  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.
