Gyrokinetic Vlasov-Poisson model derived by hybrid-coordinate transform of the distribution function

This paper points out that the full-orbit density obtained in the standard electrostatic gyrokinetic model is not truly accurate at the order $\varepsilon^{\sigma-1}$ with respect to the equilibrium distribution $e^{-\alpha \mu}$ with $\mu \in (0, \mu_{\max})$, where $\varepsilon$ is the order of the normalized Larmor radius, $\varepsilon^{\sigma}$ the order of the amplitude of the normalized electrostatic potential, and $\alpha$ a factor of $O(1)$. This error makes the exact order of the full-orbit density not consistent with that of the approximation of the full-orbit distribution function. By implementing a hybrid coordinate frame to get the full-orbit distribution, specifically, by replacing the magnetic moment on the full-orbit coordinate frame with the one on the gyrocenter coordinate frame to derive the full-orbit distribution transformed from the gyrocenter distribution, it's proved that the full-orbit density can be approximated with the exact order being $\varepsilon^{\sigma-1}$. The numerical comparison between the new gyrokinetic model and the standard one was carried out using Selalib code for an initial distribution proportional to $\exp(\frac{-\mu B}{T_i})$ in constant cylindrical magnetic field configuration with the existence of electrostatic perturbations. In such a configuration, the simulation results exhibit similar performance between the two models.