  We derive asymptotic freedom and the $SU(3)$ Yang-Mills $\beta$-function
using the renormalization group procedure for effective particles. In this
procedure, the concept of effective particles of size $s$ is introduced.
Effective particles in the Fock space build eigenstates of the effective
Hamiltonian $H_s$, which is a matrix written in a basis that depend on the
scale (or size) parameter $s$. The effective Hamiltonians $H_s$ and the
(regularized) canonical Hamiltonian $H_{0}$ are related by a similarity
transformation. We calculate the effective Hamiltonian by solving its
renormalization-group equation perturbatively up to third order and calculate
the running coupling from the three-gluon-vertex function in the effective
Hamiltonian operator.
