Thermodynamics of dissipative quantum system by effective potential

Classical-like formulas are given in order to evaluate thermal averages of observables belonging to a quantum nonlinear system with dissipation described by the Caldeira-Leggett model [Phys. Rev. Lett. 46, 211 (1981); Ann. Phys. (N.Y.) 149, 374 (1983)]. The underlying scheme is the pure-quantum self-consistent harmonic approximation, which leads to expressions with a Boltzmann factor involving an effective potential and with a Gaussian average. The latter describes the effect of the fluctuations of purely quantum origin. As an illustration we calculate the coordinate probability distribution for a double-well potential in the presence of various degrees of Ohmic dissipation.