Evolution of The Proton Velocity Distribution due to Stochastic Heating in the Near-Sun Solar Wind

We investigate how the proton distribution function evolves when the protons undergo stochastic heating by strong, low-frequency, Alfv\'en-wave turbulence under the assumption that $\beta$ is small. We apply our analysis to protons undergoing stochastic heating in the supersonic fast solar wind and obtain proton distributions at heliocentric distances ranging from 4 to 30 solar radii. We find that the proton distribution develops non-Gaussian structure with a flat core and steep tail. For $r >5 \ R_{\rm S}$, the proton distribution is well approximated by a modified Moyal distribution. Comparisons with future measurements from \emph{Solar Probe Plus} could be used to test whether stochastic heating is occurring in the solar-wind acceleration region.