  It is commonly known that the dephasing in open quantum systems is due to the
establishment of bipartite correlations with an environment. Recently, a new
approach of average over disordered Hamiltonian ensemble is developed and shown
to capable of describing both the incoherent dynamical behavior and the
nonclassicality of dynamical processes. Here we further extend the approach of
Hamiltonian ensemble in the canonical form to the realm of structural disorder.
Under the separation of the probability distribution within the Hamiltonian
ensemble, the geometrical structural is easily visualized and can be
characterized according to the degree of symmetry. We demonstrate four degrees
and investigate the effects of different types of symmetry breaking on the
incoherent dynamics. With these effects, we obtain rather general master
equations, going beyond the previous frameworks of pure dephasing or isotropic
depolarization. The practicality of the Hamiltonian ensemble and the theory of
process nonclassicality is significantly enhanced.
