From convolutionless generalized master to finite-coupling Pauli master equations

Time-convolutionless Generalized Master Equations (TCL-GME) for probabilities of finding a system in a general state, irrespective of the state of the thermodynamic bath, are investigated. For general systems interacting with a genuine bath with a continuous spectrum described by time-independent system + bath Hamiltonians and after the thermodynamic limit for the bath, the long-time asymptotics of time-dependent coefficients can be taken as counterparts of Pauli-Master-Equation (PME) transfer rates. Here, within TCL-GME, asymptotics of these coefficients is calculated without resorting to any approximation. In the lowest order, these coefficients are known to turn to the usual Fermi Golden Rule transfer rates. Anyway, we argue that if the exact form of these coefficients has a long-time limit, this limit is inevitably equal to zero. This makes the possibility to derive standard markovian finite-coupling Pauli, rate or balance equations as long-time asymptotics to TCL-GME illusory. Invaluable discussions with Prof. P. Grigolini are also highly appreciated.