The ‘Inverse-Gaussian’ SW proton populations: Do they tell something about heating/acceleration by turbulence ?

The solar wind proton distributions measured by PAS (Proton Alfa Sensor -Solar Orbiter)  are often constituted by a core and a beam. If the core is generally gaussian, the beam is asymmetric and non-gaussian with a more populated high-energy side (‘heavy tail’) in the magnetic field direction. It appears that the ‘Inverse Gaussian Distribution’ (IG), a type of hyperbolic statistical distributions, provides good fits of these skewed distributions. Then, excellent models of the whole proton distribution are obtained by superposing a gaussian (or almost gaussian distribution) for the core and an IG for the beam.  This modelling (Gaussian + Inverse Gaussian) applies to different situations: relatively slow and fast winds, single and double-bump populations, low or high level of turbulence. An interpretation is given, inspired by the ‘Normal-Inverse Gaussian’ (NIG) process, common in finance applications. Our ‘toy’ model assumes that the acceleration/heating is modelled as a drifting gaussian process in velocity space controlled (or subordinated) by an independent time-control process that follows an IG distribution. It is proposed that this control process is linked to the time of residence of the particles within accelerating structures of finite size, the relative motion between the particles and the structures being a drifting random walk (problem of the 'first passage time' of a random walk). Some applications of the model are discussed, as the estimate of the relative importance of heating and acceleration or the possible role of ambipolar fields.