  The paper deals with the problem of open systems out of equilibrium. An
analytical expression for time-dependent density matrix of two arbitrary
coupled identical quantum oscillators interacting with separate reservoirs is
derived using path integral methods. The temporal behavior of spatial variances
and of covariance from given initial values up to stationary values is
investigated. It was shown that at comparatively low coupling strengths the
asymptotic variances in the long-time limit achieve steady states despite on
initial conditions. Stationary values of variances differ from the case of
total equilibrium due to their coupling simultaneously with thermal reservoirs
of different temperatures. The larger the difference in temperatures of thermal
baths, the larger is the difference of the stationary values of variances of
coupled oscillators comparing with values given by the fluctuation dissipation
theorem. At strong couplings the variances have divergent character. Otherwise,
in the weak coupling limit the asymptotic stationary variances are always in
accordance with the fluctuation dissipation theorem despite of the difference
in temperatures within the whole system.
