  In the present paper, the proton velocity distribution function (VDF)in the
solar wind is determined by solving numerically the kinetic evolution equation.
We compare the results obtained when considering the effects of ex- ternal
forces and Coulomb collisions with those obtained by adding effects of Alfven
wave turbulence. We use Fokker-Planck diffusion terms due to Alfvenic
turbulence, which take into account observed turbulence spectra and kinetic
effects of finite proton gyroradius. Assuming a displaced Maxwellian for the
proton VDF at the simulation boundary at 14 solar radii, we show that the
turbulence leads to a fast (within several solar radii) development of the
anti-sunward tail in the proton VDF. Our results provide a natural explanation
for the nonthermal tails in the proton VDFs, which are often observed in-situ
in the solar wind beyond 0.3 AU.
