Nakajima–Zwanzig equation

The Nakajima–Zwanzig equation (named after the physicists who developed it, Sadao Nakajima and Robert Zwanzig) is an integral equation describing the time evolution of the "relevant" part of a quantum-mechanical system. It is formulated in the density matrix formalism and can be regarded as a generalization of the master equation.

The equation belongs to the Mori-Zwanzig formalism within the statistical mechanics of irreversible processes (named after Hazime Mori). By means of a projection operator, the dynamics is split into a slow, collective part (relevant part) and a rapidly fluctuating irrelevant part. The goal is to develop dynamical equations for the collective part.

The Nakajima-Zwanzig (NZ) generalized master equation is a formally exact approach for simulating quantum dynamics in condensed phases. This framework is particularly designed to address the dynamics of a reduced system interact with a larger environment, often represented as a system coupled to a bath.  Within the NZ framework, one can choose between time convolution (TC) and time convolution less (TCL) forms of the quantum master equations.

The TC approach involves memory effects, where the future state of the system depends on its entire history (Non-Markovian dynamics). The TCL approach formulates the dynamics where the system's rate of change at any moment depends only on its current state, simplifying calculations by neglecting memory effects (Markovian dynamics).