Analytical solution of the evolution equations of the parton distribution functions in the small-x region through the Kramers-Moyal expansion of the master equation of Markov processes

Recently, we generated the evolution equations of the parton distribution functions (PDF) usually used in the hadrons phenomenology using the stochastic modeling of the non-equilibrium statistical mechanics in the momentum space. The evolution equations obtained from stochastic modeling are the same as the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations, but are obtained by a more simplistic mathematical procedure based on the non-equilibrium statistical mechanics and the theory of Markov processes. In this paper, we analytically solve the parton evolution equation for the non-singlet quark distribution function in the small-x (the longitudinal momentum fraction) region through the Kramers-Moyal expansion of the master equation. Finally, we compare the cutoff dependent non-singlet quark distribution function obtained from the analytical solution by considering the strong ordering and the angular ordering constraints with the ordinary non-singlet quark distribution function produced by the MMHT2014 group. In general, we show that our results at the small x and moderate Q2 (the energy scale) are in good agreement with the results of the MMHT2014 group.