When there are explicit system-environment interactions ( ĤI ̸ = 0), however, the dynamics of the reduced density matrix of the system ρS (t) can no longer be described by the unitary evolution of Eq. 4. An exact calculation of ρS (t) could be performed using Eq. 3 if ρT (t) is fully known, but this is generally infeasible to obtain due to the large number of environmental DOFs. Approximations of the environment and system-environment interactions can be made to calculate ρS (t) under certain conditions.