  We consider a quantum mechanical system represented in phase space (referred
to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We
derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time,
but also in imaginary time, which represents an inverse temperature. This is an
extension of a previous work, in which we studied a spin-boson system, to a
Brownian system. It is shown that the QHFP in real time obtained from a
correlated thermal equilibrium state of the total system possess the same form
as those obtained from a factorized initial state. A modified terminator for
the hierarchal equations of motion is introduced to treat the non-Markovian
case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic
quantities, including the free energy, entropy, internal energy, heat capacity,
and susceptibility can be evaluated for any potential. These equations allow us
to treat non-Markovian, non-perturbative system-bath interactions at finite
temperature. Through numerical integration of the real-time QHFP for a harmonic
system, we obtain the equilibrium distributions, the auto-correlation function,
and the first- and second-order response functions. These results are compared
with analytically exact results for the same quantities. This provides a
critical test of the formalism for a non-factorized thermal state, and
elucidates the roles of fluctuation, dissipation, non-Markovian effects, and
system-bath coherence. Employing numerical solutions of the imaginary-time
QHFP, we demonstrate the capability of this method to obtain thermodynamic
quantities for any potential surface. It is shown that both types of QHFP
equations can produce numerical results of any desired accuracy.
