Quantum fluctuation theorem for initial near-equilibrium system

Quantum work fluctuation theorem (FT) commonly requires the system initially prepared in an equilibrium state. Whether there exists universal exact quantum work FT for initial state beyond equilibrium needs further discussions. Here, I initialize the system in a near-equilibrium state, and derive the corresponding modified Jarzynski equality by using the perturbation theory. The correction is nontrivial because it directly leads to the principle of maximum work or the second law of thermodynamics for near-equilibrium system and also gives a much tighter bound of work for a given process. I also verify my theoretical results by considering a concrete many-body system, and reveal a fundamental connection between quantum critical phenomenon and near-equilibrium state at really high temperature.