Context. In the solar wind (SW), the particle distribution functions are generally not Gaussian. They present nonthermal features that are related to underlying acceleration and heating processes. These processes are critical in the overall dynamics of this expanding astrophysical fluid. Aims. The Proton Alpha Sensor (PAS) on board Solar Orbiter commonly observes skewed proton distributions, with a more populated high-energy side in the magnetic field direction than the Gaussian distribution. Our objectives are: (1) to identify a theoretical statistical function that adequately models the observed distributions and (2) to use its statistical interpretation to constrain the acceleration and heating processes. Methods. We analyzed the 3D velocity distribution functions (VDFs) measured by PAS and compared them to model statistical functions. Results. We show that the normal inverse Gaussian (NIG), a type of hyperbolic statistical distribution, provides excellent fits of skewed and leptokurtic proton distributions. NIG can model both the core distribution and the beam, if present. We propose an interpretation that is inspired by the mathematical formulation of the NIG. It assumes that the acceleration or heating mechanism can be modeled as a drifting diffusion process in velocity space, controlled (or subordinated) by the time of interaction of the particles with “accelerating structures”. The probability function of the interaction time is an inverse Gaussian (IG), obtained by considering a random drift across structures of a given size. The control of the diffusion by interaction times that follow an IG probability function formally defines the NIG distribution. Following this model, we show that skewness and kurtosis can be used to estimate the kinetic and thermal energy gains provided by the interaction with structures. For example, in the case studies presented here, the analyzed populations would have gained kinetic energy representing approximately two to four times their thermal energy, with an increase in velocity – due to acceleration – of from one-tenth to one-third of the observed flow velocity. We also show that the model constrains the initial temperature of the populations. Conclusions. Overall, the NIG model offers excellent fits of the observed proton distributions. Combining the skewness and the kurtosis, it also leads to constraints in the part of acceleration and heating due to the interactions with structures in the formation of the proton populations. We suggest that these effects add to the classical thermal evolution of the bulk velocity and temperature resulting from SW expansion.Read less <