Hydrodynamic correlation functions of chiral active fluids

The success of spectroscopy to characterize equilibrium fluids, for example the heat-capacity ratio, suggests a parallel approach for active fluids. Here, we start from a hydrodynamic description of chiral active fluids composed of spinning constituents and derive their low-frequency, long-wavelength response functions using the Kadanoff-Martin formalism. We find that the presence of odd (equivalently, Hall) viscosity leads to mixed density-vorticity response even at linear order. Such response, prohibited in time-reversal-invariant fluids, is a large-scale manifestation of the microscopic breaking of time-reversal symmetry. Our work suggests possible experimental probes that can measure anomalous transport coefficients in active fluids through dynamic light scattering.

Figure 1

(a) A schematic representation of a chiral active fluid composed of granular rotors. The red arrows indicate the intrinsic rotation field for each of the fluid's constituents around their own center of mass, and the black arrow represents the linear velocity of the center of mass for each particle. The frictional coupling between rotors is represented by the gear-like shape. (b) Schematic illustration of the mechanism that couples intrinsic rotation and fluid vorticity. When two particles collide, the frictional coupling is responsible for the generation of an angular momentum in the fluid (in terms of the center-of-mass velocities) due to intrinsic rotation.

Figure 2

Contour plots of correlation functions $S_{a,b}$ versus frequency of response $z$ and wave number of response $q$ (up to overall prefactors). (a) The density-density correlation function $S_{\rho ,\rho }$ from Eqs. (57), (62) for ${\nu }^o=0$ and (b) ${\nu }^o=\nu /10$. Note that the color bar uses logarithmic scale. (c) The vorticity-vorticity correlation function $S_{\omega ,\omega }$ from Eqs. (58), (63) for ${\nu }^o=0$ and (d) ${\nu }^o=\nu /10$ (color bar is logarithmic). For these small values of odd viscosity ${\nu }^o$, its effect is not evident from the above density-density and vorticity-vorticity correlations. However, the effect of odd viscosity becomes apparent in the plots of off-diagonal density-vorticity correlations $S_{\rho ,\omega }$ from Eqs. (59), (64). (e) For ${\nu }^o=0$, the color bar is in linear scale, $S_{\rho ,\omega }=0$ everywhere while for (f) ${\nu }^o=\nu /10$ there is a nonzero $S_{\rho ,\omega }$ correlation. In this figure, the parameters ${\rho }_0, \nu , {\nu }^{\prime }$, and $c^2$ are all set to unity.