
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
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...
formalism and can be regarded as a generalization of the
master equation
In physics, chemistry, and related fields, master equations are used to describe the time evolution of a system that can be modeled as being in a probabilistic combination of states at any given time, and the switching between states is determi ...
.
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).
Derivation
The total Hamiltonian of a system interacting with its environment (or bath) is typically expressed in system-bath form,
:
where
is the system Hamiltonian,
is the bath Hamiltonian, and
describes the coupling between them.
The starting point
[A derivation analogous to that presented here is found, for instance, in Breuer, Petruccione ''The theory of open quantum systems'', Oxford University Press 2002, S.443ff] is the quantum mechanical version of the
von Neumann equation
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those ...
, also known as the Liouville equation:
:
where the Liouville operator
is defined as