The above expressions for E-FGR can also be directly obtained from the standard frequency-domain FGR derivation, see details in the supplementary material, Sec. III. The essential scaling of these rates with respect to N remains the as in the NE-FGR expression, and the same scaling relations are also discovered using Lindblad Equations and recently through the 1/N expansion approach . The pre-factor N -1 (degeneracy of the dark states manifold) can be interpreted either as the density of state for the dark states or as the effective entropy of the dark states manifold. To demonstrate that, consider the E-FGR rate |-⟩ ↔ {|D j } under resonance condition ω