Quantum thermodynamics in nonequilibrium

Understanding thermodynamics far from equilibrium at the quantum scale remains a fundamental challenge, particularly in the presence of quantum coherence. Here we develop a first-principles framework for nonequilibrium quantum thermodynamics by integrating quantum resource theory of coherence with thermodynamic laws. We derive a previously unexplored entropy balance relation that explicitly separates entropy flux due to heat exchange from entropy production arising from the loss of quantum coherence. This formulation identifies the appropriate thermodynamic entropy in nonequilibrium quantum processes as the energy entropy associated with energy measurements, demonstrating that the von Neumann entropy does not, in general, represent thermodynamic entropy away from equilibrium. Within this framework, dynamical temperature, free energy, work, and heat are consistently defined, and both the first and second laws are shown to hold far from equilibrium. Applying the theory to an exactly solvable open quantum system, we reveal how equilibrium thermodynamics emerges dynamically in the weak-coupling limit. Our results establish a unified and operational foundation for nonequilibrium quantum thermodynamics and clarify the fundamental thermodynamic role of quantum coherence.