  We describe the theory of Fermi-type acceleration, including first-order
Fermi acceleration at a parallel shock front and second-order Fermi
acceleration in a test particle limit. Including the theory of the turbulent
acceleration and the derivation of the general Fokker-Planck equation, we take
into account the basic particle-transport equation and the Fokker-Planck
equation with spatially homogeneous isotropic distribution. In the cases of
some special physical processes, we construct the particle-transport equations,
and compile the analytical or semi-analytical solutions in 11 different cases.
Even though, traditionally, one of the electron energy distributions for these
different cases can be used to reproduce the multi-wavelength emission of a
blazar, due to very long integration times of high-energy observations, a
time-integrated electron energy distribution should be introduced in approaches
to modeling the multi-wavelength spectral energy distribution of blazars,
especially in modeling the flaring state.
