# Hierarchy of second-order gyrokinetic Hamiltonian models for particle-in-cell codes

## TL;DR

This paper derives and compares second-order gyrokinetic Hamiltonian particle models for particle-in-cell simulations, including full FLR and long-wavelength approximations, enhancing the theoretical foundation for plasma simulation accuracy.

## Contribution

It provides explicit derivations of second-order gyrokinetic Hamiltonian models suitable for particle-in-cell codes, including a comparison of full FLR and long-wavelength models.

## Key findings

- Derived the full second-order FLR Hamiltonian model.
- Presented the long-wavelength approximation model.
- Showed the equivalence of models in the long-wavelength limit.

## Abstract

The reduced-particle model is the central element for the systematic derivation of the gyrokinetic Vlasov-Maxwell equations from first principles. Coupled to the fields inside the gyrokinetic field-particle Lagrangian, the reduced-particle model defines polarization and magnetization effects appearing in the gyrokinetic Maxwell equations. It is also used for the reconstruction of the gyrokinetic Vlasov equation from the particle characteristics. Various representations of reduced-particle models are available according to the choice of the gyrokinetic phase space coordinates. In this paper, the Hamiltonian representation of the reduced particle dynamics at an order suitable for the implementation in particle-in-cell simulations is explicitly derived from the general reduction procedure. The second-order (with respect to the fluctuating electromagnetic fields), full Finite Larmor Radius (FLR) Hamiltonian gyrokinetic particle model as well as the second-order model suitable specifically for the long-wavelength approximation (i.e., containing up to the second-order FLR corrections), are derived and compared to the model recently implemented in the particle-in-cell code ORB5. We show that the same long-wavelength approximate equations can also be derived by taking the proper limit of the full FLR model.