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We investigate how the proton distribution function evolves when the protonsundergo stochastic heating by strong, low-frequency, Alfv\'en-wave turbulenceunder the assumption that $\beta$ is small. We apply our analysis to protonsundergoing stochastic heating in the supersonic fast solar wind and obtainproton distributions at heliocentric distances ranging from 4 to 30 solarradii. We find that the proton distribution develops non-Gaussian structurewith a flat core and steep tail. For $r >5 \ R_{\rm S}$, the protondistribution is well approximated by a modified Moyal distribution. Comparisonswith future measurements from \emph{Solar Probe Plus} could be used to testwhether stochastic heating is occurring in the solar-wind acceleration region.

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