Universal winding properties of chiral active motion

We propose the area swept $A(t)$ and the winding angle $\Omega(t)$ as the key observables to characterize chiral active motion. We find that the distributions of the scaled area and the scaled winding angle are described by universal scaling functions across all well-known models of active particles, parametrized by the chirality $\omega$, along with a self-propulsion speed $v_0$, and the persistence time $\tau$. In particular, we show that, at late times, the average winding angle grows logarithmically with time $\la\Omega \ra\sim(\omega\tau/2)\,\ln t$, while the average area swept has a linear temporal growth $\la A(t)\ra\simeq(\omega\tau D_{\text{eff}})\,t$, where $D_{\text{eff}}=v_0^2 \tau /[2(1+ \omega^2 \tau^2)]$ is the effective diffusion coefficient. Moreover, we find that the distribution of the scaled area $z=[A-\la A\ra]/(2D_{\text{eff}}t)$ is described by the universal scaling function $F_{\text{ch}}(z)=\text{sech}(\pi z)$. From extensive numerical evidence, we conjecture the emergence of a new universal scaling function $G_{\text{ch}}(z)=\mathcal {N}/[e^{\alpha z} + e^{-\beta z}]$ for the distribution of the scaled winding angle $z=\Omega/[\ln t]$, where the parameters $\alpha$ and $\beta$ are model-dependent and $\mathcal{N}$ is the normalization constant. In the absence of chirality, i.e., $\omega=0$, the scaling function becomes $G_{\text{ch}}(z)=(\alpha/\pi)\,\mathrm{sech}(\alpha z)$.