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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, :\hat = \hat_S + \hat_B + \hat_ where \hat_S is the system Hamiltonian, \hat_B is the bath Hamiltonian, and \hat_ describes the coupling between them. The starting pointA 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: :\partial_t \rho = \frac rho,H= L \rho, where the Liouville operator L is defined as L A = \frac ,H/math>. In the Nakajima-Zwanzig formulation, a key step involves defining a projection operator \mathcal that projects the total density operator \rho onto the subspace of the system of interest. The complementary operator \mathcal\equiv 1-\mathcal projects onto the orthogonal subspace, effectively separating the system from the bath. The Liouville – von Neumann equation can thus be represented as :\left( \begin \mathcal \\ \mathcal \\ \end \right)\rho =\left( \begin \mathcal \\ \mathcal \\ \end \right)L\left( \begin \mathcal \\ \mathcal \\ \end \right)\rho +\left( \begin \mathcal \\ \mathcal \\ \end \right)L\left( \begin \mathcal \\ \mathcal \\ \end \right)\rho. The dynamics of the projected state \mathcal\rho, under any idempotent projection operator (where\mathcal^2=\mathcal), is described by the NZ generalized master equation (GQME). This equation can be used to obtain a closed equation of motion for the reduced system dynamics, focusing solely on the dynamics within the subsystem of interest. \frac\mathcal\hat(t) = -\frac\mathcal L\mathcal\hat(t)-\frac\int_^ d\tau\mathcalLe^\mathcalL\mathcal\hat(t-\tau)-\frac\mathcalLe^\mathcal\hat(0) In practice, the specific form of the projection operator \mathcal can be chosen based on the problem at hand. One common choice involves defining \mathcal using a reference nuclear density operator \hat_^ such that \text_ \ = 1. \mathcal(\rho) = \hat_n^ \otimes \text_n \ This ensures that \mathcal remains idempotent. Using this projection, tracing over the nuclear Hilbert space leads to a generalized quantum master equation that describes the reduced electronic density operator which accounts for both Markovian dynamics generated by the Hamiltonian and non-Markovian dynamics due to coupling between electronic and nuclear degrees of freedom. \frac \hat(t) = -\frac (L)_n^ \hat(t) - \int_0^t d\tau \,\mathcal(\tau) \hat(t-\tau) + \hat(t) This \langle L \rangle_^ = \text_ \ describes the dynamics driven by the Hamiltonian, \langle \hat \rangle_n^ = \text_n \which are Hamiltonian and Markovian in nature, while the other two terms on the right-hand side represent the non-Hamiltonian and non-Markovian dynamics that arise from the interactions between the electronic and nuclear degrees of freedom. The memory kernel \mathcal(\tau) captures the effects of the bath on the system over the time interval from (0, t), reflecting non-Markovian dynamics where the system's history influences its future evolution. \mathcal(\tau) = \frac \operatorname_n \ The inhomogeneous term \hat(t) represents the influence of the initial state of the bath on the system at time t, which is crucial for accurately describing the
system dynamics System dynamics (SD) is an approach to understanding the nonlinear behaviour of complex systems over time using stocks, flows, internal feedback loops, table functions and time delays. Overview System dynamics is a methodology and mathematical ...
from an initial condition. \hat(t) = \frac \text_n \left\ The memory kernel is crucial for simulating the dynamics of the electronic degrees of freedom. However, calculating \mathcal(\tau) presents difficulties due to its time-dependent nature. Additionally, the time dependency of \mathcal(\tau) is complex because it is governed by the projection-dependent propagator, e^. Therefore, the exact memory kernel is difficult to calculate except for several analytically solvable models proposed by Shi-Geva to remove the projection operator \mathcal.


See also

*
Redfield equation Redfield may refer to: People * Redfield (surname) Places ;United Kingdom *Redfield, Bristol, an area within the City of Bristol ;United States * Mount Redfield, a mountain in Essex County, New York *Redfield, Arkansas, a city in northwestern ...


Notes


References

* E. Fick, G. Sauermann: ''The Quantum Statistics of Dynamic Processes'' Springer-Verlag, 1983, . * Heinz-Peter Breuer, Francesco Petruccione: ''Theory of Open Quantum Systems.'' Oxford, 2002 * Hermann Grabert ''Projection operator techniques in nonequilibrium statistical mechanics'', Springer Tracts in Modern Physics, Band 95, 1982 * R. Kühne, P. Reineker: ''Nakajima-Zwanzig's generalized master equation: Evaluation of the kernel of the integro-differential equation'', Zeitschrift für Physik B (Condensed Matter), Band 31, 1978, S. 105–110, * Xu, M.; Yan, Y.; Liu, Y.; Shi, Q. Convergence of High Order Memory Kernels in the Nakajima-Zwanzig Generalized Master Equation and Rate Constants: Case Study of the Spin-Boson Model. Journal of Chemical Physics 2018, 148 (16). https://doi.org/10.1063/1.5022761. * Mulvihill, E.; Geva, E. A Road Map to Various Pathways for Calculating the Memory Kernel of the Generalized Quantum Master Equation. Journal of Physical Chemistry B 2021, 125 (34), 9834–9852. https://doi.org/10.1021/acs.jpcb.1c05719. * Mulvihill, E.; Schubert, A.; Sun, X.; Dunietz, B. D.; Geva, E. A Modified Approach for Simulating Electronically Nonadiabatic Dynamics via the Generalized Quantum Master Equation. Journal of Chemical Physics 2019, 150 (3). https://doi.org/10.1063/1.5055756.


External links

* {{DEFAULTSORT:Nakajima-Zwanzig equation Quantum mechanics Statistical mechanics