Description
Understanding how kinetic turbulence organizes structure in phase space and affects the turbulent transport remains a central challenge for gyrokinetic theory and simulation. However, it has not been fully established due to the multidimensionality of phase space and the vast amount of data.
To overcome the complexity of the analysis for phase space, we present a data-driven diagnostic that combines singular value decomposition (SVD) with an information-theoretic measure—the von Neumann entropy (vNE) for turbulent fields [Yatomi and Nakata, Phys. Rev. Res., vol. 7, 023212 (2025)]—to quantify the effective number of velocity-space degrees of freedom required to represent perturbed distribution functions. This framework is applied to nonlinear flux-tube simulations of ion-temperature-gradient (ITG) turbulence using GKV [Watanabe and Sugama, Nucl. Fusion, vol. 46, 24-32 (2006)]. For each perpendicular wavevector, we assemble the time-evolving ion distribution function on velocity variables, compute its SVD over the quasi-steady state, and evaluate vNE from normalized modal weights. A global survey over wavenumber reveals a rapid increase of the vNE as the wavenumber increases: large-scale fluctuations are captured by a few SVD modes, whereas small-scale fluctuations require many modes.
To relate vNE to established velocity-space diagnostics, we project onto orthonormal Hermite and Laguerre bases, for parallel-velocity and magnetic-moment variables, respectively. The Hermite spectra broaden systematically with wavenumber, indicating enhanced parallel phase mixing as the principal driver of the observed increase of the vNE. In contrast, the Laguerre spectrum exhibits a prominent peak in the region of large Laguerre exponents, demonstrating the presence of finite Larmor radius phase mixing [Tatsuno et al., Phys. Rev. Lett., vol. 103, 015003 (2009)]. However, its dependence on wave number is relatively weak, suggesting that phase mixing plays only a limited role in determining the wavenumber trend of the vNE in this region.