Here the effect of the bath throughout the time evolution of the system is hidden in the memory kernel κ κ ( τ τ ) . While the Nakajima-Zwanzig equation is an exact equation that holds for almost all open quantum systems and environments, it can be very difficult to solve. This means that approximations generally need to be introduced to reduce the complexity of the problem into something more manageable. As an example, the assumption of a fast bath is required to lead to a time local equation: ∂ ∂ t ρ ρ S = L ρ ρ S . Other examples of valid approximations include the weak-coupling approximation and the single-coupling approximation.