  Equilibrium canonical distribution in statistical mechanics assumes weak
system-bath coupling (SBC). In real physical situations this assumption can be
invalid and equilibrium quantum statistics of the system may be non-canonical.
By exploiting both polaron transformation and perturbation theory in a
spin-boson model, an analytical treatment is advocated to study non-canonical
statistics of a two-level system at arbitrary temperature and for arbitrary SBC
strength, yielding theoretical results in agreement with exact Monte-Carlo
simulations. In particular, the eigen-representation of system's reduced
density matrix is used to quantify non-canonical statistics as well as the
quantumness of the open system. For example, it is found that irrespective of
SBC strength, non-canonical statistics enhances as temperature decreases but
vanishes at high temperature.
