Extended Data Fig. 5: Power spectra of the velocity–velocity correlation functions \(\langle {u}_{a}({{{{{\bf{k}}}}}},\omega ){u}_{b}^{* }({{{{{\bf{k}}}}}},\omega )\rangle\).
Here we compare the measured velocity–velocity correlation functions with the empirical prediction using fluctuating hydrodynamic theory. a. Correlation functions directly measured in the particle-based simulations of our chiral active fluid. b. Correlation functions predicted using the fluctuating hydrodynamic theory with the measured stress–stress correlation functions and viscosity tensor. The empirical prediction matches with the simulation results expect at very high k-modes, where the linear response approximation is no longer valid. Although the comparison is made at a given wave frequency ω0 = 0.055π/Δt, the consistency between direct measurements in simulations and predictions using fluctuating hydrodynamic theory generally holds at all wave frequencies.
