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The role of average time dependence on the relaxation of excited electron populations in nonequilibrium many-body physics

We examine the exact equation of motion for the relaxation of populations of strongly correlated electrons after a nonequilibrium excitation by a pulsed field, and prove that the populations do not change when the Green's functions have no average time dependence. We show how the average time dependence enters into the equation of motion to lowest order and describe what governs the relaxation process of the electron populations in the long-time limit. While this result may appear, on the surface, to be required by any steady-state solution, the proof is nontrivial, and provides new critical insight into how nonequilibrium populations relax, which goes beyond the assumption that they thermalize via a simple relaxation rate determined by the imaginary part of the self-energy, or that they can be described by a quasi-equilibrium condition with a Fermi-Dirac distribution and a time-dependent temperature. We also discuss the implications of this result to approximate theories, which may not satisfy the exact relation in the equation of motion.