  It is commonly known that the dephasing in open quantum systems is due to the
establishment of bipartite correlations with ambient environments, which are
typically difficult to be fully characterized. Recently, a new approach of
average over disordered Hamiltonian ensemble is developed and shown to be
capable of describing the nonclassicality of incoherent dynamics based on
inferring the nonclassical nature of the correlations. Here we further extend
the approach of Hamiltonian ensemble in the canonical form to the realm of
structural disorder. Under the variable separation of the probability
distribution within the Hamiltonian ensemble, the geometrical structure 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. We show that these effects are
easily understood from the emergences of additional terms in the master
equations, leading to rather general master equations and, consequently, going
beyond the previous frameworks of pure dephasing or isotropic depolarization.
