  We introduce a generalized classical model of Brownian motion for describing
thermal relaxation processes which is thermodynamically consistent. Applying
the canonical quantization to this model, a quantum equation for the density
operator is obtained. This equation has a thermal equilibrium state as its
stationary solution, but the time evolution is not necessarily a Completely
Positive and Trace-Preserving (CPTP) map. In the application to the harmonic
oscillator potential, however, the requirement of the CPTP map is shown to be
satisfied by choosing parameters appropriately and then our equation reproduces
a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation satisfying the detailed
balance condition. This result suggests a quantum-classical correspondence in
thermal relaxation processes and will provide a new insight to the study of
decoherence.
