Reduced Dynamics with Initial Correlations, and Time-Dependent Environment and Hamiltonians

We consider a small system s in a bath b, in the case that the state of b and all Hamiltonians are possibly time dependent. We obtain for the reduced density matrix of s an exact evolution equation ₒ (t) 0ex{0ex}=0ex{0ex} (t, ) ₒ (), with (/) (t, ) 0ex{0ex}=0ex{0ex} (t, ) (t, ), where (t, ) depends on the system-bath correlations at time. The open evolutor (t, ) can (but need not) be chosen completely positive. It is argued that as t- increases, (t, ) ₀ (t) forgets the initial correlations and tends to Lindblad form in time-coarse-grained weak coupling limits.