Kinetic Turing Instability and Emergent Spectral Scaling in Chiral Active Turbulence

TL;DR

The paper addresses how chaotic active chiral matter self-organizes into coherent turbulence and identifies a kinetic Turing instability as the mechanism that selects a finite structural wavelength. It combines a 2D polar chiral agent model with localized Kuramoto coupling and a continuum Vlasov-Fokker-Planck description to derive a dispersion relation yielding $A_{\mathrm{crit}}(k)$ and $k_{\mathrm{crit}}$, with a minimum at $k_{\mathrm{crit}} \approx 0.86$ and $A_{\mathrm{crit}} \approx 0.2$. Simulations reveal robust power-law spectra in the spatial distribution, with spectral indices in the range $\alpha \in [-3, -1.5]$, and emergent quantized loop currents with winding number $-1$. The results bridge discrete chimera-like states and continuous active turbulence, suggesting universal scaling laws across active matter and kinetic plasmas and providing a mechanistic, semi-analytical foundation for length-scale selection in active turbulence.

Abstract

The spontaneous emergence of coherent structures from chaotic backgrounds is a hallmark of active biological swarms. We investigate this self-organization by simulating an ensemble of polar chiral active agents that couple locally via a Kuramoto interaction. We demonstrate that the system's transition from chaos to active turbulence is characterized by quantized loop phase currents and coherent clustering, and that this transition is strictly governed by a kinetic Turing instability. By deriving the continuum kinetic theory for the model, we identify that the competition between local phase-locking and active agent motility selects a critical structural wavenumber. The instability then drives the system into a state of developed, active turbulence that exhibits stable, robust power-laws in spectral density, suggestive of universality and consistent with observations from a broad range of turbulent phenomena. Our results bridge the gap between discrete chimera states and continuous fluid turbulence, suggesting that the statistical scaling laws of active turbulence can arise from fundamental kinetic instability criteria.

Figures (6)

• Figure 1: Slow (dashed line) and fast (solid line) distributions in intrinsic frustration $\omega_i$, exhibiting broken power-law distributions.
• Figure 2: Panel a): Linear stability analysis for the Kuramoto model implemented in the Main Paper, using $\omega$ distribution labeled "fast mode" in Figure 1 of the Main Paper. Panel b): Linear stability analysis for all "distribution multiplier numbers", that is sliding the oscillation modes from "slow" to "fast" in Figure 1 in the Main Paper. The configurations used for Figures \ref{['fig:ex']} and \ref{['fig:stats']} are indicated with black arrows.
• Figure 3: Summary of a simulation with "fast" intrinsic frequencies $\omega_i$ and coupling strength $A_0=0.2$. The localized coupling of oscillators is switched on at 2.5 min and kept on. The leftmost column (panels a, d, g, and j) show PSD (cyan for guiding centers and red for active agents), while the middle column (panels b, e, h, and k) show the locations of the oscillators's guiding centers as cyan-colored dots. The rightmost column (panels c, f, i, and l) show the 30,000 oscillators color-coded according to their phase. See Video S1 in the Supplementary Materials.
• Figure 4: A magnified part of the simulation space, showing the instantaneous velocity of chiral agents that momentarily form a vortex. Panel a) shows the motion of guiding centers while panel b) shows the motion of the oscillating agents. The motion arrows in panel a) are scaled up ($\times6.67$) as a visual aid; the motion arrows in panel b) have a fixed magnitude $v_0$, which can be partially averaged out when taking the velocity. See Video S2 in the Supplementary Materials.
• Figure 5: The results of a Monte Carlo estimation of the average spectral indices seen in the active chiral matter model. In each simulation, the coupling is initiated at 5% of the total number of simulation steps and "turned off" at 67%. Spectral index is shown with a colorscale (and black contour lines), achieved by systematically varying the base coupling strength $A_0 \in [10^{-2},\pi]$ in logarithmic steps ($y$-axis), using a factor 100 for the natural phase multiplier. Green contour lines indicate the median global order, while a black dash-dotted line indicates the kinetic Turing instability threshold $A_{crit}$. Panel a) shows spectral scaling in agent structuring, whereas panel b) shows the spectral index for guiding center clustering.