Emergent hydrodynamics of chiral active fluids: vortices, bubbles and odd diffusion

Starting from a microscopic multiparticle Langevin equation, we systematically derive a hydrodynamic description in terms of density and momentum fields for chiral active particles interacting via standard repulsive and nonlocal odd forces. These odd interactions are reciprocal but non-conservative: they are non-potential forces, as they act perpendicular to the vector joining any pair of particles. As a result, the torques that two particles exert on one another are non-reciprocal. The ensuing macroscopic continuum description consists of a continuity equation for the density and a generalized compressible Navier-Stokes equation for the fluid velocity. The latter includes a chirality-induced torque density term and an odd viscosity contribution. Our theory predicts the emergence of odd diffusivity, edge currents, and an inhomogeneous phase - characterized by bubble-like structures - recently observed in simulations. Specifically, the theory exhibits a linear instability arising from the interplay between odd viscosity and torque density, and admits steady-state inhomogeneous solutions featuring bubbles and vortices, in agreement with numerical simulations. Our findings can be tested experimentally in systems of granular spinners or rotating microorganisms suspended in a fluid.

💡 Research Summary

The authors present a comprehensive theoretical framework for chiral active fluids that incorporates non‑conservative, transverse (“odd”) interactions between particles. Starting from an under‑damped Langevin description of N interacting particles, they separate the forces into a conventional central repulsion derived from a short‑range potential and an odd force that acts perpendicular to the inter‑particle vector. Although the odd force respects Newton’s third law (action–reaction), it generates non‑reciprocal torques, breaking parity and time‑reversal symmetry at the microscopic level. By constructing the corresponding Kramers–Fokker–Planck equation and closing the BBGKY hierarchy at the single‑particle level, they derive a Boltzmann‑type kinetic equation for the one‑particle distribution. Projecting this kinetic equation onto the slow fields – the number density n(r,t) and the velocity field u(r,t) – yields a continuity equation and a modified Navier–Stokes equation.

The momentum equation contains the usual pressure gradient, a Korteweg term (∝∇∇²n) arising from inter‑particle forces, standard shear (η) and bulk (ζ) viscosities, and two distinct chiral contributions. The first is an odd‑viscosity term ηₒ ẑ × ∇²u, which is reactive rather than dissipative and produces a stress perpendicular to the flow direction. The second is a torque‑density term χₒ ẑ × ∇n, proportional to the density gradient and antisymmetric in its Cartesian components. This torque density acts tangentially to density interfaces and is the first explicit macroscopic manifestation of the non‑reciprocal torques generated by odd interactions.

Linear stability analysis of the homogeneous state (constant density, zero flow) shows that the coupling between odd viscosity and torque density can render the system unstable when ηₒ χₒ exceeds a threshold set by friction γ and wave‑number k. The resulting growth rate predicts a band of unstable modes, in quantitative agreement with earlier molecular‑dynamics simulations that observed the emergence of bubble‑like low‑density regions surrounded by vortical flows.

Beyond the linear regime, the authors explore steady‑state solutions in the inviscid limit (η = ζ = 0). They find a family of non‑uniform solutions featuring circular cavities (the “BIO” phase) where the density is depleted and a circulating velocity field encircles the cavity. These solutions reproduce the characteristic bubble‑vortex structures reported in simulations of granular spinners and rotating microorganisms.

The theory also predicts odd diffusion (a transverse contribution to the diffusion tensor) and persistent edge currents along confining walls, both of which are hallmarks of odd hydrodynamics. The authors discuss experimental platforms where these predictions could be tested, including granular spinners driven by airflow or vibration, rotating algae or starfish embryos in fluid, and electronic fluids where Hall viscosity has been measured.

Overall, the paper bridges the gap between microscopic odd forces and macroscopic fluid behavior, providing a unified description that captures odd diffusion, edge currents, linear instability, and the formation of bubble‑vortex patterns in chiral active matter.