  We investigate a total thermodynamic entropy production rate of an isolated
quantum system. In particular, we consider a quantum model of coupled harmonic
oscillators in a star configuration, where a central harmonic oscillator
(system) is coupled to a finite number of surrounding harmonic oscillators
(bath). In this model, when the initial state of the total system is given by
the tensor product of the Gibbs states of the system and the bath, every
harmonic oscillator is always in a Gibbs state with a time-dependent
temperature. This enables us to define time-dependent thermo-dynamic entropy
for each harmonic oscillator and total non-equilibrium thermodynamic entropy as
the summation of them. We analytically confirm that the total thermodynamic
entropy satisfies the third law of thermodynamics. Our numerical solutions show
that, even when the dynamics of the system is well approximated by the
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)-type Markovian master equation,
the total thermodynamic entropy production rate can be negative, while the
total thermodynamic entropy satisfies the second law of thermodynamics. This
result is a counterexample to the common belief that the total entropy
production rate is non-negative when the system is under the GKSL-type
Markovian dynamics.
