On the phase-space dynamics of a time-dependent harmonic oscillator at finite temperature

We propose a generalization, valid at finite temperatures, of the treatment of quantal damped motion at zero temperature. First we infer that the existence of a quadratic constant of the motion for a quantal, time-dependent, damped harmonic oscillator allows us to develop, in the moving framework, a thermodynamic theory for the “representative” system that possesses constant energy eigenvalues. In this way, the semiclassical distribution functions associated with the pure nonstationary states (gaussons) of the original oscillator can be calculated. Secondly, we show that a gaussian shape-preserving distribution in phase-space, that corresponds to a statistical mixture of gaussons, gives a possible description of the irreversible approach to equilibrium of the oscillator coupled to the heat reservoir. This distribution is shown to satisfy a time-dependent Fokker-Planck equation, provided that the thermal fluctuations follow the same equations of motion as those of a gaussian wave packed that moves under the influence of Kostin's friction at zero temperature.