Nonlocal adiabatic theory. I. The action distribution function

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, I, remains nearly constant. We insist here on the fact that I is different from the dynamical action, I. In particular, we show that 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.