Kinetic Theory of Chiral Active Disks: Odd Transport and Torque Density

Abstract

Parity-odd transport is a central signature of chiral fluids, yet analytical predictions are sparse. Here, we introduce a minimal two-dimensional hard-disk gas in which chirality arises solely from a collision-induced transverse impulse. Motivated by granular spinners, collisions are dissipative and inject orbital angular momentum through a fixed tangential ``kick'' at contact. Starting from a Boltzmann-Enskog description, we derive nonlinear hydrodynamic equations for density, momentum, and temperature, and show that chirality generates an antisymmetric homogeneous stress corresponding to a nonzero torque density. In the dilute limit, a Chapman-Enskog expansion yields analytical predictions for transport coefficients, including odd viscosity, odd thermal conductivity, and odd self-diffusivity, in good agreement with numerical simulations. This minimal kinetic model can serve as a foundation for systematic coarse-graining of chiral fluids and as a tractable benchmark for gaining insight into odd transport across a broader class of chiral systems.

Figures (6)

• Figure 1: Illustration of a typical collision between two chiral active particles, labeled $1$ and $2$, following the collision rule \ref{['eq: collision rule']}. The post-collisional velocities $\bm v'_1$ and $\bm v'_2$ are characterized by (i) a reduced normal magnitude due to dissipation, as in conventional granular particles ($\alpha < 1$), and (ii) an additional velocity component transverse to the line connecting the particle centers, generated by chirality.
• Figure 2: Comparison between numerical measurements and theoretical predictions [Eq. \ref{['eq: theo tau']}] for the torque density $\tau$ as a function of the restitution coefficient $\alpha$. Results are obtained for $N = 10^4$ particles, with each data point averaged over at least $10^3$ independent snapshots. Here, we include only data showing a homogeneous configuration. Therefore, numerical measurements at low $\alpha$ are not included since the system exhibits an inhomogeneous phase reminiscent of the bubble phase numerically observed in Refs. caprini2025Bubbleshen2023collectivedigregorio2025phaseguo2025chirality.
• Figure 3: Odd transport coefficients as functions of the restitution coefficient $\alpha$—which also sets the strength of chirality---at three different packing fractions, $\phi = 0.01, 0.05, 0.1$. (a) Odd viscosity measured from an imposed shear flow, as detailed in Appendix \ref{['app: viscosity measurement']}. The missing data points for $\phi = 0.1$ correspond to systems in which the smallest numerically accessible shear rate was still too large to yield reliable measurements. (b) Odd thermal conductivity measured using a Green–Kubo relation (see Appendix \ref{['app: conductivity measurement']}). (c) Odd self-diffusivity measured using a Green–Kubo relation (see Appendix \ref{['sec: diffusivity measurement']}). Solid lines show theoretical predictions [Eqs. \ref{['eq: 1']}, \ref{['eq: 2']}, and \ref{['eq: 3']}], while dashed lines indicate the corresponding values for a nonchiral system.
• Figure 4: Comparison of the steady-state velocity probability distribution $f(v)$ between simulations ($N = 5000$) and theory. (a) Distribution of the normalized velocity in a dilute system ($\phi = 0.01$) for several values of the restitution coefficient $\alpha$. We use $v_{\mathrm{th}} = \sqrt{2T/m}$ so that the distributions have unit variance. (b) Excess kurtosis $\kappa_{\rm ex}$ as a function of $\alpha$ at different packing fractions. Solid and dashed lines show theoretical predictions: $\kappa_{\rm ex} = 0$ corresponds to a Gaussian distribution, while a Sonine expansion predicts $\kappa_{\rm ex} \neq 0$ in the dilute limit where molecular chaos holds. Data at low $\alpha$ are omitted for some values of $\phi$ when the system develops inhomogeneous configurations, reminiscent of those observed in Refs. caprini2025Bubbleshen2023collectivedigregorio2025phaseguo2025chirality.
• Figure 5: Homogeneous temperature $T$ [(a),(b)] and pressure $p$ [(c),(d)] as functions of the restitution coefficient $\alpha$ for different values of the density $\phi$. Panels (a) and (c) show the raw quantities, while panels (b) and (d) display the same data normalized by their Gaussian theoretical predictions (Eq. \ref{['eq: temperature iso']} and Eq. \ref{['eq: pressure iso']} with $T$ replaced by its Gaussian prediction). Results are obtained for $N = 10^4$ particles, with each data point averaged over at least $10^3$ independent snapshots. Only data corresponding to homogeneous configurations are shown; data exhibiting inhomogeneous states---typically occurring at large $\phi$ and low $\alpha$ and reminiscent of the nonuniform phase observed in Refs. caprini2025Bubbleshen2023collectivedigregorio2025phaseguo2025chirality---are omitted.