We consider the non-equilibrium steady state induced by two infinitely extended quantum thermal reservoirs at different inverse temperatures β + Δβ, β − Δβ and derive a work relation. We consider global cyclic operations and derive an upper bound of the work density in one-dimensional quantum lattice systems. This relation reproduces the second law of thermodynamics in the equilibrium limit Δβ → 0. A free fermion system is discussed as an example, and the physical interpretation is given to the upper bound.
We consider the non-equilibrium steady state induced by two infinitely extended quantum thermal reservoirs at different inverse temperatures β + Δβ, β − Δβ and derive a work relation. We consider global cyclic operations and derive an upper bound of the work density in one-dimensional quantum lattice systems. This relation reproduces the second law of thermodynamics in the equilibrium limit Δβ → 0. A free fermion system is discussed as an example, and the physical interpretation is given to the upper bound.
Work extraction is a fundamental aspect in thermodynamics. In the context of quantum physics, ergotropy quantifies the maximum amount of work that can be obtained from quantum system through cyclic unitary process. In this work, the steady-state ergotropy of two coupled qubit, each interacting locally with its individual boson or fermion reservoir, will be examined. In this work, both equilibrium and non-equilibrium scenarios for bosonic and fermionic environments interacting with the qubits will be considered. In scenarios where two coupled qubits embedded within equilibrium boson reservoirs, it has been observed that the temperature of the reservoirs and the inter-qubits interaction strangth act as detrimental factors in work extraction. In the case of fermionic equilibrium reservoirs, it will be observed that ergotropy grows monotonically with the reservoirs chemical potential. In the non-equilibrium boson reservoirs, the temperature difference between the two reservoirs is a destructive factor for ergotropy. In non-equilibrium fermion reservoirs, the situation is somewhat more complicated. For r base chemical potential values that are smaller than the qubit transition frequency, the behavior of ergotropy is non-monotonic. However, for base chemical potential values that are larger than the transition frequency, ergotropy grows monotonically with the reservoirs chemical potential difference. Also, we study the situation in which the coupled qubits are asymmetric. It is observed that the maximum work will be extracted in the situation where the coupled qubits within both boson and fermion reservoirs be symmetric .
The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.
We consider open quantum systems consisting of a finite system of independent fermions with arbitrary Hamiltonian coupled to one or more equilibrium fermion reservoirs (which need not be in equilibrium with each other). A strong form of the third law of thermodynamics, S(T) → 0 as T → 0, is proven for fully open quantum systems in thermal equilibrium with their environment, defined as systems where all states are broadened due to environmental coupling. For generic open quantum systems, it is shown that S(T) → g ln 2 as T → 0, where g is the number of localized states lying exactly at the chemical potential of the reservoir. For driven open quantum systems in a nonequilibrium steady state, it is shown that the local entropy Sx;T→0 as T(x) → 0, except for cases of measure zero arising due to localized states, where T(x) is the temperature measured by a local thermometer.
This paper has been withdrawn by the author. For the reason, see the bottom paragraph of this abstract. By generalizing Tasaki's work on the second law of thermodynamics for an adiabatic process between two equilibrium states of a macroscopic quantum compound system, we obtain an extension of the second law to a transient adiabatic process that takes a macroscopic quantum compound system consisting of a system of interest and two heat reservoirs from an initial equilibrium state to a final non-equilibrium state, where the system of interest is in a steady heat conduction state. For the system of interest, we define its generalized entropy so that it becomes an experimentally measurable quantity and exactly satisfies an extension of the Clausius relation. This paper has been withdrawn by the author because the generalized entropy presented in this article violates the second law when the system in a steady heat conduction state is detached from the two reservoirs and then attached to a reservoir at the temperature T^bar defined in Sec.1.2 so that it will reach an equilibrium state at T^bar. In this irreversible process, the total entropy for the system and the reservoir remains constant and thus violates the second law.
We derive the fluctuation theorem for quantum-state statistics that can be obtained when we initially measure the total energy of a quantum system at thermal equilibrium, let the system evolve unitarily, and record the quantum-state data reconstructed at the end of the process. The obtained theorem shows that the quantum-state statistics for the forward and backward processes is related to the equilibrium free-energy difference through an infinite series of independent relations, which gives the quantum work fluctuation theorem as a special case, and reproduces the out-of-time-order fluctuation-dissipation theorem near thermal equilibrium. The quantum-state statistics exhibits a system-size scaling behavior that differs between integrable and non-integrable (quantum chaotic) systems as demonstrated numerically for one-dimensional quantum lattice models.