A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
In quantum thermodynamics, the decomposition of energy exchanges into heat and work remains an open problem beyond weak-coupling and slow-driving regimes. Recent formulations have shown that quantum coherence introduces additional energy contributions whose thermodynamic interpretation is still under debate, raising fundamental questions about the structure of the quantum first law. In this work, we investigate this problem through a time-dependent perturbative framework applied to the first law of quantum thermodynamics. By expanding the thermodynamic quantities up to second order, we derive explicit perturbative corrections for work, heat, and coherence contributions. Our results show that the coherence term can be consistently decomposed into coherent heat and coherent work, demonstrating that quantum coherence does not require the introduction of an independent energetic contribution beyond heat and work. The formalism resolves inconsistencies associated with previous formulations of the quantum first law, including the interpretation of coherence contributions and their connection with entropy fluxes. At second order, the perturbative corrections become directly connected to transition rates governed by Fermi's golden rule, establishing a bridge between microscopic quantum transitions and macroscopic thermodynamic quantities. These results provide a physically transparent framework to investigate coherence-driven thermodynamic processes and offer new perspectives for the analysis of driven quantum systems and nonequilibrium quantum technologies.
This paper presents a comprehensive exposition of quantum thermodynamics, aiming to bridge the foundational concepts of classical thermodynamics and quantum mechanics.Beginning with a rigorous overview of classical thermodynamic laws and statistical ensembles, the work transitions into the quantum domain by introducing quantum states, density operators, and von Neumann entropy.Core thermodynamic quantities such as work, heat, energy, and entropy are redefined in the quantum regime, with emphasis on operational frameworks like the twopoint measurement scheme.The study explores the role of quantum resource theories in formulating generalizations of the second law and delves into the intricacies of nonequilibrium thermodynamics through fluctuation theorems, such as the Jarzynski equality and Crooks theorem.Special attention is given to the influence of coherence, entanglement, and measurement backaction, which contribute uniquely to quantum thermodynamic behavior.The treatise concludes with a discussion on quantum heat engines, thermodynamic consistency of quantum master equations, and information-theoretic approaches, establishing quantum thermodynamics as a vital framework for understanding energy and information flow at the quantum scale.
Quantum thermodynamics has emerged as a central field for understanding how energy conversion processes occur in microscopic systems. In these systems, effects such as coherence, entanglement, and non-Markovianity play key roles. In this thesis, we explore different ways to describe quantum thermodynamics, using two main approaches: one based on entropy and the other on ergotropy. First, we introduce a generalized approach to quantify non-Markovianity through the breakdown of monotonicity in thermodynamic functions. In this context, the entropy-based heat flow serves as a practical tool to witness and measure quantum memory in unital maps that do not reverse the sign of the internal energy. Next, we analyze the dynamics of ergotropy in open qubits under both Markovian and non-Markovian evolutions. We identify phenomena such as freezing and sudden death of ergotropy, and we establish an analytical relation between the change in ergotropy and the environment-induced work . This provides a clear physical interpretation for the additional term in the first law in the entropy-based formulation. Finally, we propose an ergotropy-based thermodynamic formulation, in which heat is reinterpreted in terms of the change of the passive state associated with the density operator governing the quantum dynamics. This approach allows one to measure non-Markovianity of unital maps more generally and accurately than the entropy-based heat flow. This advantage comes from the direct link between heat and von Neumann entropy, a property ensured by invariance under passive transformations. Moreover, the out-of-equilibrium temperature naturally remains non-negative, similarly to equilibrium thermodynamics.
We develop a general theory describing the thermodynamical behavior of open quantum systems coupled to thermal baths beyond perturbation theory. Our approach is based on the exact time-local quantum master equation for the reduced open-system states, and on a principle of minimal dissipation. This principle leads to a unique prescription for the decomposition of the master equation into a Hamiltonian part representing coherent time evolution and a dissipator part describing dissipation and decoherence. Employing this decomposition we demonstrate how to define work, heat, and entropy production, formulate the first and second laws of thermodynamics, and establish the connection between violations of the second law and quantum non-Markovianity.
Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics.This tutorial focuses on the derivation of the first and second law for isolated and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies.The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat, and temperature.These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of states and processes.The consistency with previous results is demonstrated.The framework applies to multiple baths including particle transport and accounts for processes with, e.g., a changing temperature of the bath, which is determined microscopically.An integral fluctuation theorem for entropy production is satisfied.In summary, this tutorial introduces a consistent and versatile framework to understand and apply the laws of thermodynamics in the quantum regime and beyond.
Deriving the laws of thermodynamics from an underlying microscopic picture is a central quest of statistical mechanics. The present article focuses on the derivation of the first and second law of thermodynamics for closed and open quantum systems, where such foundational questions in addition become practically relevant for emergent nanotechnologies. Recent progress is presented in a self-contained way, with an emphasis on general ideas and useful tools instead of particular applications. We identify limits and conceptual shortcomings of previous approaches. Then, we propose a novel, unifying perspective, which starts from a microscopic definition of nonequilibrium thermodynamic entropy. The change of this entropy is identified with the entropy production, which satisfies a fluctuation theorem for a large class of initial states. Moreover, this entropy production can be naturally separated into a quantum and a classical component. Within our framework, we introduce the notions of recoverable work and remaining heat. In the case of a weakly perturbed ideal thermal bath this approach is in quantitative agreement with previous ones, thus ensuring their thermodynamic consistency.