We study a quantum harmonic oscillator undergoing thermalization. To describe
    the thermalization process, we generalize the Ermakov-Lewis-Riesenfeld (ELR)
    invariant method for the oscillator. After imposing appropriate conditions on
    the thermalization process, we introduce an ansatz equation that describes the
    time evolution effectively. We write down the first law for thermalization in
    the same form as that for ordinary thermodynamics. Here, the thermalization
    effect appears through a change of the ELR frequency. Finally, we obtain the
    oscillator’s energy undergoing thermalization as a function of entropy and its
    time derivative.

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