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

TL;DR

This work derives a macroscopic hydrodynamic theory for a two-dimensional chiral active fluid from a microscopic Langevin model with central and odd transverse interactions. The resulting compressible Navier–Stokes-like equations include an odd viscosity term and a chirality-induced torque density, which together generate odd diffusion and destabilize the homogeneous phase, giving rise to a bubble-rich inhomogeneous BIO phase with edge currents and vortical structures. A linear stability analysis identifies the parameter regime for the BIO transition, while a nonlinear inviscid solution provides a self-consistent single-bubble configuration and a route to understanding cavity–vortex patterns observed in simulations. The framework connects microscopic odd-interaction physics to macroscopic pattern formation and offers experimentally testable predictions in granular spinners and rotating microorganisms suspended in fluids.

Abstract

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.

Figures (3)

• Figure 1: Hydrodynamics of a chiral active fluid. The panels of the figure illustrate the main logical steps of this work and highlight the key results. Starting from a particle-based model of a chiral active fluid subject to odd interactions, we derive the Fokker–Planck equation for the $N$-particle distribution and close the BBGKY hierarchy at the single-particle level, obtaining a Boltzmann-like equation for the single-body distribution. Projecting onto the subspace spanned by the density $n = n(\mathbf{r}, t)$ and the velocity field $\mathbf{u} = \mathbf{u}(\mathbf{r}, t)$, we obtain a hydrodynamic description. The resulting chiral active fluid is governed by a compressible odd Navier–Stokes equation, featuring odd viscosity and a chirality-induced torque density. Our theory predicts that (i) odd interactions generate transverse diffusion; (ii) the interplay between odd viscosity and torque density destabilizes the homogeneous state; and (iii) this instability gives rise to an inhomogeneous phase characterized by bubbles, termed BIO caprini2025bubble.
• Figure 2: Linear instability induced by the interplay between odd viscosity and torque density strength. (a)-(b) Real parts of the eigenvalues $\text{Re}[\lambda_T]$ (blue), $\text{Re}[\lambda_L]$ (red), and $\text{Re}[\lambda_D]$ (green) as functions of the wavevector magnitude $q$, normalized by the particle diameter $\sigma$. Panels (a) and (b) correspond to parameter regimes in which the homogeneous state is linearly unstable and stable, respectively. Specifically, the effective chirality is set to $n_b \chi_o \nu_o/(m \nu c_s^2) = 40$ in panel (a) and $0.4$ in panel (b), while the reduced inertia is fixed to $\nu/(\sigma^2 \gamma) = 0.13$ in both cases. The horizontal dashed line serves as a guide to the eye indicating the zero value, while the vertical line in panel (a) marks the critical wavevector $q_c$ at which the instability sets in, i.e., where $\text{Re}[\lambda_T]$ becomes positive. (c) Phase diagram in the plane defined by the reduced inertia, $\nu/(\sigma^2 \gamma)$, and the effective chirality, $n_b \chi_o \nu_o /(m \nu c_s^2)$. The reduced inertia is given by the ratio of the inertial time $1/\gamma$ to the viscous time $\sigma^2/\nu$. The effective chirality corresponds to the product of the torque-density strength $\chi_o$ and the odd viscosity $\nu_o$, appropriately normalized. Colors in the phase diagram indicate homogeneous phases with vanishing velocity: stable (grey) and unstable (yellow). The two colored stars mark the parameter values used in panels (a) and (b). Dashed black lines in panel (c) serve as guides to the eye: the horizontal line corresponds to $1 = n_b \chi_o \nu_o/(m \nu c_s^2)$, which is the necessary condition for the linear instability, while the diagonal line denotes the condition $q_c \sigma = 1$ (Eq. \ref{['eq:stability_condition_2']}), ensuring that the instability occurs on a length scale larger than the particle diameter $\sigma$. (d) Phase diagram in the plane of reduced inertia ($\propto 1/\gamma$) and chirality, i.e. the strength, of odd interactions ($\propto \chi_o$), obtained from particle-based simulations of Eqs. \ref{['Langevin1']}. Here, the color code is the same as in panel (c): grey denotes the homogeneous phase, while yellow indicates the inhomogeneous phase, termed BIO (bubbles induced by odd interactions). The remaining dimensionless parameters used in panels (a)–(c) are $\nu_b/\nu = 0$, $\kappa n_b^2/(m c_s^2) = 0$, $\nu_o/\nu = 1$, and $n_b \nu^2/c_s^2 = 1$. Panel (d) is adapted with permission from Ref. caprini2025bubble; copyright (2025) AIP Publishing.
• Figure 3: Non-linear phase diagram (a) Predicted bubble radius $R_c$ (Eq. \ref{['eq:Rcn_expr']}) as a function of the bulk average number density $n_b$ for different values of the chirality, quantified by the torque-density strength $\chi_o$. The horizontal dashed line indicates the threshold used for bubble identification, set to $R_c = \sigma$, where $\sigma = 1$ is the particle diameter. (b) Phase diagram in the plane of number density $n_b$ and chirality, represented by the torque-density strength $\chi_o$. Colors denote the different phases: the homogeneous phase (grey) and the BIO phase caprini2025bubble (yellow), corresponding to bubbles induced by odd interactions. Red points indicate configurations for which $R_c = \sigma$, corresponding to the minimal radius used to identify a bubble. Consequently, the phase diagram is constructed using the straight-line construction illustrated in panel (a). The panels are obtained from Eq. \ref{['eq:Rcn_expr']} using the parameters $\sigma \gamma / c_s = 100$, $\ell/\sigma = 0.1$, $T /(m c_s^2) = 0.02$, and $\kappa /(m \sigma^4 c_s^2) = 10$, where $m = 1$ is the particle mass and $c_s = 1$ is the sound speed of the fluid. We note that all kinematic viscosities are set to zero, $\nu_b = \nu = \nu_o = 0$.