# Nonlocal adiabatic theory. I. The action distribution function

## TL;DR

This paper develops a nonlocal adiabatic theory for charged particle motion in a slowly varying electrostatic wave, emphasizing the difference between the adiabatic invariant and the action, with numerical and analytical insights.

## Contribution

It introduces a nonlocal adiabatic framework that accounts for the entire wave history, extending traditional local action concepts and including weak inhomogeneity effects.

## Key findings

- The adiabatic invariant depends on wave history, not just local parameters.
- Numerical results show how the action distribution function evolves over time.
- Analytical derivation of the action distribution function is provided.

## Abstract

In this paper, we address the motion of charged particles acted upon by a sinusoidal electrostatic wave, whose amplitude and phase velocity vary slowly enough in time for neo-adiabatic theory to apply. Moreover, we restrict to the situation when only few separatrix crossings have occurred, so that the adiabatic invariant, $\mathcal{I}$, remains nearly constant. We insist here on the fact that $\mathcal{I}$ is different from the dynamical action, $I$. In particular, we show that $\mathcal{I}$ depends on the whole time history of the wave variations, while the action is usually defined as a local function of the wave amplitude and phase velocity. Moreover, we provide several numerical results showing how the action distribution function, $f(I)$, varies with time, and we explain how to derive it analytically. The derivation is then generalized to the situation when the wave is weakly inhomogeneous.