**4. Temporally-Local Equilibrium State, Power Output and Work along Isoenergetic Process**

A nonequilibrium thermodynamic system with a slowly-varying Hamiltonian is dominantly described by a locally-equilibrium state, in which the thermodynamic variables also vary slowly. In the present case, what is relevant is the temporally-local equilibrium state given by

$$\rho^{(0)} = \frac{1}{Z\_{\,\,\,t}} \exp\left(-\beta(t)H(t)\right),\tag{19}$$

where *Z*(*t*) is the partition function

$$Z(t) = \text{tr } \exp(-\beta(t) \, H(t)). \tag{20}$$

β(*t*) is the time-dependent inverse temperature that slowly varies in time [see Equation (26) below]. For the Hamiltonian in Equation (7), the explicit expression of Equation (19) is the familiar one

$$\rho^{(0)} = \frac{1}{2} \langle I - (\mathbf{n}(t) \cdot \boldsymbol{\sigma}) \tanh(\beta(t) \, B(t)) \rangle\_{\prime} \tag{21}$$

where *I* is the 2 × 2 unit matrix and **n**(*t*) is a unit vector defined by **n**(*t*) = **B**(*t*)/*B*(*t*).

The full density matrix may be expanded around the state in Equation (21) as follows:

$$
\rho = \rho^{(0)} + \rho^{(1)} + \cdots,\tag{22}
$$

where the expansion should be performed in terms of the elements of **. B**(*t*) and its higher-order derivatives. It is natural to assume that all of the correction terms are traceless. Substituting Equation (22) into Equation (14), the first-order correction is found to satisfy

$$\left[H(t),\,\rho^{(1)}\right] = i\,\frac{\partial}{\partial t}\frac{\rho^{(0)}}{t} + \frac{i}{2}\sum\_{i=1}^{3}a\_{i}\left[\sigma\_{i\prime}\left[\sigma\_{i\prime}\,\,\rho^{(0)}\right]\right],\tag{23}$$

*Entropy* **2019**, *21*, 503

for instance.

In accordance with Equation (22), the internal energy is also expanded as follows:

$$\mathcal{U} = \mathcal{U}^{(0)} + \mathcal{U}^{(1)} + \cdots \, . \tag{24}$$

The isoenergeticity condition should be satisfied in each order. For the leading-order term

$$\mathcal{U}^{(0)} = \text{tr}\{H(t)\,\rho\,\,^{(0)}\} = -B(t)\,\tanh(\beta(t)\,B(t)),\tag{25}$$

we find that the isoenergeticity condition, *dU*(0)/*d t* = 0, in the leading order gives rise to

$$
\dot{B}(t) \left[ \sinh(2\beta(t)\,B(t)) + 2\beta(t)\,B(t) \right] = -2B^{\frac{\alpha}{2}}(t)\,\dot{\beta}(t). \tag{26}
$$

Since . *<sup>B</sup>*(*t*) is positive due to Equation (12) resulting from the Lindblad equation, . β(*t*) is necessarily negative, implying that the temperature monotonically increases in time.

Now, let us discuss the power output defined by

$$P(t) = -\text{tr}\left(\dot{H}(t)\,\rho\right). \tag{27}$$

The leading-order contribution comes from the temporally-local equilibrium state and is calculated to be

$$P(t) = -\dot{\mathbf{B}}(t) \cdot \text{tr}\left(\sigma \,\rho^{(0)}\right) = \dot{\mathbf{B}}(t) \cdot \mathbf{n}(t) \,\tanh(\beta(t) \,\mathcal{B}(t)).\tag{28}$$

An interesting point is that, similarly to the case of the time-dependent harmonic oscillator [20], this quantity can be expressed in terms of the internal energy as follows:

$$P(t) = -\frac{\dot{B}(t)}{B(t)} \ll 1^{(0)}.\tag{29}$$

From this expression, the work done during the time interval *t <sup>i</sup>* ≤ *t* ≤ *tf* along the isoenergetic process is given, in the leading order, by

$$\mathcal{W} = \int\_{t\_i}^{t\_f} dt \, P(t) = -\mathcal{U}^{(0)}(t\_i) \int\_{t\_i}^{t\_f} dt \, \frac{\dot{B}(t)}{B(t)} = B(t\_i) \tanh(\beta(t\_i) \, B(t\_i)) \cdot \ln\left[\frac{B(t\_f)}{B(t\_i)}\right],\tag{30}$$

where the initial value is used for the conserved internal energy.

#### **5. Concluding Remarks**

Toward clarifications of the physical properties of exotic baths present in quantum thermodynamics but absent in classical thermodynamics, we have focused our attention on the isoenergetic process that is connected with the energy bath. For this purpose, we have studied a single spin in a magnetic field slowly varying in time based on the Lindblad equation. We have shown how the isoenergeticity condition can determine the Lindbladian operators without recourse to detailed knowledge of the interaction between the subsystem and the energy bath. We have developed a discussion about finite-time thermodynamics of such a system and evaluated the power output and the work in the leading order.

The result given in Equation (30) implies that the work is determined only by the initial and final values of the variables without depending on paths of the magnetic field in the parameter **B**-space. This would be a common feature of the isoenergetic processes in the leading order.

In the present work, we have treated a single Pauli spin. As known, by virtue of the structure of spin algebra, any density matrix of a multispin system can be expanded in terms of the spin matrices, their tensor products as well as the unit matrix. Therefore, the temporally-local equilibrium state in Equation (21) can be generalized to the multispin case [21].

Here, we have considered only the energy bath, which is nothing but one of the exotic baths in quantum thermodynamics. Existence of various baths indicates that there may be ensemble-dependent dynamics. In this point, it is worth mentioning that such dynamics are also suggested in nanothermodynamics of classical small systems, in which fluctuations can be very large [22,23]. Much is yet to be clarified about the quantum-classical correspondence in thermodynamics.

**Author Contributions:** S.A. has provided the idea of the study, performed the calculations and written the manuscript. C.O. and Y.Y. have checked the calculations, made the critical comments and contributed to the improvement of the manuscript. All authors have read and approved the final manuscript.

**Funding:** The authors would like to thank the support by a grant from the National Natural Science Foundation of China (No. 11775084). The work of S.A. has also been supported in part by the programs of Fujian Province, China, and of Competitive Growth of Kazan Federal University from the Ministry of Education and Science, Russian Federation.

**Acknowledgments:** This work has been completed while S.A. has stayed at the Wigner Research Centre for Physics with the support of the Distinguished Guest Fellowship of the Hungarian Academy of Sciences. He would like to express his sincere thanks to the Wigner Research Centre for Physics for the warm hospitality extended to him.

**Conflicts of Interest:** The authors declare no conflict of interest.
