active forces via Active Brownian (ABP) or Active Ornstein-Uhlenbeck (AOUP) dynamics.
• Define mesoscopic hydrodynamic fields as coarse-grained moments of the distribution: density, polarization, velocity, and kinetic temperature.
• Derive a closed hierarchy of hydrodynamic balance equations by truncating the Born-Bogolubov-Green-Kirkwood (BBGK) hierarchy and applying a suitable closure approximation.
• Apply linear hydrodynamic fluctuation theory to study small deviations from equilibrium averages, enabling the calculation of correlation functions.
• Use the resulting framework to compute the dynamical structure factors and intermediate scattering functions, analyzing temporal evolution of velocity order.
• Relate the spatial decay of velocity correlations to system parameters such as persistence time and sound speed through analytical expressions derived from the hydrodynamic equations.

Experimental results

Research questions

• RQ1 What hydrodynamic fields are necessary to describe the collective behavior of active liquids beyond standard passive hydrodynamics?
• RQ2 How do velocity correlations emerge in active liquids in the absence of explicit alignment forces or velocity-velocity interactions?
• RQ3 What is the spatial dependence of equal-time velocity correlations, and how do longitudinal and transverse components differ in their correlation lengths?
• RQ4 How does the persistence time of active particles affect the spatial extent and temporal persistence of velocity order?
• RQ5 What is the characteristic relaxation time of velocity fluctuations in active liquids, and how does it compare to passive systems?

Key findings

• The hydrodynamic theory predicts the emergence of equal-time, spatially correlated velocity fluctuations in active liquids even without explicit alignment interactions, driven by the interplay of active forces and steric repulsion.
• Longitudinal velocity correlations exhibit a correlation length that increases with persistence time and scales with the sound speed, indicating long-range order in the flow direction.
• Transverse velocity correlations have a significantly shorter correlation length than longitudinal ones, reflecting weaker collective behavior perpendicular to flow.
• The velocity field decorrelates on a timescale much longer than in passive fluids, indicating a slow relaxation of collective motion in active liquids.
• The dynamical structure factor reveals a non-exponential decay of velocity fluctuations, consistent with persistent collective motion and long-lived correlations.
• The intermediate scattering function shows a stretched-exponential decay, confirming the slow relaxation of velocity order and distinguishing active liquids from passive fluids.