  We investigate heating and acceleration of protons from a thermal gas with a
generic diffusion and acceleration model, and subject to Coulomb scattering and
energy loss, as was carried out in Petrosian & East (2008) for electrons. As
protons gain energy their loss to electrons becomes important. Thus, we need to
solve the coupled proton-electron kinetic equation. We numerically solve the
coupled Fokker-Plank equations and computes the time evolution of the spectra
of both particles. We show that this can lead to a quasi-thermal component plus
a high energy nonthermal tail. We determine the evolution of nonthermal tail
and the quasi-thermal component. The results may be used to explore the
possibility of inverse bremsstrahlung radiation as a source of hard X-ray
emissions from hot sources such as solar flares, accretion disk coronas and the
intracluster medium of galaxy clusters. We find that emergence of nonthermal
protons is accompanied by excessive heating of the entire plasma, unless the
turbulence needed for scattering and acceleration is steeper than Kolmogorov
and the acceleration parameters, the duration of the acceleration, and/or the
initial distributions are significantly fine-tuned. These results severely
constraint the feasibility of nonthermal inverse bremsstrahlung process
producing hard X-ray emissions. However the nonthermal tail may be the seed
particles for further re-acceleration to relativistic energies, say by a shock.
In the Appendix we present some tests of the integrity of the algorithm used
and present a new formula for the energy loss rate due to inelastic
proton-proton interactions.
