  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.
