Thermodynamics Beyond State Functions from Quantum Relaxation

In standard thermodynamics, internal energy is a state function, independent of process rates. We show that this structure breaks down in open quantum systems undergoing thermalization. Within Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) dynamics with detailed balance, relaxation at the generator level promotes a dynamical invariant to an emergent thermodynamic coordinate. As a result, the internal energy acquires an intrinsic dependence on the rate of entropy change, $$E = E(S,\dot{S}),$$ implying that thermalization enlarges the thermodynamic state space. This mechanism is generic in the Gaussian regime, where dynamics admits an effective quadratic description, and extends to quantum fields, where each mode contributes a rate-dependent term to the energy. It also applies to physically relevant interacting systems, such as a photon field coupled to an electronic bath. Our results show that detailed-balance relaxation induces a dynamical extension of thermodynamics, in which thermodynamic potentials depend on both state variables and their rates.