von Neumann entropy of phase space structures in gyrokinetic plasma turbulence

TL;DR

The paper introduces a data-driven von Neumann entropy (vNE) diagnostic that combines singular value decomposition (SVD) with information-theoretic weighting to quantify velocity-space complexity in gyrokinetic turbulence. Applying this framework to nonlinear flux-tube simulations, the authors map the wavenumber dependence of velocity-space structure and find a sharp increase in vNE around $|oldsymbol{k}| ho_{t i}\sim 1$, indicating a transition from low-rank to high-rank velocity-space dynamics as scales become finer. Hermite-Laguerre analyses point to parallel phase mixing (Landau resonance) as the primary mechanism driving this rise, with FLR phase mixing contributing but less sensitivity to $|oldsymbol{k}|$ in the studied regime. The vNE approach provides a compact, basis-agnostic measure of kinetic complexity and enables global, data-driven insight into phase-space mixing processes, offering a path toward linking phase-space structure to transport and turbulence modeling.

Abstract

We introduce a data-driven diagnostic that combines the singular value decomposition (SVD) with an information-theoretic entropy to quantify the phase-space complexity of perturbed distribution functions in gyrokinetic turbulence. Applying this framework to nonlinear flux-tube simulations that solve the time evolution of the ion distribution function represented by Fourier modes with the wavenumber for real space, we define the von Neumann entropy (vNE) to analyze the velocity-space structure. A global survey in the wavenumber space reveals a wavenumber-dependent variation of the vNE in velocity-space structure: the vNE remains low at low wavenumber but increases across $k_\perpρ_{t\mathrm{i}}\sim 1$. Hermite/Laguerre decompositions revealed that the finite Larmor radius (FLR) phase mixing in the perpendicular (magnetic-moment) direction is active. Simultaneously, the systematic increase in vNE for $k_\perpρ_{t\mathrm{i}}$ correlates with the broadening of the Hermite spectrum, suggesting enhanced parallel phase mixing (Landau resonance) as the primary mechanism for the observed wave number dependence. These results demonstrate that the SVD-based vNE provides a compact measure of kinetic complexity without assuming a predefined basis and enables a global mapping of its wavenumber dependence of phase-mixing processes in gyrokinetic turbulence.

Figures (10)

• Figure 1: The temporal evolution of the electrostatic potential amplitude for the total value (purple), at $k_y=0.3$ (green), and $k_y=1.35$ (cyan).
• Figure 2: The snapshot of the turbulent fields in real space. (a) the electrostatic potential $\phi(x,y)$ and (b) the density fluctuation $n(x,y)$ at the time $t=80\,v_\mathrm{ref}/L_\mathrm{ref}$ and $z=0$.
• Figure 3: The snapshot of the distribution function $\mathrm{Re}[\delta f_{\mathrm{i}\bm k}(z,v_\parallel,\mu,t)]$. (a) the low-wavenumber case $(k_x,k_y)=(0,0.3)$ and (b) the high-wavenumber case $(k_x,k_y)=(0,1.35)$ at the time $t=80\,v_\mathrm{ref}/L_\mathrm{ref}$ and $z=0$.
• Figure 4: The spectra of the singular value $s_i$ in the SVD of the gyrokinetic distribution function $\delta f_{\mathrm{i}\bm k}(z,v_\parallel,\mu,t)$ at $(k_x,k_y)=(0,0.3)$ (black) and $(k_x,k_y)=(0,1.35)$ (red) normalized so that $s_1=1$.
• Figure 5: The velocity-space structure of the basis functions $\psi_i(v_\parallel,v_\perp)$ in the SVD of the gyrokinetic distribution function.