Statistical description of small quantum systems beyond the weak-coupling limit

An explicit expression is derived for the statistical description of small quantum systems, which are relatively weakly and directly coupled to only small parts of their environments. The derived expression has a canonical form, but is given by a renormalized self-Hamiltonian of the studied system, which appropriately takes into account the influence of the system-environment interaction. In the case that the system has a narrow spectrum and the environment is sufficiently large, the modification to the self-Hamiltonian usually has a mean-field feature, given by an environmental average of the interaction Hamiltonian. In other cases, the modification may be beyond the mean-field approximation.