A Statistical Treatment to the Evolution Equation for Quark Distributions

The evolution of polarized quark distribution functions is taken into account the gluon emission and absorption, quark pair production and annihilation processes and treated by a statistical method which provides quark distribution functions at small $Q^2$. The evolution function of quark distributions is derived and solved by numerical methods. Then the $Q^2$ dependent and $\alpha_{s}$ dependent structure functions $F_{2}(x, Q^{2})$ are obtained and the example of it, the proton structure function, is calculated, whose increase and decrease tendency with $\alpha_{s}$ at different $x$ is consistent with experiments. This is also an evidence that the statistical methods for the nucleon is reasonable.